Workshop on Free Divisors, Warwick
May 31st-June 4th 2011
Linearity properties on Jacobian ideals
Luis Narváez Macarro
Departamento de Álgebra & Instituto de
Matemáticas, Universidad de Sevilla
(Partially supported by MTM2010-19298 and FEDER)
Ideals of linear type
Ideals of linear type
Let A be a commutative ring and I = (a1 , . . . , ar ) ⊂ A
un ideal.
Ideals of linear type
Let A be a commutative ring and I = (a1 , . . . , ar ) ⊂ A
un ideal.
D EFINITION .- We say that I is an ideal of linear type
if any homogeneous polynomial F (x1 , . . . , xr ) ∈ A[x]
vanishing on (a1 , . . . , ar ) is a linear combination of homogeneous polynomials of degree 1 vanishing on (a1 , . . . , ar ).
Ideals of linear type
Let A be a commutative ring and I = (a1 , . . . , ar ) ⊂ A
un ideal.
D EFINITION .- We say that I is an ideal of linear type
if any homogeneous polynomial F (x1 , . . . , xr ) ∈ A[x]
vanishing on (a1 , . . . , ar ) is a linear combination of homogeneous polynomials of degree 1 vanishing on (a1 , . . . , ar ).
The above definition does not depend on the generators
ai of I.
Ideals of linear type
Let A be a commutative ring and I = (a1 , . . . , ar ) ⊂ A
un ideal.
D EFINITION .- We say that I is an ideal of linear type
if any homogeneous polynomial F (x1 , . . . , xr ) ∈ A[x]
vanishing on (a1 , . . . , ar ) is a linear combination of homogeneous polynomials of degree 1 vanishing on (a1 , . . . , ar ).
The above definition does not depend on the generators
ai of I.
R EMARK .- The homogeneous polynomials of degree 1
vanishing on (a1 , . . . , ar ) are the “same” as the syzygies
of a1 , . . . , ar .
Ideals of linear type
Let A be a commutative ring and I = (a1 , . . . , ar ) ⊂ A
un ideal.
D EFINITION .- We say that I is an ideal of linear type
if any homogeneous polynomial F (x1 , . . . , xr ) ∈ A[x]
vanishing on (a1 , . . . , ar ) is a linear combination of homogeneous polynomials of degree 1 vanishing on (a1 , . . . , ar ).
The above definition does not depend on the generators
ai of I.
R EMARK .- The homogeneous polynomials of degree 1
vanishing on (a1 , . . . , ar ) are the “same” as the syzygies
of a1 , . . . , ar .
The above definition means that the kernel of the (graded)
map
A[x] → A[a1 t, . . . , ar t] ⊂ A[t],
xi 7→ ai t
is generated by homogeneous polynomials of degree 1,
or in fancy words: I is of linear type if and only if the
canonical map SymA I → R(I) is an isomorphism.
Ideals of linear type
Let A be a commutative ring and I = (a1 , . . . , ar ) ⊂ A
un ideal.
D EFINITION .- We say that I is an ideal of linear type
if any homogeneous polynomial F (x1 , . . . , xr ) ∈ A[x]
vanishing on (a1 , . . . , ar ) is a linear combination of homogeneous polynomials of degree 1 vanishing on (a1 , . . . , ar ).
The above definition does not depend on the generators
ai of I.
R EMARK .- The homogeneous polynomials of degree 1
vanishing on (a1 , . . . , ar ) are the “same” as the syzygies
of a1 , . . . , ar .
The above definition means that the kernel of the (graded)
map
A[x] → A[a1 t, . . . , ar t] ⊂ A[t],
xi 7→ ai t
is generated by homogeneous polynomials of degree 1,
or in fancy words: I is of linear type if and only if the
canonical map SymA I → R(I) is an isomorphism.
E XAMPLE .- If a1 , . . . , ar is a regular sequence in A, then
I = (a1 , . . . , ar ) is of linear type.
Divisors of linear Jacobian type (LJT)
Divisors of linear Jacobian type (LJT)
D EFINITION .- We say that a germ of divisor (D, 0) ⊂
(Cd , 0), with a reduced equation f = 0, is of linear Jacobian type (LJT) if the Jacobian ideal of (D, 0), J =
(f, fx′ 1 , . . . , fx′ d ), is of linear type.
Divisors of linear Jacobian type (LJT)
D EFINITION .- We say that a germ of divisor (D, 0) ⊂
(Cd , 0), with a reduced equation f = 0, is of linear Jacobian type (LJT) if the Jacobian ideal of (D, 0), J =
(f, fx′ 1 , . . . , fx′ d ), is of linear type.
E XAMPLE .- Any quasi-homogeneous (germ of) divisor
with an isolated singularity is (LJT).
Divisors of linear Jacobian type (LJT)
D EFINITION .- We say that a germ of divisor (D, 0) ⊂
(Cd , 0), with a reduced equation f = 0, is of linear Jacobian type (LJT) if the Jacobian ideal of (D, 0), J =
(f, fx′ 1 , . . . , fx′ d ), is of linear type.
E XAMPLE .- Any quasi-homogeneous (germ of) divisor
with an isolated singularity is (LJT).
P ROPOSITION .- For a divisor (D, 0) ⊂ (Cd , 0), the following properties are equivalent:
(a) (D, 0) ⊂ (Cd , 0) is (LJT).
(b) (D × C, (0, 0)) ⊂ (Cd × C, (0, 0)) is (LJT).
Divisors of linear Jacobian type (LJT)
D EFINITION .- We say that a germ of divisor (D, 0) ⊂
(Cd , 0), with a reduced equation f = 0, is of linear Jacobian type (LJT) if the Jacobian ideal of (D, 0), J =
(f, fx′ 1 , . . . , fx′ d ), is of linear type.
E XAMPLE .- Any quasi-homogeneous (germ of) divisor
with an isolated singularity is (LJT).
P ROPOSITION .- For a divisor (D, 0) ⊂ (Cd , 0), the following properties are equivalent:
(a) (D, 0) ⊂ (Cd , 0) is (LJT).
(b) (D × C, (0, 0)) ⊂ (Cd × C, (0, 0)) is (LJT).
D EFINITION .- We say that a germ f ∈ OCd ,0 with f (0) =
0 is Euler homogeneous (EH) (resp. strongly Euler homogeneous (SEH)) if there is a (germ of) vector field χ
(resp. vanishing on 0) such that χ(f ) = f .
Divisors of linear Jacobian type (LJT)
D EFINITION .- We say that a germ of divisor (D, 0) ⊂
(Cd , 0), with a reduced equation f = 0, is of linear Jacobian type (LJT) if the Jacobian ideal of (D, 0), J =
(f, fx′ 1 , . . . , fx′ d ), is of linear type.
E XAMPLE .- Any quasi-homogeneous (germ of) divisor
with an isolated singularity is (LJT).
P ROPOSITION .- For a divisor (D, 0) ⊂ (Cd , 0), the following properties are equivalent:
(a) (D, 0) ⊂ (Cd , 0) is (LJT).
(b) (D × C, (0, 0)) ⊂ (Cd × C, (0, 0)) is (LJT).
D EFINITION .- We say that a germ f ∈ OCd ,0 with f (0) =
0 is Euler homogeneous (EH) (resp. strongly Euler homogeneous (SEH)) if there is a (germ of) vector field χ
(resp. vanishing on 0) such that χ(f ) = f .
R EMARK .- If f is (SEH), then uf is also (SEH) for any
unit u. In particular the notion of “strongly Euler homogeneous” also applies to germs of divisors. The situation
with the notion of “Euler homogeneous” is different.
Why are (LJT) divisors interesting?
Why are (LJT) divisors interesting?
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced equation f = 0, f ∈ O = OCd ,0 and consider the Bernstein
module O[f −1 , s]f s . It is a module over D[s]: ∂i (f s ) =
sf −1 fx′ i f s .
Why are (LJT) divisors interesting?
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced equation f = 0, f ∈ O = OCd ,0 and consider the Bernstein
module O[f −1 , s]f s . It is a module over D[s]: ∂i (f s ) =
sf −1 fx′ i f s .
Over D[s] we consider the “total order filtration”:
ord ∂i , ord s = 1.
The corresponding graded ring is O[s, ξ1 , . . . , ξd ].
Why are (LJT) divisors interesting?
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced equation f = 0, f ∈ O = OCd ,0 and consider the Bernstein
module O[f −1 , s]f s . It is a module over D[s]: ∂i (f s ) =
sf −1 fx′ i f s .
Over D[s] we consider the “total order filtration”:
ord ∂i , ord s = 1.
The corresponding graded ring is O[s, ξ1 , . . . , ξd ].
The (total) order one operators of D[s] annihilating f s
are of the form δ−αs where δ is a logarithmic derivation
and δ(f ) = αf .
Why are (LJT) divisors interesting?
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced equation f = 0, f ∈ O = OCd ,0 and consider the Bernstein
module O[f −1 , s]f s . It is a module over D[s]: ∂i (f s ) =
sf −1 fx′ i f s .
Over D[s] we consider the “total order filtration”:
ord ∂i , ord s = 1.
The corresponding graded ring is O[s, ξ1 , . . . , ξd ].
The (total) order one operators of D[s] annihilating f s
are of the form δ−αs where δ is a logarithmic derivation
and δ(f ) = αf .
D EFINITION .- We say that (D, 0) (or f ) is of differential
linear type (DLT) if annD[s] f s is generated by (total)
order one operators.
Why are (LJT) divisors interesting?
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced equation f = 0, f ∈ O = OCd ,0 and consider the Bernstein
module O[f −1 , s]f s . It is a module over D[s]: ∂i (f s ) =
sf −1 fx′ i f s .
Over D[s] we consider the “total order filtration”:
ord ∂i , ord s = 1.
The corresponding graded ring is O[s, ξ1 , . . . , ξd ].
The (total) order one operators of D[s] annihilating f s
are of the form δ−αs where δ is a logarithmic derivation
and δ(f ) = αf .
D EFINITION .- We say that (D, 0) (or f ) is of differential
linear type (DLT) if annD[s] f s is generated by (total)
order one operators.
P ROPOSITION .- (LJT) ⇒ (DLT).
Why are (LJT) divisors interesting?
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced equation f = 0, f ∈ O = OCd ,0 and consider the Bernstein
module O[f −1 , s]f s . It is a module over D[s]: ∂i (f s ) =
sf −1 fx′ i f s .
Over D[s] we consider the “total order filtration”:
ord ∂i , ord s = 1.
The corresponding graded ring is O[s, ξ1 , . . . , ξd ].
The (total) order one operators of D[s] annihilating f s
are of the form δ−αs where δ is a logarithmic derivation
and δ(f ) = αf .
D EFINITION .- We say that (D, 0) (or f ) is of differential
linear type (DLT) if annD[s] f s is generated by (total)
order one operators.
P ROPOSITION .- (LJT) ⇒ (DLT).
P ROOF : Let us consider ϕ : O[s, ξ1 , . . . , ξd ] → R(J),
ϕ(s) = f t, ϕ(ξi ) = fx′ i t.
If an operator P ∈ D[s] annihilates f s , then σT (P ) ∈
ker ϕ . . .
(LJT) ⇒ strongly Euler homogeneous
(LJT) ⇒ strongly Euler homogeneous
Let (D, 0) ⊂ (Cd , 0) be a divisor of linear Jacobian type
with a reduced equation f = 0, f ∈ O = OCd ,0 . Let
I = (fx′ 1 , . . . , fx′ d ) be the gradient ideal and J = (f ) + I
the Jacobian ideal, and consider ϕ : O[s, ξ1 , . . . , ξd ] →
R(J), ϕ(s) = f t, ϕ(ξi ) = fx′ i t.
(LJT) ⇒ strongly Euler homogeneous
Let (D, 0) ⊂ (Cd , 0) be a divisor of linear Jacobian type
with a reduced equation f = 0, f ∈ O = OCd ,0 . Let
I = (fx′ 1 , . . . , fx′ d ) be the gradient ideal and J = (f ) + I
the Jacobian ideal, and consider ϕ : O[s, ξ1 , . . . , ξd ] →
R(J), ϕ(s) = f t, ϕ(ξi ) = fx′ i t.
Since D is (LJT), ker ϕ is generated by ∆i = −αi s +
ai1 ξ1 + · · · + aid ξd , i = 1, . . . , m corresponding to a
system of generators of the syzygies of f, fx′ 1 , . . . , fx′ d
(LJT) ⇒ strongly Euler homogeneous
Let (D, 0) ⊂ (Cd , 0) be a divisor of linear Jacobian type
with a reduced equation f = 0, f ∈ O = OCd ,0 . Let
I = (fx′ 1 , . . . , fx′ d ) be the gradient ideal and J = (f ) + I
the Jacobian ideal, and consider ϕ : O[s, ξ1 , . . . , ξd ] →
R(J), ϕ(s) = f t, ϕ(ξi ) = fx′ i t.
Since D is (LJT), ker ϕ is generated by ∆i = −αi s +
ai1 ξ1 + · · · + aid ξd , i = 1, . . . , m corresponding to a
system of generators of the syzygies of f, fx′ 1 , . . . , fx′ d
P
Notice that the δi =
aij ∂j form a system of generators
of the logarithmic derivations, with δi (f ) = αi f .
(LJT) ⇒ strongly Euler homogeneous
Let (D, 0) ⊂ (Cd , 0) be a divisor of linear Jacobian type
with a reduced equation f = 0, f ∈ O = OCd ,0 . Let
I = (fx′ 1 , . . . , fx′ d ) be the gradient ideal and J = (f ) + I
the Jacobian ideal, and consider ϕ : O[s, ξ1 , . . . , ξd ] →
R(J), ϕ(s) = f t, ϕ(ξi ) = fx′ i t.
Since D is (LJT), ker ϕ is generated by ∆i = −αi s +
ai1 ξ1 + · · · + aid ξd , i = 1, . . . , m corresponding to a
system of generators of the syzygies of f, fx′ 1 , . . . , fx′ d
P
Notice that the δi =
aij ∂j form a system of generators
of the logarithmic derivations, with δi (f ) = αi f .
We know that f ∈ I, i.e. there is a homogeneous polynomial F (s, ξ) with F (s, 0) = sN , F (f, fx′ 1 , . . . , fx′ d ) =
0.
(LJT) ⇒ strongly Euler homogeneous
Let (D, 0) ⊂ (Cd , 0) be a divisor of linear Jacobian type
with a reduced equation f = 0, f ∈ O = OCd ,0 . Let
I = (fx′ 1 , . . . , fx′ d ) be the gradient ideal and J = (f ) + I
the Jacobian ideal, and consider ϕ : O[s, ξ1 , . . . , ξd ] →
R(J), ϕ(s) = f t, ϕ(ξi ) = fx′ i t.
Since D is (LJT), ker ϕ is generated by ∆i = −αi s +
ai1 ξ1 + · · · + aid ξd , i = 1, . . . , m corresponding to a
system of generators of the syzygies of f, fx′ 1 , . . . , fx′ d
P
Notice that the δi =
aij ∂j form a system of generators
of the logarithmic derivations, with δi (f ) = αi f .
We know that f ∈ I, i.e. there is a homogeneous polynomial F (s, ξ) with F (s, 0) = sN , F (f, fx′ 1 , . . . , fx′ d ) =
0.
So, F must be a linear combination of the ∆i and making
ξ1 = · · · = ξd = 0, s = 1 we deduce that 1 belongs to
the ideal generated by the αi . We conclude that some of
the αi is a unit and f is Euler homogeneous.
(LJT) ⇒ strongly Euler homogeneous
Let (D, 0) ⊂ (Cd , 0) be a divisor of linear Jacobian type
with a reduced equation f = 0, f ∈ O = OCd ,0 . Let
I = (fx′ 1 , . . . , fx′ d ) be the gradient ideal and J = (f ) + I
the Jacobian ideal, and consider ϕ : O[s, ξ1 , . . . , ξd ] →
R(J), ϕ(s) = f t, ϕ(ξi ) = fx′ i t.
Since D is (LJT), ker ϕ is generated by ∆i = −αi s +
ai1 ξ1 + · · · + aid ξd , i = 1, . . . , m corresponding to a
system of generators of the syzygies of f, fx′ 1 , . . . , fx′ d
P
Notice that the δi =
aij ∂j form a system of generators
of the logarithmic derivations, with δi (f ) = αi f .
We know that f ∈ I, i.e. there is a homogeneous polynomial F (s, ξ) with F (s, 0) = sN , F (f, fx′ 1 , . . . , fx′ d ) =
0.
So, F must be a linear combination of the ∆i and making
ξ1 = · · · = ξd = 0, s = 1 we deduce that 1 belongs to
the ideal generated by the αi . We conclude that some of
the αi is a unit and f is Euler homogeneous.
To prove strong Euler homogeneity we eliminate trivial
factors of our divisor by integrating non-singular Euler
vector fields and so on. . .
Koszul free divisors
Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0 and let δ1 , . . . , δd be a basis of the logarithmic derivations, with δi (f ) = αi f . Let us call σi =
σ(δi ) the symbol of δi in the graded ring O[ξ1 , . . . , ξd ] of
the ring of differential operators D = O[∂1 , . . . , ∂d ].
Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0 and let δ1 , . . . , δd be a basis of the logarithmic derivations, with δi (f ) = αi f . Let us call σi =
σ(δi ) the symbol of δi in the graded ring O[ξ1 , . . . , ξd ] of
the ring of differential operators D = O[∂1 , . . . , ∂d ].
D EFINITION .- We say that (D, 0) is a Koszul (free) divisor if σ1 , . . . , σd is a regular sequence in O[ξ1 , . . . , ξd ] (it
does not depend on the chosen basis δ1 , . . . , δd ).
Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0 and let δ1 , . . . , δd be a basis of the logarithmic derivations, with δi (f ) = αi f . Let us call σi =
σ(δi ) the symbol of δi in the graded ring O[ξ1 , . . . , ξd ] of
the ring of differential operators D = O[∂1 , . . . , ∂d ].
D EFINITION .- We say that (D, 0) is a Koszul (free) divisor if σ1 , . . . , σd is a regular sequence in O[ξ1 , . . . , ξd ] (it
does not depend on the chosen basis δ1 , . . . , δd ).
P ROPOSITION .- For a free divisor (D, 0) ⊂ (Cd , 0), the
following properties are equivalent:
(a) (D, 0) is Koszul.
(b) dim T (log D) = d, where T (log D) ⊂ T ∗ Cd is
the “logarithmic characteristic variey” of D, defined as V (σ1 , . . . , σd ).
(c) (D, 0) is “holonomic” in Saito’s sense.
Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0 and let δ1 , . . . , δd be a basis of the logarithmic derivations, with δi (f ) = αi f . Let us call σi =
σ(δi ) the symbol of δi in the graded ring O[ξ1 , . . . , ξd ] of
the ring of differential operators D = O[∂1 , . . . , ∂d ].
D EFINITION .- We say that (D, 0) is a Koszul (free) divisor if σ1 , . . . , σd is a regular sequence in O[ξ1 , . . . , ξd ] (it
does not depend on the chosen basis δ1 , . . . , δd ).
P ROPOSITION .- For a free divisor (D, 0) ⊂ (Cd , 0), the
following properties are equivalent:
(a) (D, 0) is Koszul.
(b) dim T (log D) = d, where T (log D) ⊂ T ∗ Cd is
the “logarithmic characteristic variey” of D, defined as V (σ1 , . . . , σd ).
(c) (D, 0) is “holonomic” in Saito’s sense.
E XAMPLE .- Any plane curve (D, 0) ⊂ (C2 , 0) is a Koszul
free divisor.
Free (LJT) ⇒ Koszul
Free (LJT) ⇒ Koszul
Simis, Torrelli.
Free (LJT) ⇒ Koszul
Simis, Torrelli.
P ROOF : Since D is (strongly) Euler homogeneous, we
can take δ1 (f ) = · · · = δd−1 (f ) = 0, δd (f ) = f and by
(LJT)
R(J) = O[ξ]/(σ1 , . . . , σd−1 )
(J = (fx′ 1 , . . . , fx′ d ) and δ1 , . . . , δd−1 correspond to a
basis of the syzygies of fx′ 1 , . . . , fx′ d ).
Free (LJT) ⇒ Koszul
Simis, Torrelli.
P ROOF : Since D is (strongly) Euler homogeneous, we
can take δ1 (f ) = · · · = δd−1 (f ) = 0, δd (f ) = f and by
(LJT)
R(J) = O[ξ]/(σ1 , . . . , σd−1 )
(J = (fx′ 1 , . . . , fx′ d ) and δ1 , . . . , δd−1 correspond to a
basis of the syzygies of fx′ 1 , . . . , fx′ d ).
dim
O[ξ1 , . . . , ξd ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
and so σ1 , . . . , σd−1 is a regular sequence.
Free (LJT) ⇒ Koszul
Simis, Torrelli.
P ROOF : Since D is (strongly) Euler homogeneous, we
can take δ1 (f ) = · · · = δd−1 (f ) = 0, δd (f ) = f and by
(LJT)
R(J) = O[ξ]/(σ1 , . . . , σd−1 )
(J = (fx′ 1 , . . . , fx′ d ) and δ1 , . . . , δd−1 correspond to a
basis of the syzygies of fx′ 1 , . . . , fx′ d ).
dim
O[ξ1 , . . . , ξd ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
and so σ1 , . . . , σd−1 is a regular sequence.
If F σd ∈ (σ1 , . . . , σd−1 ) then 0 = ϕ(F σd ) = ϕ(F )ϕ(σd ) =
ϕ(F )f , and so F ∈ ker ϕ = (σ1 , . . . , σd−1 ).
Free (LJT) ⇒ Koszul
Simis, Torrelli.
P ROOF : Since D is (strongly) Euler homogeneous, we
can take δ1 (f ) = · · · = δd−1 (f ) = 0, δd (f ) = f and by
(LJT)
R(J) = O[ξ]/(σ1 , . . . , σd−1 )
(J = (fx′ 1 , . . . , fx′ d ) and δ1 , . . . , δd−1 correspond to a
basis of the syzygies of fx′ 1 , . . . , fx′ d ).
dim
O[ξ1 , . . . , ξd ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
and so σ1 , . . . , σd−1 is a regular sequence.
If F σd ∈ (σ1 , . . . , σd−1 ) then 0 = ϕ(F σd ) = ϕ(F )ϕ(σd ) =
ϕ(F )f , and so F ∈ ker ϕ = (σ1 , . . . , σd−1 ).
We conclude that σ1 , . . . , σd−1 , σd is a regular sequence.
Free (LJT) ⇒ Koszul
Simis, Torrelli.
P ROOF : Since D is (strongly) Euler homogeneous, we
can take δ1 (f ) = · · · = δd−1 (f ) = 0, δd (f ) = f and by
(LJT)
R(J) = O[ξ]/(σ1 , . . . , σd−1 )
(J = (fx′ 1 , . . . , fx′ d ) and δ1 , . . . , δd−1 correspond to a
basis of the syzygies of fx′ 1 , . . . , fx′ d ).
dim
O[ξ1 , . . . , ξd ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
and so σ1 , . . . , σd−1 is a regular sequence.
If F σd ∈ (σ1 , . . . , σd−1 ) then 0 = ϕ(F σd ) = ϕ(F )ϕ(σd ) =
ϕ(F )f , and so F ∈ ker ϕ = (σ1 , . . . , σd−1 ).
We conclude that σ1 , . . . , σd−1 , σd is a regular sequence.
Better:
(LJT) ⇒ holonomic
Locally quasi-homogeneous free divisors
Locally quasi-homogeneous free divisors
D EFINITION .- We say that a divisor (D, 0) ⊂ (Cd , 0) is
locally quasi-homogeneous (LQH) if for any point p ∈ D
there are local coordinates. . .
Locally quasi-homogeneous free divisors
D EFINITION .- We say that a divisor (D, 0) ⊂ (Cd , 0) is
locally quasi-homogeneous (LQH) if for any point p ∈ D
there are local coordinates. . .
R EMARK .- For any divisor (D, 0) ⊂ (Cd , 0), if (D, p) ⊂
(Cd , p) is holonomic for all p 6= 0, then (D, 0) ⊂ (Cd , 0)
is also holonomic.
Locally quasi-homogeneous free divisors
D EFINITION .- We say that a divisor (D, 0) ⊂ (Cd , 0) is
locally quasi-homogeneous (LQH) if for any point p ∈ D
there are local coordinates. . .
R EMARK .- For any divisor (D, 0) ⊂ (Cd , 0), if (D, p) ⊂
(Cd , p) is holonomic for all p 6= 0, then (D, 0) ⊂ (Cd , 0)
is also holonomic.
P ROPOSITION .- Any (LQH) divisor is holonomic. In
particular, any (LQH) free divisor is Koszul.
Locally quasi-homogeneous free divisors
D EFINITION .- We say that a divisor (D, 0) ⊂ (Cd , 0) is
locally quasi-homogeneous (LQH) if for any point p ∈ D
there are local coordinates. . .
R EMARK .- For any divisor (D, 0) ⊂ (Cd , 0), if (D, p) ⊂
(Cd , p) is holonomic for all p 6= 0, then (D, 0) ⊂ (Cd , 0)
is also holonomic.
P ROPOSITION .- Any (LQH) divisor is holonomic. In
particular, any (LQH) free divisor is Koszul.
T HEOREM .- (Calderón-Moreno, NM) Any (LQH) free
divisor is of linear Jacobian type.
Locally quasi-homogeneous free divisors
D EFINITION .- We say that a divisor (D, 0) ⊂ (Cd , 0) is
locally quasi-homogeneous (LQH) if for any point p ∈ D
there are local coordinates. . .
R EMARK .- For any divisor (D, 0) ⊂ (Cd , 0), if (D, p) ⊂
(Cd , p) is holonomic for all p 6= 0, then (D, 0) ⊂ (Cd , 0)
is also holonomic.
P ROPOSITION .- Any (LQH) divisor is holonomic. In
particular, any (LQH) free divisor is Koszul.
T HEOREM .- (Calderón-Moreno, NM) Any (LQH) free
divisor is of linear Jacobian type.
P ROOF :
Strongly Koszul free divisors
Strongly Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with δi (f ) = αi f . Let us call
σi = σ(δi ).
Strongly Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with δi (f ) = αi f . Let us call
σi = σ(δi ).
Granger and Schulze have introduced a notion of “strongly
Koszul” for linear free divisors.
Strongly Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with δi (f ) = αi f . Let us call
σi = σ(δi ).
Granger and Schulze have introduced a notion of “strongly
Koszul” for linear free divisors.
Their notion can be extended for any free divisor:
Strongly Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with δi (f ) = αi f . Let us call
σi = σ(δi ).
Granger and Schulze have introduced a notion of “strongly
Koszul” for linear free divisors.
Their notion can be extended for any free divisor:
D EFINITION .- We say that (D, 0) is strongly Koszul (SK)
if f, σ1 − α1 s, . . . , σd − αd s is a regular sequence in
O[s, ξ].
Strongly Koszul free divisors
Let (D, 0) ⊂ (Cd , 0) be a free divisor with reduced
equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with δi (f ) = αi f . Let us call
σi = σ(δi ).
Granger and Schulze have introduced a notion of “strongly
Koszul” for linear free divisors.
Their notion can be extended for any free divisor:
D EFINITION .- We say that (D, 0) is strongly Koszul (SK)
if f, σ1 − α1 s, . . . , σd − αd s is a regular sequence in
O[s, ξ].
T HEOREM .- For a free divisor (D, 0) ⊂ (Cd , 0), the following properties are equivalent:
(a) (D, 0) is (SK).
(b) (D, 0) is (LJT).
Proof of: free (SK) ⇒ (LJT)
Proof of: free (SK) ⇒ (LJT)
Let
δi =
X
aij ∂j ,
i = 1, . . . , d
be a basis of Der(log D), with δi (f ) = αi f and ∆i =
−αi s + σ(δi ).
Proof of: free (SK) ⇒ (LJT)
Let
δi =
X
aij ∂j ,
i = 1, . . . , d
be a basis of Der(log D), with δi (f ) = αi f and ∆i =
−αi s + σ(δi ).
We can work at the level of coherent sheaves on a neighborhood X ⊂ Cd of 0.
Proof of: free (SK) ⇒ (LJT)
Let
δi =
X
aij ∂j ,
i = 1, . . . , d
be a basis of Der(log D), with δi (f ) = αi f and ∆i =
−αi s + σ(δi ).
We can work at the level of coherent sheaves on a neighborhood X ⊂ Cd of 0.
K := ker
OX [ξ, s]
surj.
−−→ R(J) .
(∆1 , . . . , ∆d )
Proof of: free (SK) ⇒ (LJT)
Let
δi =
X
aij ∂j ,
i = 1, . . . , d
be a basis of Der(log D), with δi (f ) = αi f and ∆i =
−αi s + σ(δi ).
We can work at the level of coherent sheaves on a neighborhood X ⊂ Cd of 0.
K := ker
OX [ξ, s]
surj.
−−→ R(J) .
(∆1 , . . . , ∆d )
Since supp K ⊂ D sing ⇒ any local section F of K is
killed by a power of f .
Proof of: free (SK) ⇒ (LJT)
Let
δi =
X
aij ∂j ,
i = 1, . . . , d
be a basis of Der(log D), with δi (f ) = αi f and ∆i =
−αi s + σ(δi ).
We can work at the level of coherent sheaves on a neighborhood X ⊂ Cd of 0.
K := ker
OX [ξ, s]
surj.
−−→ R(J) .
(∆1 , . . . , ∆d )
Since supp K ⊂ D sing ⇒ any local section F of K is
killed by a power of f .
f N F ∈ (∆1 , . . . , ∆d ), but f, ∆1 , . . . , ∆d is a regular
sequence ⇒ F ∈ (∆1 , . . . , ∆d ) ⇒ F = 0.
Proof of: free (SK) ⇒ (LJT)
Let
δi =
X
aij ∂j ,
i = 1, . . . , d
be a basis of Der(log D), with δi (f ) = αi f and ∆i =
−αi s + σ(δi ).
We can work at the level of coherent sheaves on a neighborhood X ⊂ Cd of 0.
K := ker
OX [ξ, s]
surj.
−−→ R(J) .
(∆1 , . . . , ∆d )
Since supp K ⊂ D sing ⇒ any local section F of K is
killed by a power of f .
f N F ∈ (∆1 , . . . , ∆d ), but f, ∆1 , . . . , ∆d is a regular
sequence ⇒ F ∈ (∆1 , . . . , ∆d ) ⇒ F = 0.
So K = 0 and D is (LJT).
Proof of: free (LJT) ⇒ (SK)
Proof of: free (LJT) ⇒ (SK)
We know that D is (strongly) Euler homogeneous, and
we can assume that
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
Proof of: free (LJT) ⇒ (SK)
We know that D is (strongly) Euler homogeneous, and
we can assume that
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
(LJT) ⇒
→ R(J),
ϕ : O[ξ]/(σ1 , . . . , σd−1 ) −
is an isomorphism
ϕ(ξi ) = fx′ i t
Proof of: free (LJT) ⇒ (SK)
We know that D is (strongly) Euler homogeneous, and
we can assume that
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
(LJT) ⇒
→ R(J),
ϕ : O[ξ]/(σ1 , . . . , σd−1 ) −
ϕ(ξi ) = fx′ i t
is an isomorphism
⇒
dim
O[ξ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
Proof of: free (LJT) ⇒ (SK)
We know that D is (strongly) Euler homogeneous, and
we can assume that
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
(LJT) ⇒
→ R(J),
ϕ : O[ξ]/(σ1 , . . . , σd−1 ) −
ϕ(ξi ) = fx′ i t
is an isomorphism
⇒
dim
O[ξ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
⇒ σ1 , . . . , σd−1 is a regular sequence.
Proof of: free (LJT) ⇒ (SK)
We know that D is (strongly) Euler homogeneous, and
we can assume that
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
(LJT) ⇒
→ R(J),
ϕ : O[ξ]/(σ1 , . . . , σd−1 ) −
ϕ(ξi ) = fx′ i t
is an isomorphism
⇒
dim
O[ξ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
⇒ σ1 , . . . , σd−1 is a regular sequence.
On the other hand: (σ1 , . . . , σd−1 ) is a prime ideal not
containing f ⇒ f, σ1 , . . . , σd−1 is a regular sequence in
O[ξ] ⇒ f, σ1 , . . . , σd−1 , σd − s is a regular sequence in
O[s, ξ].
Proof of: free (LJT) ⇒ (SK)
We know that D is (strongly) Euler homogeneous, and
we can assume that
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
(LJT) ⇒
→ R(J),
ϕ : O[ξ]/(σ1 , . . . , σd−1 ) −
ϕ(ξi ) = fx′ i t
is an isomorphism
⇒
dim
O[ξ]
(σ1 , . . . , σd−1 )
= dim R(J) = d + 1
⇒ σ1 , . . . , σd−1 is a regular sequence.
On the other hand: (σ1 , . . . , σd−1 ) is a prime ideal not
containing f ⇒ f, σ1 , . . . , σd−1 is a regular sequence in
O[ξ] ⇒ f, σ1 , . . . , σd−1 , σd − s is a regular sequence in
O[s, ξ].
So, D is (SK).
Other linearity conditions on the Jacobian
ideal
Other linearity conditions on the Jacobian
ideal
Let (D, 0) ⊂ (Cd , 0) be a divisor with reduced equation f = 0, and let δ1 , . . . , δe be a system of generators
of the logarithmic derivations, with δi (f ) = αi f . Let us
call σi = σ(δi ).
Other linearity conditions on the Jacobian
ideal
Let (D, 0) ⊂ (Cd , 0) be a divisor with reduced equation f = 0, and let δ1 , . . . , δe be a system of generators
of the logarithmic derivations, with δi (f ) = αi f . Let us
call σi = σ(δi ).
D EFINITION : We say that (D, 0) is weakly of linear Jacobian type (WLJT) if the canonical map
O[s, s−1 , ξ]/(σ1 − α1 s, . . . , σe − αe s) −
→ R(J)f t
is an isomorphism, or equivalently, if for any homogeneous polynomial F (s, ξ) ∈ O[s, ξ] vanishing on (f, fx′ )
there is an integer N ≥ 0 such that sN F ∈ (σ1 −
α1 s, . . . , σe − αe s).
Other linearity conditions on the Jacobian
ideal
Let (D, 0) ⊂ (Cd , 0) be a divisor with reduced equation f = 0, and let δ1 , . . . , δe be a system of generators
of the logarithmic derivations, with δi (f ) = αi f . Let us
call σi = σ(δi ).
D EFINITION : We say that (D, 0) is weakly of linear Jacobian type (WLJT) if the canonical map
O[s, s−1 , ξ]/(σ1 − α1 s, . . . , σe − αe s) −
→ R(J)f t
is an isomorphism, or equivalently, if for any homogeneous polynomial F (s, ξ) ∈ O[s, ξ] vanishing on (f, fx′ )
there is an integer N ≥ 0 such that sN F ∈ (σ1 −
α1 s, . . . , σe − αe s).
P ROPOSITION : If (D, 0) is (WLJT), then it is strongly
Euler homogeneous.
Other linearity conditions on the Jacobian
ideal
Let (D, 0) ⊂ (Cd , 0) be a divisor with reduced equation f = 0, and let δ1 , . . . , δe be a system of generators
of the logarithmic derivations, with δi (f ) = αi f . Let us
call σi = σ(δi ).
D EFINITION : We say that (D, 0) is weakly of linear Jacobian type (WLJT) if the canonical map
O[s, s−1 , ξ]/(σ1 − α1 s, . . . , σe − αe s) −
→ R(J)f t
is an isomorphism, or equivalently, if for any homogeneous polynomial F (s, ξ) ∈ O[s, ξ] vanishing on (f, fx′ )
there is an integer N ≥ 0 such that sN F ∈ (σ1 −
α1 s, . . . , σe − αe s).
P ROPOSITION : If (D, 0) is (WLJT), then it is strongly
Euler homogeneous.
E XAMPLE : D = {x1 x2 (x1 + x2 )(x1 + x2 x3 ) = 0} is a
(WLJT) free divisor which is not Koszul (either (LJT)),
and satisfies (LCT).
Other linearity conditions on the Jacobian
ideal
Let (D, 0) ⊂ (Cd , 0) be a divisor with reduced equation f = 0, and let δ1 , . . . , δe be a system of generators
of the logarithmic derivations, with δi (f ) = αi f . Let us
call σi = σ(δi ).
D EFINITION : We say that (D, 0) is weakly of linear Jacobian type (WLJT) if the canonical map
O[s, s−1 , ξ]/(σ1 − α1 s, . . . , σe − αe s) −
→ R(J)f t
is an isomorphism, or equivalently, if for any homogeneous polynomial F (s, ξ) ∈ O[s, ξ] vanishing on (f, fx′ )
there is an integer N ≥ 0 such that sN F ∈ (σ1 −
α1 s, . . . , σe − αe s).
P ROPOSITION : If (D, 0) is (WLJT), then it is strongly
Euler homogeneous.
E XAMPLE : D = {x1 x2 (x1 + x2 )(x1 + x2 x3 ) = 0} is a
(WLJT) free divisor which is not Koszul (either (LJT)),
and satisfies (LCT).
All the examples of free divisors which satisfy the “Logarithmic Comparison Theorem” and which I have been
able to compute are (WLJT).
Question: Are (LQH) divisors (LJT)?
Question: Are (LQH) divisors (LJT)?
One can try to adapt the proof in the case or free divisors
(F. Calderón Moreno, LNM, Compositio Math. 2002).
Question: Are (LQH) divisors (LJT)?
One can try to adapt the proof in the case or free divisors
(F. Calderón Moreno, LNM, Compositio Math. 2002).
Let (D, 0) ⊂ (Cd , 0) be a (LQH) free divisor with reduced equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
Question: Are (LQH) divisors (LJT)?
One can try to adapt the proof in the case or free divisors
(F. Calderón Moreno, LNM, Compositio Math. 2002).
Let (D, 0) ⊂ (Cd , 0) be a (LQH) free divisor with reduced equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
We know that the σi = σ(δi ) form a regular sequence.
Question: Are (LQH) divisors (LJT)?
One can try to adapt the proof in the case or free divisors
(F. Calderón Moreno, LNM, Compositio Math. 2002).
Let (D, 0) ⊂ (Cd , 0) be a (LQH) free divisor with reduced equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
We know that the σi = σ(δi ) form a regular sequence.
Let us work on a neighborhood X ⊂ Cd of 0 and consider the augmented Koszul complex of σ1 , . . . , σd−1
ϕ
→ R(J) → 0.
0 → K −(d−1) → · · · → K −1 → OX [ξ] −
Question: Are (LQH) divisors (LJT)?
One can try to adapt the proof in the case or free divisors
(F. Calderón Moreno, LNM, Compositio Math. 2002).
Let (D, 0) ⊂ (Cd , 0) be a (LQH) free divisor with reduced equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
We know that the σi = σ(δi ) form a regular sequence.
Let us work on a neighborhood X ⊂ Cd of 0 and consider the augmented Koszul complex of σ1 , . . . , σd−1
ϕ
→ R(J) → 0.
0 → K −(d−1) → · · · → K −1 → OX [ξ] −
By induction of the dimension d of the ambient space and
by integration of the Euler vector field around any point
p ∈ D, p 6= 0, we deduce that the above complex is exact
at any p 6= 0.
Question: Are (LQH) divisors (LJT)?
One can try to adapt the proof in the case or free divisors
(F. Calderón Moreno, LNM, Compositio Math. 2002).
Let (D, 0) ⊂ (Cd , 0) be a (LQH) free divisor with reduced equation f = 0, and let δ1 , . . . , δd be a basis of the
logarithmic derivations, with
δ1 (f ) = · · · = δd−1 (f ) = 0,
δd (f ) = f.
We know that the σi = σ(δi ) form a regular sequence.
Let us work on a neighborhood X ⊂ Cd of 0 and consider the augmented Koszul complex of σ1 , . . . , σd−1
ϕ
→ R(J) → 0.
0 → K −(d−1) → · · · → K −1 → OX [ξ] −
By induction of the dimension d of the ambient space and
by integration of the Euler vector field around any point
p ∈ D, p 6= 0, we deduce that the above complex is exact
at any p 6= 0.
We conclude by an argument of local cohomology (the
above complex have the good length, H0i (OX ) = 0 for
i 6= d and ϕ is surjective everywhere).
How to generalize the above proof?
How to generalize the above proof?
Let (D, 0) ⊂ (Cd , 0) be a (LQH) divisor with reduced
equation f = 0, and let δ1 , . . . , δe be a system of generators of the logarithmic derivations, with
δ1 (f ) = · · · = δe−1 (f ) = 0,
δe (f ) = f.
How to generalize the above proof?
Let (D, 0) ⊂ (Cd , 0) be a (LQH) divisor with reduced
equation f = 0, and let δ1 , . . . , δe be a system of generators of the logarithmic derivations, with
δ1 (f ) = · · · = δe−1 (f ) = 0,
δe (f ) = f.
What we need is a (locally) free resolution of
O[ξ]/(σ1 , . . . , σe−1 ) of the good length d − 1 over O[ξ].
How to generalize the above proof?
Let (D, 0) ⊂ (Cd , 0) be a (LQH) divisor with reduced
equation f = 0, and let δ1 , . . . , δe be a system of generators of the logarithmic derivations, with
δ1 (f ) = · · · = δe−1 (f ) = 0,
δe (f ) = f.
What we need is a (locally) free resolution of
O[ξ]/(σ1 , . . . , σe−1 ) of the good length d − 1 over O[ξ].
We consider the augmented complex to R(J) as before.
How to generalize the above proof?
Let (D, 0) ⊂ (Cd , 0) be a (LQH) divisor with reduced
equation f = 0, and let δ1 , . . . , δe be a system of generators of the logarithmic derivations, with
δ1 (f ) = · · · = δe−1 (f ) = 0,
δe (f ) = f.
What we need is a (locally) free resolution of
O[ξ]/(σ1 , . . . , σe−1 ) of the good length d − 1 over O[ξ].
We consider the augmented complex to R(J) as before.
By induction on the ambient dimension d we can deduce
as in the free case that we have exactness outside the origin, and we conclude we have exactness everywhere. In
particular
(σ1 ,...,σe−1 )
ϕ
→ R(J) → 0
O[ξ]e−1 −−−−−−−−→ O[ξ] −
would be exact and so J would of linear type.
How to generalize the above proof?
Let (D, 0) ⊂ (Cd , 0) be a (LQH) divisor with reduced
equation f = 0, and let δ1 , . . . , δe be a system of generators of the logarithmic derivations, with
δ1 (f ) = · · · = δe−1 (f ) = 0,
δe (f ) = f.
What we need is a (locally) free resolution of
O[ξ]/(σ1 , . . . , σe−1 ) of the good length d − 1 over O[ξ].
We consider the augmented complex to R(J) as before.
By induction on the ambient dimension d we can deduce
as in the free case that we have exactness outside the origin, and we conclude we have exactness everywhere. In
particular
(σ1 ,...,σe−1 )
ϕ
→ R(J) → 0
O[ξ]e−1 −−−−−−−−→ O[ξ] −
would be exact and so J would of linear type.
So, the key point seems to be that Sym J = O[ξ]/(σ1 , . . . , σe−1 )
is Cohen-Macaulay.