A QUEST FOR
I RREDUCIBLE H OMOGENEOUS
F REE
DIVISORS
Aron Simis
Universidade Federal de Pernambuco, Brazil
FreeDivisors, Warwick, May 2011
()
June 3, 2011
1/1
Outline
()
June 3, 2011
1/1
Abstract
We tackle the following aspects:
Exact sequences nurtured by the Jacobian ideal
The interplay between the Jacobian ideal and the differential
idealizer (Der(− log D))
Irreducible homogeneous free divisors of arbitrary degree in
dimension 3
Various special classes: homaloidal, linear type, Koszul
(With partial cooperation of A. N. Nejad.)
()
June 3, 2011
2/1
Abstract
We tackle the following aspects:
Exact sequences nurtured by the Jacobian ideal
The interplay between the Jacobian ideal and the differential
idealizer (Der(− log D))
Irreducible homogeneous free divisors of arbitrary degree in
dimension 3
Various special classes: homaloidal, linear type, Koszul
(With partial cooperation of A. N. Nejad.)
()
June 3, 2011
2/1
Abstract
We tackle the following aspects:
Exact sequences nurtured by the Jacobian ideal
The interplay between the Jacobian ideal and the differential
idealizer (Der(− log D))
Irreducible homogeneous free divisors of arbitrary degree in
dimension 3
Various special classes: homaloidal, linear type, Koszul
(With partial cooperation of A. N. Nejad.)
()
June 3, 2011
2/1
Abstract
We tackle the following aspects:
Exact sequences nurtured by the Jacobian ideal
The interplay between the Jacobian ideal and the differential
idealizer (Der(− log D))
Irreducible homogeneous free divisors of arbitrary degree in
dimension 3
Various special classes: homaloidal, linear type, Koszul
(With partial cooperation of A. N. Nejad.)
()
June 3, 2011
2/1
Abstract
We tackle the following aspects:
Exact sequences nurtured by the Jacobian ideal
The interplay between the Jacobian ideal and the differential
idealizer (Der(− log D))
Irreducible homogeneous free divisors of arbitrary degree in
dimension 3
Various special classes: homaloidal, linear type, Koszul
(With partial cooperation of A. N. Nejad.)
()
June 3, 2011
2/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
Polynomial setup of Saito’s
R = k[x1 , . . . , xn ]: a ring of polynomials over a field k of characteristic
zero.
f ∈ R: a polynomial of positive degree
∂f
∂f
I = If := ( ∂x
, . . . , ∂x
): the gradient ideal of f
n
1
J = (I, f ): the Jacobian ideal of f
Derf (R) = {δ ∈ Derk (R) | δ(f ) ∈ (f )}: the differential idealizer
(Der(− log D)) of f
Ωf (R) = (Derf (R))∗ : the R-dual (“logarithmic differentials”)
Z (∂): the first module of syzygies of I
I : (f ): the Euler conductor of f
We say that f is Eulerian if f ∈ I.
The next exact sequences appear in greater generality in a previous
work (Lecture Notes in Pure Applied Mathematics, Chapman & Hall,
2005). I believe they have also been noted by Damon.
()
June 3, 2011
3/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
General nonsense
Proposition
There are exact sequences of R-modules
0 → Z (∂) → Derf (R) → I : (f ) → 0,
and
0 → Derf (R) → Der(R) →
J
→ 0.
(f )
P
In particular, when f is Eulerian, say f = ni=1 hi (∂/∂xi ), there is a
decomposition of R-modules Derf (R) = Z (∂) ⊕ Rǫ, with
ǫ = h1 ∂x∂ 1 + · · · + hn ∂x∂ n the Euler derivation of R.
()
June 3, 2011
4/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
P RELIMINARIES
ON DIVISORS
Some exact sequences
Next is a shortened version of the differential nature of Ωf (R) by Saito.
Proposition
Let f ∈ R be a squarefree polynomial. Then
(i) Derf (R) and Ωf (R) are reflexive R-modules and dual to each
other.
(ii) There is an exact.sequence of R-modules
0 → (Ω(R) ⊕ R) R.(df , f ) −→ Ωf (R) −→ Ext2R (R/J, R) → 0
(iii) If R/(f ) is a domain the following are equivalent:
(a) R/(f ) is normal
(b) Ωf (R) = Ω(R) +
df
f
R (as an R-submodule of
1
f
Ω(R)).
Moreover, any one of these conditions implies that Ωf (R) is
n + 1-generated and has projective dimension at most one.
()
June 3, 2011
5/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
Free divisors according to Saito
Definition
f is said to be a free divisor if f is squarefree and Derf (R) is a free R-module.
Any smooth f (i.e., J = R) is a free divisor: this follows trivially from an
isomorphism of Derf (R) with a syzygy module of J in R n+1
No homogeneous f of degree ≥ 2 is smooth since it is a cone.
Any quasi-homogeneous divisor is Eulerian.
If f ∈ R is an irreducible homogeneous free divisor of degree ≥ 2 then
the singular locus of the associated projective hypersurface has
codimension 2.
()
June 3, 2011
6/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
A propedeutic example
Example
(Explicit determinantal representation of the hypersphere)
Consider the polynomial f = x12 + · · · xn2 − 1 (over any perfect field of
characteristic 6= 2). Then f is smooth and
 2

x1 − 1 x1 x2 . . . x1 xn
 x1 x2 x 2 − 1 . . . x2 xn 
2


n−1
f = (−1)
det 
.
..
..


.
.
x1 xn
x2 xn
. . . xn2 − 1
An amusing straightforward calculation shows that
every column of the matrix is a vector in Derf (R) and
the Koszul relations are combinations thereof.
()
June 3, 2011
7/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
Homological characterization
The next result is due to Terao in the analytic setup.
Proposition
Let R/(f ) be reduced.
(i) f is a free divisor if and only if either f is smooth or else the
Jacobian ideal J is a codimension two perfect ideal
(ii) If f is a non-smooth Eulerian divisor then f is a free divisor if and
only if I is a codimension two perfect ideal.
A glamor of a free divisor is seen by one of the previous
differential/homological
exact sequences:
P
0 → ( ni=1 R dxi ) ⊕ R/R.(df , f ) −→ Ωf (R) ≃ R n −→ ωR/J → 0,
where ωR/J denotes a canonical module of the singular locus R/J.
It can be read as a free R-presentation of the latter.
()
June 3, 2011
8/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation
Assume that f is a non-smooth free divisor.
As a submodule of Derk (R) ≃ R n , the idealizer Derf (R) is freely
generated by the column vectors of an n × n matrix


g11 . . . g1n

.. 
M =  ...
. 
gn1 . . . gnn
n+1 → J) generated by the
^
and admits a “lifting” Der
f (R) = ker (R
matrix by
column vectors of the (n + 1) × n matrix obtained from
P the
∂f
stacking to it the row vector (−h1 , . . . , −hn ), where i gij ∂xi = hj f for
1 ≤ j ≤ n.
Then the other n × n signed minors of the latter matrix are the partial
derivatives of f (Cramer rule).
()
June 3, 2011
9/1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
F REE
DIVISORS GALORE
The nature of the determinantal representation, 2
Assume now that f is a non-smooth Eulerian free divisor.
Then Derf (R) ⊂ Derk (R) ≃ R n is freely generated by the column
vectors of an n × n matrix such as the first one before, only this time
around the coefficients of an Euler vector will form one of the columns,
while the n − 1 remaining columns can be taken to be syzygies of the
gradient ideal I.
Since I is a codimension 2 perfect ideal, the (n − 1) × (n − 1) signed
minors of these n × (n − 1) columns will be the partial derivatives of f
whereas the total determinant gives back f , as is seen by expansion
along the Euler column.
()
June 3, 2011
10 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Plane, low degree
When n = 3, detecting a homogeneous free divisor is the equivalent of
asking whether the gradient I ideal of f is saturated, i.e., has no
(x, y, z)-primary component.
Since, for that matter, Proj(k[x, y, z]/(f )) cannot be smooth, we are
naturally led to look at singular projective curves behavior.
We naturally stick to irreducible such curves as there is plenty of
non-irreducible reduced homogeneous polynomials in k[x, y, z], of
arbitrary degrees, which are free divisors.
An example with just two factors is f = gh, where g = y r z d−r − x d is a
highercusp singularity and h = y is its multiple tangent line at
(0 : 0 : 1); so, such instances seem to abound.
By these preliminary remarks, we see that there are no
non-degenerate quadric free divisors. Moreover, no irreducible plane
projective cubics are free divisors as one readily computes:
Node y 2 z − x 2 (x + z): xz ∈ I : (x, y, z) \ I
Cusp y 2 z − x 3 : xy ∈ I : (x, y, z) \ I
()
June 3, 2011
11 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 4
The situation in degree 4 is slightly unstable, though morally
satisfactory.
Namely, based on the well-known classification by C. T. C. Wall, a work
in collaboration with A. Nejad has described a finite set of families or
rational quartics with fixed singular type.
Using this, it is possible to analyze the homological behavior of the
gradient ideal of the generic member of each family. As it turns out, the
gradient ideal is not saturated.
The verification recurs to a quite painful mix of theory and computer
checking.
For higher genus, there is a rough similar classification in Nejad’s PhD
thesis. Alas, here one really has to entirely resort to a long case by
case hand/computer checking.
On the bright side, we were actually looking for another property of the
gradient ideal (coming soon!). Thus, it might be the case that we are
missing some fine point that rules out the free divisor possibility.
()
June 3, 2011
12 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
13 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
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I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
13 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
13 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
13 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
13 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
13 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Degree 5: monoidal free divisors
Proposition
Consider the following parameterized family of monoidal quintics
F = Fa (x, y, z) = y 4 z + a1 x 5 + a2 x 2 y 3 + a3 xy 4 + a4 y 5
Then the general member of the family is an irreducible free divisor of
linear type ; more precisely, any member with a1 6= 0 and a2 6= 0 is
such a divisor.
Note that any member of the family with a1 6= 0 is an irreducible
polynomial in k[x, y, z]. This is because on the affine piece z = 1 the
resulting polynomial is the sum of two polynomials of successive
degrees 4 and 5 with no common factor.
We next sketch the proof.
()
June 3, 2011
13 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: proof
Consider the parameters as additional variables of degree 0.
In other words, work in the S0 -standard graded polynomial ring
S = S0 [x, y, z] with S0 = k[a1 , a2 , a3 , a4 ].
Take the homogeneous ideal I ⊂ S generated by the
{x, y, z}-derivatives of f ∈ S:
I = (x(5a1 x 3 + 2a2 y 3 ), y 2 (3a2 x 2 + 4a3 xy + 5a4 yz), y 4 )
It has an obvious syzygy of S-degree 2 involving only the last two
generators.
Yet another syzygy of S-degree 2 exists – a verification done by
computer.
Dualizing over S a graded minimal free presentation of I, the latter is
generated by the 2-minors of the above two columns dividing these
minors by their gcd. A direct calculation of these minors shows that the
gcd is a22 .
()
June 3, 2011
14 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: end of proof
Now pass to the ring of fractions SM = (S0 )M [x, y, z], where M ⊂ S0
is the multiplicative set generated by {a1 , a2 }.
Thereof ISM is a perfect ideal of codimension 2.
Next specialize to k[x, y, z] by evaluating both a1 and a2 to nonzero
elements of k (the other parameters can be freely evaluated).
Taking partial derivatives with respect to x, y, z commutes with this
specialization, which factors through SM .
The final step is to observe that, since ISM is generated by the
(maximal) minors of a matrix, then the partial derivatives of f are the
(maximal) minors of the specialized matrix.
()
June 3, 2011
15 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: end of proof
Now pass to the ring of fractions SM = (S0 )M [x, y, z], where M ⊂ S0
is the multiplicative set generated by {a1 , a2 }.
Thereof ISM is a perfect ideal of codimension 2.
Next specialize to k[x, y, z] by evaluating both a1 and a2 to nonzero
elements of k (the other parameters can be freely evaluated).
Taking partial derivatives with respect to x, y, z commutes with this
specialization, which factors through SM .
The final step is to observe that, since ISM is generated by the
(maximal) minors of a matrix, then the partial derivatives of f are the
(maximal) minors of the specialized matrix.
()
June 3, 2011
15 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: end of proof
Now pass to the ring of fractions SM = (S0 )M [x, y, z], where M ⊂ S0
is the multiplicative set generated by {a1 , a2 }.
Thereof ISM is a perfect ideal of codimension 2.
Next specialize to k[x, y, z] by evaluating both a1 and a2 to nonzero
elements of k (the other parameters can be freely evaluated).
Taking partial derivatives with respect to x, y, z commutes with this
specialization, which factors through SM .
The final step is to observe that, since ISM is generated by the
(maximal) minors of a matrix, then the partial derivatives of f are the
(maximal) minors of the specialized matrix.
()
June 3, 2011
15 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: end of proof
Now pass to the ring of fractions SM = (S0 )M [x, y, z], where M ⊂ S0
is the multiplicative set generated by {a1 , a2 }.
Thereof ISM is a perfect ideal of codimension 2.
Next specialize to k[x, y, z] by evaluating both a1 and a2 to nonzero
elements of k (the other parameters can be freely evaluated).
Taking partial derivatives with respect to x, y, z commutes with this
specialization, which factors through SM .
The final step is to observe that, since ISM is generated by the
(maximal) minors of a matrix, then the partial derivatives of f are the
(maximal) minors of the specialized matrix.
()
June 3, 2011
15 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: end of proof
Now pass to the ring of fractions SM = (S0 )M [x, y, z], where M ⊂ S0
is the multiplicative set generated by {a1 , a2 }.
Thereof ISM is a perfect ideal of codimension 2.
Next specialize to k[x, y, z] by evaluating both a1 and a2 to nonzero
elements of k (the other parameters can be freely evaluated).
Taking partial derivatives with respect to x, y, z commutes with this
specialization, which factors through SM .
The final step is to observe that, since ISM is generated by the
(maximal) minors of a matrix, then the partial derivatives of f are the
(maximal) minors of the specialized matrix.
()
June 3, 2011
15 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: end of proof
Now pass to the ring of fractions SM = (S0 )M [x, y, z], where M ⊂ S0
is the multiplicative set generated by {a1 , a2 }.
Thereof ISM is a perfect ideal of codimension 2.
Next specialize to k[x, y, z] by evaluating both a1 and a2 to nonzero
elements of k (the other parameters can be freely evaluated).
Taking partial derivatives with respect to x, y, z commutes with this
specialization, which factors through SM .
The final step is to observe that, since ISM is generated by the
(maximal) minors of a matrix, then the partial derivatives of f are the
(maximal) minors of the specialized matrix.
()
June 3, 2011
15 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors: end of proof
Now pass to the ring of fractions SM = (S0 )M [x, y, z], where M ⊂ S0
is the multiplicative set generated by {a1 , a2 }.
Thereof ISM is a perfect ideal of codimension 2.
Next specialize to k[x, y, z] by evaluating both a1 and a2 to nonzero
elements of k (the other parameters can be freely evaluated).
Taking partial derivatives with respect to x, y, z commutes with this
specialization, which factors through SM .
The final step is to observe that, since ISM is generated by the
(maximal) minors of a matrix, then the partial derivatives of f are the
(maximal) minors of the specialized matrix.
()
June 3, 2011
15 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
I RREDUCIBLE
HOMOGENEOUS FREE DIVISORS
Monoidal free divisors in any degree ≥ 5
The previous family of quintics generalizes to the following family in
any degree d ≥ 5:
F = Fa (x, y, z) = y d−1 z + a1 x d + a2 x 2 y d−2 + a3 xy d−1 + a4 y d ∈
S0 [x, y, z].
A similar argument shows that specializing F outside a proper closed
subset of Spec(S0 ) still yields a free divisor.
This gives a 4-dimensional family of irreducible homogeneous free
divisors of any (admissible) degree in P2k .
Further these divisors are Koszul free, which is the general case of
such divisors in P2k .
However, these divisors are not of linear type except for d = 5.
We will soon comment on this property, along with other examples.
()
June 3, 2011
16 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Ideals of linear type
I ⊂ R: an ideal
SR (I) ։ RR (I): the structural graded R-algebra surjective
homomorphism from the symmetric algebra of I to its Rees algebra
Definition
I is of linear type if this map is injective.
An ideal I ⊂ R of linear type satisfies the Artin–Nagata condition G∞
stating that the minimal number of generators of I locally at any prime
p ∈ Spec(R) is at most the codimension of p.
It can be seen that G∞ is equivalent to requiring that
cod(It (ϕ)) ≥ rank(ϕ) − t + 2,
for
1 ≤ t ≤ rank(ϕ),
where It (ϕ) denotes the determinantal ideal of the t × t minors of a
presentation matrix of ϕ.
()
June 3, 2011
17 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Ideals of linear type
I ⊂ R: an ideal
SR (I) ։ RR (I): the structural graded R-algebra surjective
homomorphism from the symmetric algebra of I to its Rees algebra
Definition
I is of linear type if this map is injective.
An ideal I ⊂ R of linear type satisfies the Artin–Nagata condition G∞
stating that the minimal number of generators of I locally at any prime
p ∈ Spec(R) is at most the codimension of p.
It can be seen that G∞ is equivalent to requiring that
cod(It (ϕ)) ≥ rank(ϕ) − t + 2,
for
1 ≤ t ≤ rank(ϕ),
where It (ϕ) denotes the determinantal ideal of the t × t minors of a
presentation matrix of ϕ.
()
June 3, 2011
17 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Ideals of linear type
I ⊂ R: an ideal
SR (I) ։ RR (I): the structural graded R-algebra surjective
homomorphism from the symmetric algebra of I to its Rees algebra
Definition
I is of linear type if this map is injective.
An ideal I ⊂ R of linear type satisfies the Artin–Nagata condition G∞
stating that the minimal number of generators of I locally at any prime
p ∈ Spec(R) is at most the codimension of p.
It can be seen that G∞ is equivalent to requiring that
cod(It (ϕ)) ≥ rank(ϕ) − t + 2,
for
1 ≤ t ≤ rank(ϕ),
where It (ϕ) denotes the determinantal ideal of the t × t minors of a
presentation matrix of ϕ.
()
June 3, 2011
17 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Ideals of linear type
I ⊂ R: an ideal
SR (I) ։ RR (I): the structural graded R-algebra surjective
homomorphism from the symmetric algebra of I to its Rees algebra
Definition
I is of linear type if this map is injective.
An ideal I ⊂ R of linear type satisfies the Artin–Nagata condition G∞
stating that the minimal number of generators of I locally at any prime
p ∈ Spec(R) is at most the codimension of p.
It can be seen that G∞ is equivalent to requiring that
cod(It (ϕ)) ≥ rank(ϕ) − t + 2,
for
1 ≤ t ≤ rank(ϕ),
where It (ϕ) denotes the determinantal ideal of the t × t minors of a
presentation matrix of ϕ.
()
June 3, 2011
17 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Ideals of linear type
I ⊂ R: an ideal
SR (I) ։ RR (I): the structural graded R-algebra surjective
homomorphism from the symmetric algebra of I to its Rees algebra
Definition
I is of linear type if this map is injective.
An ideal I ⊂ R of linear type satisfies the Artin–Nagata condition G∞
stating that the minimal number of generators of I locally at any prime
p ∈ Spec(R) is at most the codimension of p.
It can be seen that G∞ is equivalent to requiring that
cod(It (ϕ)) ≥ rank(ϕ) − t + 2,
for
1 ≤ t ≤ rank(ϕ),
where It (ϕ) denotes the determinantal ideal of the t × t minors of a
presentation matrix of ϕ.
()
June 3, 2011
17 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Ideals of linear type
I ⊂ R: an ideal
SR (I) ։ RR (I): the structural graded R-algebra surjective
homomorphism from the symmetric algebra of I to its Rees algebra
Definition
I is of linear type if this map is injective.
An ideal I ⊂ R of linear type satisfies the Artin–Nagata condition G∞
stating that the minimal number of generators of I locally at any prime
p ∈ Spec(R) is at most the codimension of p.
It can be seen that G∞ is equivalent to requiring that
cod(It (ϕ)) ≥ rank(ϕ) − t + 2,
for
1 ≤ t ≤ rank(ϕ),
where It (ϕ) denotes the determinantal ideal of the t × t minors of a
presentation matrix of ϕ.
()
June 3, 2011
17 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Ideals of linear type
I ⊂ R: an ideal
SR (I) ։ RR (I): the structural graded R-algebra surjective
homomorphism from the symmetric algebra of I to its Rees algebra
Definition
I is of linear type if this map is injective.
An ideal I ⊂ R of linear type satisfies the Artin–Nagata condition G∞
stating that the minimal number of generators of I locally at any prime
p ∈ Spec(R) is at most the codimension of p.
It can be seen that G∞ is equivalent to requiring that
cod(It (ϕ)) ≥ rank(ϕ) − t + 2,
for
1 ≤ t ≤ rank(ϕ),
where It (ϕ) denotes the determinantal ideal of the t × t minors of a
presentation matrix of ϕ.
()
June 3, 2011
17 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Divisor with Jacobian ideal of linear type
Definition
A divisor whose Jacobian ideal is of linear type will be said to be of
differentiable linear type.
(This has been dubbed of linear Jacobian type by Luis Narváez.)
Proposition
(Conjectured) Let Cn ⊂ Pn (n ≥ 3) be a rational normal curve and let
fn ∈ R = k[x0 , . . . , xn ] denote its dual hypersurface. Then
The partial derivatives of fn generate a codimension two perfect
ideal In ⊂ R of linear type
fn is homaloidal if and only if n = 3.
The first part has been verified for n ≤ 5 by computer calculation.
The second part follows from the first and recent work on the algebraic
side of Cremona transformations.
()
June 3, 2011
18 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Koszul free divisors
The notion of a Koszul free divisor was introduced by Calderón and
Narváez, in terms of differential operators.
A different approach to this notion is possible through the use of the
symmetric algebra, seemingly a much simpler gadget – it remains to
entice a full comparison between the two notions in the polynomial
case.
In order to examine the next
Pnfew examples it suffices to state the case
of an Eulerian divisor f = i=1 gi (∂f /∂xi ).
Proposition
If f is a free divisor then it is Koszul free if and only
The symmetric algebra SR (I) is Cohen–Macaulay
The Euler relation is a non-zero-divisor regarded as an element of
degree 1 in SR (I).
()
June 3, 2011
19 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Koszul free divisors
The notion of a Koszul free divisor was introduced by Calderón and
Narváez, in terms of differential operators.
A different approach to this notion is possible through the use of the
symmetric algebra, seemingly a much simpler gadget – it remains to
entice a full comparison between the two notions in the polynomial
case.
In order to examine the next
Pnfew examples it suffices to state the case
of an Eulerian divisor f = i=1 gi (∂f /∂xi ).
Proposition
If f is a free divisor then it is Koszul free if and only
The symmetric algebra SR (I) is Cohen–Macaulay
The Euler relation is a non-zero-divisor regarded as an element of
degree 1 in SR (I).
()
June 3, 2011
19 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Koszul free divisors
The notion of a Koszul free divisor was introduced by Calderón and
Narváez, in terms of differential operators.
A different approach to this notion is possible through the use of the
symmetric algebra, seemingly a much simpler gadget – it remains to
entice a full comparison between the two notions in the polynomial
case.
In order to examine the next
Pnfew examples it suffices to state the case
of an Eulerian divisor f = i=1 gi (∂f /∂xi ).
Proposition
If f is a free divisor then it is Koszul free if and only
The symmetric algebra SR (I) is Cohen–Macaulay
The Euler relation is a non-zero-divisor regarded as an element of
degree 1 in SR (I).
()
June 3, 2011
19 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Koszul free divisors
The notion of a Koszul free divisor was introduced by Calderón and
Narváez, in terms of differential operators.
A different approach to this notion is possible through the use of the
symmetric algebra, seemingly a much simpler gadget – it remains to
entice a full comparison between the two notions in the polynomial
case.
In order to examine the next
Pnfew examples it suffices to state the case
of an Eulerian divisor f = i=1 gi (∂f /∂xi ).
Proposition
If f is a free divisor then it is Koszul free if and only
The symmetric algebra SR (I) is Cohen–Macaulay
The Euler relation is a non-zero-divisor regarded as an element of
degree 1 in SR (I).
()
June 3, 2011
19 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Koszul free divisors
The notion of a Koszul free divisor was introduced by Calderón and
Narváez, in terms of differential operators.
A different approach to this notion is possible through the use of the
symmetric algebra, seemingly a much simpler gadget – it remains to
entice a full comparison between the two notions in the polynomial
case.
In order to examine the next
Pnfew examples it suffices to state the case
of an Eulerian divisor f = i=1 gi (∂f /∂xi ).
Proposition
If f is a free divisor then it is Koszul free if and only
The symmetric algebra SR (I) is Cohen–Macaulay
The Euler relation is a non-zero-divisor regarded as an element of
degree 1 in SR (I).
()
June 3, 2011
19 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Koszul free divisors
The notion of a Koszul free divisor was introduced by Calderón and
Narváez, in terms of differential operators.
A different approach to this notion is possible through the use of the
symmetric algebra, seemingly a much simpler gadget – it remains to
entice a full comparison between the two notions in the polynomial
case.
In order to examine the next
Pnfew examples it suffices to state the case
of an Eulerian divisor f = i=1 gi (∂f /∂xi ).
Proposition
If f is a free divisor then it is Koszul free if and only
The symmetric algebra SR (I) is Cohen–Macaulay
The Euler relation is a non-zero-divisor regarded as an element of
degree 1 in SR (I).
()
June 3, 2011
19 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Koszul free divisors
The notion of a Koszul free divisor was introduced by Calderón and
Narváez, in terms of differential operators.
A different approach to this notion is possible through the use of the
symmetric algebra, seemingly a much simpler gadget – it remains to
entice a full comparison between the two notions in the polynomial
case.
In order to examine the next
Pnfew examples it suffices to state the case
of an Eulerian divisor f = i=1 gi (∂f /∂xi ).
Proposition
If f is a free divisor then it is Koszul free if and only
The symmetric algebra SR (I) is Cohen–Macaulay
The Euler relation is a non-zero-divisor regarded as an element of
degree 1 in SR (I).
()
June 3, 2011
19 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples
For convenience the relevant features of the following examples will be
collected in the form of propositions.
Proposition
Let
f = 256z 3 −128x 2z 2 +16x 4 z+144xy 2 z−4x 3 y 2 −27y 4 ∈ R = C[x, y, z].
Then f is an irreducible quasi-homogenous free divisor with Euler
equality 12f = 2xfx + 3yfy + 4zfz . Moreover:
(i) f is a Koszul free divisor
(ii) The gradient ideal I ⊂ R of is an ideal of linear type – i.e., the
symmetric SR (I) algebra is an integral domain.
For the proof, since I is an almost complete intersection, to get (ii) it
suffices to verify that the coordinates of the 3 × 2 matrix, whose
(signed) 2-minors are the partial derivatives of f , generate an
(x, y, z)-primary ideal.
()
June 3, 2011
20 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 2
Remark
Let IF ⊂ R[t] denote the ideal generated by the
x, y , z-partial derivatives of the homogenization F ∈ R[t]
of f relative to a new variable t. A calculation with
Macaulay shows that the symmetric algebra of IF on R[t]
is a Cohen–Macaulay domain. However F is not a free
divisor, i.e., its full gradient ideal is not a perfect ideal (of
codimension 2). This should be confronted with the next
example.
()
June 3, 2011
21 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 2
Remark
Let IF ⊂ R[t] denote the ideal generated by the
x, y , z-partial derivatives of the homogenization F ∈ R[t]
of f relative to a new variable t. A calculation with
Macaulay shows that the symmetric algebra of IF on R[t]
is a Cohen–Macaulay domain. However F is not a free
divisor, i.e., its full gradient ideal is not a perfect ideal (of
codimension 2). This should be confronted with the next
example.
()
June 3, 2011
21 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 2
Remark
Let IF ⊂ R[t] denote the ideal generated by the
x, y , z-partial derivatives of the homogenization F ∈ R[t]
of f relative to a new variable t. A calculation with
Macaulay shows that the symmetric algebra of IF on R[t]
is a Cohen–Macaulay domain. However F is not a free
divisor, i.e., its full gradient ideal is not a perfect ideal (of
codimension 2). This should be confronted with the next
example.
()
June 3, 2011
21 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 2
Remark
Let IF ⊂ R[t] denote the ideal generated by the
x, y , z-partial derivatives of the homogenization F ∈ R[t]
of f relative to a new variable t. A calculation with
Macaulay shows that the symmetric algebra of IF on R[t]
is a Cohen–Macaulay domain. However F is not a free
divisor, i.e., its full gradient ideal is not a perfect ideal (of
codimension 2). This should be confronted with the next
example.
()
June 3, 2011
21 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 2
Remark
Let IF ⊂ R[t] denote the ideal generated by the
x, y , z-partial derivatives of the homogenization F ∈ R[t]
of f relative to a new variable t. A calculation with
Macaulay shows that the symmetric algebra of IF on R[t]
is a Cohen–Macaulay domain. However F is not a free
divisor, i.e., its full gradient ideal is not a perfect ideal (of
codimension 2). This should be confronted with the next
example.
()
June 3, 2011
21 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Revisiting known examples, 3
Proposition
Let f = xy (x + y )(x + yz) ∈ R = C[x , y , z] and let I ⊂ R denote the
corresponding gradient ideal. Then f is a reduced free Eulerian divisor –
f = 1/4xfx + 1/4yfy – but not quasihomogeneous in the sense of positive
weights.
(i) The symmetric algebra SR (I) is a Cohen–Macaulay domain, but the
Euler equation E viewed in degree one is a zero-divisor on SR (I), so f is
not Koszul free.
(ii) More precisely, if D and J respectively denote the defining ideals of
equations of the symmetric algebra SR (I/(f )) and of the Rees algebra
RR (I) then (J1 ) = J ∩ D, where J = (J1 , Q), D = (J1 , E) with
Q∈
/ (J1 ); in particular, I is not an ideal of linear type
(iii) The homogenization F ∈ R[t] of f relative to a new variable t is a
homaloidal free divisor on R[t] but not Koszul free.
()
June 3, 2011
22 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Final remarks
In the second example f is a fairly degenerated
divisor, sharing a proper factor with one of its
derivatives – something that cannot happen if f is
irreducible.
As to the fact that F is homaloidal, this is quite
frequent for non irreducible divisors such as this.
One notes that the same form Q above, responsible
for I not being of linear type, is responsible for F
being homaloidal.
Now, in general if the homogenization F ∈ R[t] of f is
perfect of codimension 2 then so is f (by simply
specializing at t = 1).
()
June 3, 2011
23 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Final remarks
In the second example f is a fairly degenerated
divisor, sharing a proper factor with one of its
derivatives – something that cannot happen if f is
irreducible.
As to the fact that F is homaloidal, this is quite
frequent for non irreducible divisors such as this.
One notes that the same form Q above, responsible
for I not being of linear type, is responsible for F
being homaloidal.
Now, in general if the homogenization F ∈ R[t] of f is
perfect of codimension 2 then so is f (by simply
specializing at t = 1).
()
June 3, 2011
23 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Final remarks
In the second example f is a fairly degenerated
divisor, sharing a proper factor with one of its
derivatives – something that cannot happen if f is
irreducible.
As to the fact that F is homaloidal, this is quite
frequent for non irreducible divisors such as this.
One notes that the same form Q above, responsible
for I not being of linear type, is responsible for F
being homaloidal.
Now, in general if the homogenization F ∈ R[t] of f is
perfect of codimension 2 then so is f (by simply
specializing at t = 1).
()
June 3, 2011
23 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Final remarks
In the second example f is a fairly degenerated
divisor, sharing a proper factor with one of its
derivatives – something that cannot happen if f is
irreducible.
As to the fact that F is homaloidal, this is quite
frequent for non irreducible divisors such as this.
One notes that the same form Q above, responsible
for I not being of linear type, is responsible for F
being homaloidal.
Now, in general if the homogenization F ∈ R[t] of f is
perfect of codimension 2 then so is f (by simply
specializing at t = 1).
()
June 3, 2011
23 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Final remarks
In the second example f is a fairly degenerated
divisor, sharing a proper factor with one of its
derivatives – something that cannot happen if f is
irreducible.
As to the fact that F is homaloidal, this is quite
frequent for non irreducible divisors such as this.
One notes that the same form Q above, responsible
for I not being of linear type, is responsible for F
being homaloidal.
Now, in general if the homogenization F ∈ R[t] of f is
perfect of codimension 2 then so is f (by simply
specializing at t = 1).
()
June 3, 2011
23 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Final remarks
In the second example f is a fairly degenerated
divisor, sharing a proper factor with one of its
derivatives – something that cannot happen if f is
irreducible.
As to the fact that F is homaloidal, this is quite
frequent for non irreducible divisors such as this.
One notes that the same form Q above, responsible
for I not being of linear type, is responsible for F
being homaloidal.
Now, in general if the homogenization F ∈ R[t] of f is
perfect of codimension 2 then so is f (by simply
specializing at t = 1).
()
June 3, 2011
23 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
Final remarks
In the second example f is a fairly degenerated
divisor, sharing a proper factor with one of its
derivatives – something that cannot happen if f is
irreducible.
As to the fact that F is homaloidal, this is quite
frequent for non irreducible divisors such as this.
One notes that the same form Q above, responsible
for I not being of linear type, is responsible for F
being homaloidal.
Now, in general if the homogenization F ∈ R[t] of f is
perfect of codimension 2 then so is f (by simply
specializing at t = 1).
()
June 3, 2011
23 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
D IVISORS
OF DIFFERENTIABLE LINEAR TYPE
What can be left?
It seems natural to ask for a sort of converse.
Question
Let f ∈ C[x1 , . . . , xn ] be a reduced Eulerian divisor and let F ∈ R[t] denote
its homogenization. If f is free (respectively, Koszul free), when is F free
(respectively, Koszul free)?
If one does not assume the Euler condition then there is a huge class of
counter-examples.
Namely, take a homogeneous irreducible F ∈ C[x1 , . . . , xn+1 ](n ≥ 2) whose
associated projective hypersurface is smooth and let f ∈ C[x1 , . . . , xn ] denote
one of its dehomogenizations. This is because the partial derivatives of F
generate a complete intersection of codimension n + 1 ≥ 3.
On the other hand, if the hypersurface defined by f is smooth but its projective
closure has singular points then the issue remains – see the quintic example
above, where dehomogenization f at y = 1 is smooth, for which the question
is then affirmative.
()
June 3, 2011
24 / 1
B IBLIOGRAPHY
Selected references
F.J. Calderón-Moreno, Logarithmic Differential Operators and
Logarithmic De Rham Complexes Relative to a Free Divisor, Ann.
Sci. E.N.S., 32 (1999), 577–595.
F, J. Calderón-Moreno and L. Narváez-Macarro, The module Dfs
for locally quasi-homogeneous free divisors, Compositio Math. 134
(2002), 59–74.
A. Doria, H. Hassanzadeh and A. Simis, A characteristic free
criterion of birationality, arXiv:1101.0197v1 [math.AC] 31 Dec
2010.
A. N. Nejad and A. Simis, The Aluffi algebra, J. of Singularities, 3
(2011), 20–47.
C. B. Miranda Neto, Vector fields and a family of linear type
modules related to free divisors, J. Pure Appl. Algebra, in press.
()
June 3, 2011
25 / 1
B IBLIOGRAPHY
Selected references
F.J. Calderón-Moreno, Logarithmic Differential Operators and
Logarithmic De Rham Complexes Relative to a Free Divisor, Ann.
Sci. E.N.S., 32 (1999), 577–595.
F, J. Calderón-Moreno and L. Narváez-Macarro, The module Dfs
for locally quasi-homogeneous free divisors, Compositio Math. 134
(2002), 59–74.
A. Doria, H. Hassanzadeh and A. Simis, A characteristic free
criterion of birationality, arXiv:1101.0197v1 [math.AC] 31 Dec
2010.
A. N. Nejad and A. Simis, The Aluffi algebra, J. of Singularities, 3
(2011), 20–47.
C. B. Miranda Neto, Vector fields and a family of linear type
modules related to free divisors, J. Pure Appl. Algebra, in press.
()
June 3, 2011
25 / 1
B IBLIOGRAPHY
Selected references
F.J. Calderón-Moreno, Logarithmic Differential Operators and
Logarithmic De Rham Complexes Relative to a Free Divisor, Ann.
Sci. E.N.S., 32 (1999), 577–595.
F, J. Calderón-Moreno and L. Narváez-Macarro, The module Dfs
for locally quasi-homogeneous free divisors, Compositio Math. 134
(2002), 59–74.
A. Doria, H. Hassanzadeh and A. Simis, A characteristic free
criterion of birationality, arXiv:1101.0197v1 [math.AC] 31 Dec
2010.
A. N. Nejad and A. Simis, The Aluffi algebra, J. of Singularities, 3
(2011), 20–47.
C. B. Miranda Neto, Vector fields and a family of linear type
modules related to free divisors, J. Pure Appl. Algebra, in press.
()
June 3, 2011
25 / 1
B IBLIOGRAPHY
Selected references
F.J. Calderón-Moreno, Logarithmic Differential Operators and
Logarithmic De Rham Complexes Relative to a Free Divisor, Ann.
Sci. E.N.S., 32 (1999), 577–595.
F, J. Calderón-Moreno and L. Narváez-Macarro, The module Dfs
for locally quasi-homogeneous free divisors, Compositio Math. 134
(2002), 59–74.
A. Doria, H. Hassanzadeh and A. Simis, A characteristic free
criterion of birationality, arXiv:1101.0197v1 [math.AC] 31 Dec
2010.
A. N. Nejad and A. Simis, The Aluffi algebra, J. of Singularities, 3
(2011), 20–47.
C. B. Miranda Neto, Vector fields and a family of linear type
modules related to free divisors, J. Pure Appl. Algebra, in press.
()
June 3, 2011
25 / 1
B IBLIOGRAPHY
Selected references
F.J. Calderón-Moreno, Logarithmic Differential Operators and
Logarithmic De Rham Complexes Relative to a Free Divisor, Ann.
Sci. E.N.S., 32 (1999), 577–595.
F, J. Calderón-Moreno and L. Narváez-Macarro, The module Dfs
for locally quasi-homogeneous free divisors, Compositio Math. 134
(2002), 59–74.
A. Doria, H. Hassanzadeh and A. Simis, A characteristic free
criterion of birationality, arXiv:1101.0197v1 [math.AC] 31 Dec
2010.
A. N. Nejad and A. Simis, The Aluffi algebra, J. of Singularities, 3
(2011), 20–47.
C. B. Miranda Neto, Vector fields and a family of linear type
modules related to free divisors, J. Pure Appl. Algebra, in press.
()
June 3, 2011
25 / 1
B IBLIOGRAPHY
Selected references
F.J. Calderón-Moreno, Logarithmic Differential Operators and
Logarithmic De Rham Complexes Relative to a Free Divisor, Ann.
Sci. E.N.S., 32 (1999), 577–595.
F, J. Calderón-Moreno and L. Narváez-Macarro, The module Dfs
for locally quasi-homogeneous free divisors, Compositio Math. 134
(2002), 59–74.
A. Doria, H. Hassanzadeh and A. Simis, A characteristic free
criterion of birationality, arXiv:1101.0197v1 [math.AC] 31 Dec
2010.
A. N. Nejad and A. Simis, The Aluffi algebra, J. of Singularities, 3
(2011), 20–47.
C. B. Miranda Neto, Vector fields and a family of linear type
modules related to free divisors, J. Pure Appl. Algebra, in press.
()
June 3, 2011
25 / 1
B IBLIOGRAPHY
Selected references
K. Saito, Theory of logarithmic differential forms and logarithmic
vector fields, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 27 (1980),
265–291.
A. Simis, Differential idealizers and algebraic free divisors, in
C OMMUTATIVE A LGEBRA : G EOMETRIC, H OMOLOGICAL ,
C OMBINATORIAL AND C OMPUTATIONAL A SPECTS, Lecture Notes in
Pure and Applied Mathematics (Eds. A. Corso, P. Gimenez, M. V.
Pinto and S. Zarzuela), Chapman & Hall/CRC, Volume 244 (2005)
211–226.
A. Simis, The depth of the Jacobian ring of a homogeneous
polynomial in three variables, Proc. Amer. Math. Soc., 134 (2006),
1591–1598.
()
June 3, 2011
26 / 1
B IBLIOGRAPHY
Selected references
K. Saito, Theory of logarithmic differential forms and logarithmic
vector fields, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 27 (1980),
265–291.
A. Simis, Differential idealizers and algebraic free divisors, in
C OMMUTATIVE A LGEBRA : G EOMETRIC, H OMOLOGICAL ,
C OMBINATORIAL AND C OMPUTATIONAL A SPECTS, Lecture Notes in
Pure and Applied Mathematics (Eds. A. Corso, P. Gimenez, M. V.
Pinto and S. Zarzuela), Chapman & Hall/CRC, Volume 244 (2005)
211–226.
A. Simis, The depth of the Jacobian ring of a homogeneous
polynomial in three variables, Proc. Amer. Math. Soc., 134 (2006),
1591–1598.
()
June 3, 2011
26 / 1
B IBLIOGRAPHY
Selected references
K. Saito, Theory of logarithmic differential forms and logarithmic
vector fields, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 27 (1980),
265–291.
A. Simis, Differential idealizers and algebraic free divisors, in
C OMMUTATIVE A LGEBRA : G EOMETRIC, H OMOLOGICAL ,
C OMBINATORIAL AND C OMPUTATIONAL A SPECTS, Lecture Notes in
Pure and Applied Mathematics (Eds. A. Corso, P. Gimenez, M. V.
Pinto and S. Zarzuela), Chapman & Hall/CRC, Volume 244 (2005)
211–226.
A. Simis, The depth of the Jacobian ring of a homogeneous
polynomial in three variables, Proc. Amer. Math. Soc., 134 (2006),
1591–1598.
()
June 3, 2011
26 / 1
B IBLIOGRAPHY
Selected references
K. Saito, Theory of logarithmic differential forms and logarithmic
vector fields, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 27 (1980),
265–291.
A. Simis, Differential idealizers and algebraic free divisors, in
C OMMUTATIVE A LGEBRA : G EOMETRIC, H OMOLOGICAL ,
C OMBINATORIAL AND C OMPUTATIONAL A SPECTS, Lecture Notes in
Pure and Applied Mathematics (Eds. A. Corso, P. Gimenez, M. V.
Pinto and S. Zarzuela), Chapman & Hall/CRC, Volume 244 (2005)
211–226.
A. Simis, The depth of the Jacobian ring of a homogeneous
polynomial in three variables, Proc. Amer. Math. Soc., 134 (2006),
1591–1598.
()
June 3, 2011
26 / 1
B IBLIOGRAPHY
Selected references
H. Terao, Arrangements of hyperplanes and their freeness I, II, J.
Fac. Sci. Univ. Tokyo Sect. Math. 27 (1980), 293–320.
H. Terao, The bifurcation set and logarithmic vector fields, Math.
Ann. 263 (1983), 313–321.
H. Terao, The exponents of a free hypersurface, Singularities Part
2, Proc. Symp. Pure Math. 40 (1983), 561–566.
()
June 3, 2011
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