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Polarity
1.1 Polar hypersurfaces
1.1.1 The polar pairing
We will take C as the base field, although many constructions in this book work
over an arbitrary algebraically closed field.
We will usually denote by E a vector space of dimension n + 1. Its dual vector
space will be denoted by E ∨ .
Let S (E) be the symmetric algebra of E, the quotient of the tensor algebra
T (E) = ⊕d≥0 E ⊗d by the two-sided ideal generated by tensors of the form v ⊗ w −
w⊗v, v, w ∈ E. The symmetric algebra is a graded commutative algebra, its graded
d
components S d (E) are the images of E ⊗d in the quotient. The vector
d+nspace S (E)
is called the dth symmetric power of E. Its dimension is equal to n . The image
of a tensor v1 ⊗ · · · ⊗ vd in S d (E) is denoted by v1 · · · vd .
The permutation group Sd has a natural linear representation in E ⊗d via permuting the factors. The symmetrization operator ⊕σ∈Sd σ is a projection operator
onto the subspace of symmetric tensors S d (E) = (E ⊗d )Sd multiplied by d!. It
factors through S d (E) and defines a natural isomorphism
S d (E) → S d (E).
Replacing E by its dual space E ∨ , we obtain a natural isomorphism
pd : S d (E ∨ ) → S d (E ∨ ).
(1.1)
Under the identification of (E ∨ )⊗d with the space (E ⊗d )∨ , we will be able to identify S d (E ∨ ) with the space Hom(E d , C)Sd of symmetric d-multilinear functions
E d → C. The isomorphism pd is classically known as the total polarization map.
Next we use the fact that the quotient map E ⊗d → S d (E) is a universal symmetric d-multilinear map, i.e. any linear map E ⊗d → F with values in some vector
space F factors through a linear map S d (E) → F. If F = C, this gives a natural
isomorphism
S d (E ∨ ) → S d (E)∨ .
1
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Composing it with pd , we get a natural isomorphism
S d (E ∨ ) → S d (E)∨ .
(1.2)
It can be viewed as a perfect bilinear pairing, the polar pairing
, : S d (E ∨ ) ⊗ S d (E) → C.
(1.3)
This pairing extends the natural pairing between E and E ∨ to the symmetric
powers. Explicitly,
l1 · · · ld , w1 · · · wd =
lσ−1 (1) (w1 ) · · · lσ−1 (d) (wd ).
σ∈Sd
One can extend the total polarization isomorphism to a partial polarization map
, : S d (E ∨ ) ⊗ S k (E) → S d−k (E ∨ ),
l1 · · · ld , w1 · · · wk =
k ≤ d,
li1 · · · lik , w1 · · · wk (1.4)
l j.
ji1 ,...,ik
1≤i1 ≤···≤ik ≤n
In coordinates, if we choose a basis (ξ0 , . . . , ξn ) in E and its dual basis t0 , . . . , tn
in E ∨ , then we can identify S (E ∨ ) with the polynomial algebra C[t0 , . . . , tn ] and
S d (E ∨ ) with the space C[t0 , . . . , tn ]d of homogeneous polynomials of degree d.
Similarly, we identify S d (E) with C[ξ0 , . . . , ξn ]. The polarization isomorphism
extends by linearity of the pairing on monomials
⎧
⎪
⎪
⎨i0 ! · · · in ! if (i0 , . . . , in ) = ( j0 , . . . , jn ),
j0
jn
i0
in
t0 · · · tn , ξ0 · · · ξn = ⎪
⎪
⎩0
otherwise.
One can give an explicit formula for pairing (1.4) in terms of differential operators. Since ti , ξ j = δi j , it is convenient to view a basis vector ξ j as the partial
derivative operator ∂ j = ∂t∂ j . Hence any element ψ ∈ S k (E) = C[ξ0 , . . . , ξn ]k can
be viewed as a differential operator
Dψ = ψ(∂0 , . . . , ∂n ).
The pairing (1.4) becomes
ψ(ξ0 , . . . , ξn ), f (t0 , . . . , tn ) = Dψ ( f ).
For any monomial ∂i = ∂i00 · · · ∂inn and any monomial tj = t0j0 · · · tnjn , we have
⎧ j!
j−i
⎪
⎪
if j − i ≥ 0,
⎨ (j−i)! t
i j
(1.5)
∂ (t ) = ⎪
⎪
⎩0
otherwise.
Here and later we use the vector notation:
k!
k
i! = i0 ! · · · in !,
= ,
i
i!
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1.1 Polar hypersurfaces
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The total polarization f˜ of a polynomial f is given explicitly by the following
formula:
f˜(v1 , . . . , vd ) = Dv1 ···vd ( f ) = (Dv1 ◦ · · · ◦ Dvd )( f ).
Taking v1 = · · · = vd = v, we get
f˜(v, . . . , v) = d! f (v) = Dvd ( f ) =
d
i
ai ∂i f.
(1.6)
|i|=d
Remark 1.1.1 The polarization isomorphism was known in the classical literature as the symbolic method. Suppose f = ld is a dth power of a linear form. Then
Dv ( f ) = dl(v)d−1 and
Dv1 ◦ · · · ◦ Dvk ( f ) = d(d − 1) · · · (d − k + 1)l(v1 ) · · · l(vk )ld−k .
In classical notation, a linear form ai xi on Cn+1 is denoted by a x and the
dot-product of two vectors a, b is denoted by (ab). Symbolically, one denotes any
homogeneous form by adx and the right-hand side of the previous formula reads as
d(d − 1) · · · (d − k + 1)(ab)k ad−k
x .
Let us take E = S m (U ∨ ) for some vector space U and consider the linear space
S d (S m (U ∨ )∨ ). Using the polarization isomorphism, we can identify S m (U ∨ )∨ with
S m (U). Let (ξ0 , . . . , ξr ) be a basis in U and (t0 , . . . , tr+1 ) be the dual basis in U ∨ .
Then we can take for a basis of S m (U) the monomials ξj . The dual basis in S m (U ∨ )
is formed by the monomials i!1 xi . Thus, for any f ∈ S m (U ∨ ), we can write
m
i
(1.7)
m! f =
i ai x .
|i|=m
In symbolic form, m! f = (a x ) . Consider the matrix
⎛ (1)
⎞
. . . ξ0(d) ⎟⎟⎟
⎜⎜⎜ξ0
⎜⎜ .
..
.. ⎟⎟⎟⎟
Ξ = ⎜⎜⎜⎜⎜ ..
.
. ⎟⎟⎟⎟ ,
⎜⎝ (1)
(d) ⎠
ξr
. . . ξr
m
where (ξ0(k) , . . . , ξr(k) ) is a copy of a basis in U. Then the space S d (S m (U)) is equal
(i)
to the subspace of the polynomial algebra C[(ξ(i)
j )] in d(r + 1) variables ξ j of
polynomials that are homogeneous of degree m in each column of the matrix
and symmetric with respect to permutations of the columns. Let J ⊂ {1, . . . , d}
with #J = r + 1 and (J) be the corresponding maximal minor of the matrix
dm
such minors in
Ξ. Assume r + 1 divides dm. Consider a product of k = r+1
which each column participates exactly m times. Then a sum of such products
that is invariant with respect to permutations of columns represents an element
from S d (S m (U)) that has an additional property that it is invariant with respect
to the group SL(U) SL(r + 1, C) which acts on U by the left multiplication
with a vector (ξ0 , . . . , ξr ). The First Fundamental Theorem of invariant theory
states that any element in S d (S m (U))SL(U) is obtained in this way (see [180]).
We can interpret elements of S d (S m (U ∨ )∨ ) as polynomials in coefficients of ai
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of a homogeneous form of degree d in r + 1 variables written in the form (1.7).
We write symbolically an invariant in the form (J1 ) · · · (Jk ), meaning that it is
obtained as a sum of such products with some coefficients. If the number d is
small, we can use letters, say a, b, c, . . . , instead of numbers 1, . . . , d. For example,
(12)2 (13)2 (23)2 = (ab)2 (bc)2 (ac)2 represents an element in S 3 (S 4 (C2 )).
In a similar way, one considers the matrix
⎛ (1)
⎞
. . . ξ0(d) t0(1) . . . t0(s) ⎟⎟⎟
⎜⎜⎜ξ0
⎜⎜⎜ .
..
..
..
..
.. ⎟⎟⎟⎟
⎜⎜⎜ ..
.
.
.
.
. ⎟⎟⎟⎟ .
⎜⎜⎝
(1)
(d)
(1)
(s) ⎠
ξr
. . . ξr
tr
. . . tr
The product of k maximal minors such that each of the first d columns occurs
exactly k times and each of the last s columns occurs exactly p times represents a
covariant of degree p and order k. For example, (ab)2 a x b x represents the Hessian
determinant
⎞
⎛ ∂2 f
∂2 f ⎟
⎜⎜⎜ 2
⎟⎟⎟
∂x
∂x
∂x
1
2
⎜
⎟⎟⎟
He( f ) = det ⎜⎜⎜⎜ 2 1
2
⎟⎠
∂
f
∂
f
⎝
∂x2 ∂x1
∂x22
of a binary cubic form f .
The projective space of lines in E will be denoted by |E|. The space |E ∨ | will
be denoted by P(E) (following Grothendieck’s notation). We call P(E) the dual
projective space of |E|. We will often denote it by |E|∨ .
A basis ξ0 , . . . , ξn in E defines an isomorphism E Cn+1 and identifies |E| with
the projective space Pn := |Cn+1 |. For any nonzero vector v ∈ E we denote by
[v] the corresponding point in |E|. If E = Cn+1 and v = (a0 , . . . , an ) ∈ Cn+1 we
set [v] = [a0 , . . . , an ]. We call [a0 , . . . , an ] the projective coordinates of a point
[a] ∈ Pn . Other common notation for the projective coordinates of [a] is (a0 : a1 :
. . . : an ), or simply (a0 , . . . , an ), if no confusion arises.
The projective space comes with the tautological invertible sheaf O|E| (1) whose
space of global sections is identified with the dual space E ∨ . Its dth tensor power
is denoted by O|E| (d). Its space of global sections is identified with the symmetric
dth power S d (E ∨ ).
For any f ∈ S d (E ∨ ), d > 0, we denote by V( f ) the corresponding effective divisor from |O|E| (d)|, considered as a closed subscheme of |E|, not necessary reduced.
We call V( f ) a hypersurface of degree d in |E| defined by equation f = 0.1 A
hypersurface of degree 1 is a hyperplane. By definition, V(0) = |E| and V(1) = ∅.
The projective space |S d (E ∨ )| can be viewed as the projective space of hypersurfaces in |E|. It is equal to the complete linear system |O|E| (d)|. Using isomorphism
(1.2), we may identify the projective space |S d (E)| of hypersurfaces of degree d
in |E ∨ | with the dual of the projective space |S d E ∨ |. A hypersurface of degree d in
|E ∨ | is classically known as an envelope of class d.
1
This notation should not be confused with the notation of the closed subset in Zariski topology
defined by the ideal ( f ). It is equal to V( f )red .
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The natural isomorphisms
(E ∨ )⊗d H 0 (|E|d , O|E| (1)d ), S d (E ∨ ) H 0 (|E|d , O|E| (1)d )Sd
allow one to give the following geometric interpretation of the polarization isomorphism. Consider the diagonal embedding δd : |E| → |E|d . Then the total
polarization map is the inverse of the isomorphism
δ∗d : H 0 (|E|d , O|E| (1)d )Sd → H 0 (|E|, O|E| (d)).
We view a0 ∂0 + · · · + an ∂n 0 as a point a ∈ |E| with projective coordinates
[a0 , . . . , an ].
Definition 1.1.2 Let X = V( f ) be a hypersurface of degree d in |E| and x = [v]
be a point in |E|. The hypersurface
Pak (X) := V(Dvk ( f ))
of degree d − k is called the kth polar hypersurface of the point a with respect to
the hypersurface V( f ) (or of the hypersurface with respect to the point).
Example 1.1.3
Let d = 2, i.e.
f =
n
i=0
αii ti2 + 2
αi j ti t j
0≤i< j≤n
is a quadratic form on Cn+1 . For any x = [a0 , . . . , an ] ∈ Pn , P x (V( f )) = V(g),
where
n
∂f
ai
=2
ai αi j t j , α ji = αi j .
g=
∂ti
i=0
0≤i< j≤n
The linear map v → Dv ( f ) is a map from Cn+1 to (Cn+1 )∨ which can be identified
with the polar bilinear form associated to f with matrix 2(αi j ).
Let us give another definition of the polar hypersurfaces P xk (X). Choose two
different points a = [a0 , . . . , an ] and b = [b0 , . . . , bn ] in Pn and consider the line
= ab spanned by the two points as the image of the map
ϕ : P1 → Pn ,
[u0 , u1 ] → u0 a + u1 b := [a0 u0 + b0 u1 , . . . , an u0 + bn u1 ]
(a parametric equation of ). The intersection ∩ X is isomorphic to the positive
divisor on P1 defined by the degree d homogeneous form
ϕ∗ ( f ) = f (u0 a + u1 b) = f (a0 u0 + b0 u1 , . . . , an u0 + bn u1 ).
Using the Taylor formula at (0, 0), we can write
1
uk0 um
ϕ∗ ( f ) =
1 Akm (a, b),
k!m!
k+m=d
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where
Akm (a, b) =
Using the Chain Rule, we get
Akm (a, b) =
∂d ϕ∗ ( f )
(0, 0).
∂uk0 ∂um
1
k m
i
j
ai bj ∂i+j f = Dak bm ( f ).
(1.9)
|i|=k,|j|=m
Observe the symmetry
Akm (a, b) = Amk (b, a).
(1.10)
When we fix a and let b vary in Pn we obtain a hypersurface V(A(a, x)) of degree
d − k which is the kth polar hypersurface of X = V( f ) with respect to the point a.
When we fix b and vary a in Pn , we obtain the mth polar hypersurface V(A(x, b))
of X with respect to the point b.
Note that
Dak bm ( f ) = Dak (Dbm ( f )) = Dbm (a) = Dbm (Dak ( f )) = Dak ( f )(b).
(1.11)
This gives the symmetry property of polars
b ∈ Pak (X) ⇔ a ∈ Pbd−k (X).
(1.12)
Since we are in characteristic 0, if m ≤ d, Dam ( f ) cannot be zero for all a. To see
this we use the Euler formula:
df =
n
∂f
ti .
∂ti
i=0
Applying this formula to the partial derivatives, we obtain
k i i
d(d − 1) . . . (d − k + 1) f =
i t∂ f
(1.13)
|i|=k
(also called the Euler formula). It follows from this formula that, for all k ≤ d,
a ∈ Pak (X) ⇔ a ∈ X.
(1.14)
This is known as the reciprocity theorem.
Example 1.1.4 Let Md be the vector space of complex square matrices of size
d with coordinates ti j . We view the determinant function det : Md → C as an
element of S d (Md∨ ), i.e. a polynomial of degree d in the variables ti j . Let Ci j =
∂ det
∂ti j . For any point A = (ai j ) in Md the value of C i j at A is equal to the i jth cofactor
of A. Applying (1.6), for any B = (bi j ) ∈ Md , we obtain
d−1
DAd−1 B (det) = Dd−1
A (D B (det)) = DA
Thus Dd−1
A (det) is a linear function
S d−1 (Mn ) → Md∨ ,
bi jCi j = (d − 1)!
bi jCi j (A).
ti jCi j on Md . The linear map
A →
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Dd−1 (det),
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can be identified with the function A → adj(A), where adj(A) is the cofactor
matrix (classically called the adjugate matrix of A, but not the adjoint matrix as it
is often called in modern text-books).
1.1.2 First polars
Let us consider some special cases. Let X = V( f ) be a hypersurface of degree d.
Obviously, any 0th polar of X is equal to X and, by (1.12), the dth polar Pad (X) is
empty if a X and equals Pn if a ∈ X. Now take k = 1, d − 1. Using (1.6), we
obtain
n
∂f
ai ,
Da ( f ) =
∂ti
i=0
∂f
1
Dad−1 ( f ) =
(a)ti .
(d − 1)!
∂ti
i=0
n
Together with (1.12) this implies the following.
Theorem 1.1.5
For any smooth point x ∈ X, we have
P xd−1 (X) = T x (X).
If x is a singular point of X, P xd−1 (X) = Pn . Moreover, for any a ∈ Pn ,
X ∩ Pa (X) = {x ∈ X : a ∈ T x (X)}.
Here and later on we denote by T x (X) the embedded tangent space of a projective subvariety X ⊂ Pn at its point x. It is a linear subspace of Pn equal to the
projective closure of the affine Zariski tangent space T x (X) of X at x (see [275],
p. 181).
In classical terminology, the intersection X ∩ Pa (X) is called the apparent
boundary of X from the point a. If one projects X to Pn−1 from the point a, then
the apparent boundary is the ramification divisor of the projection map.
The following picture (Figure 1.1) makes an attempt to show what happens in
the case when X is a conic.
Pa (X)
UUUU
UUUU
UUUU
UUUU
UUUU a iiii
g̀a
f
b
e
c
d
iiUiUiUUUUU
X
iiii
i
i
i
iiii
iiii
i
i
i
i
Figure 1.1 Polar line of a conic.
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The set of first
polars P a (X) defines a linear system contained in the complete
linear system OPn (d − 1). The dimension of this linear system is ≤ n. We will
be freely using the language of linear systems and divisors on algebraic varieties
(see [279]).
Proposition 1.1.6 The dimension of the linear system of first polars is ≤ r if and
only if, after a linear change of variables, the polynomial f becomes a polynomial
in r + 1 variables.
Proof Let X = V( f ). It is obvious that the dimension of the linear system of first
polars is ≤ r if and only if the linear map E → S d−1 (E ∨ ), v → Dv ( f ) has kernel of
dimension ≥ n − r. Choosing an appropriate basis, we may assume that the kernel
is generated by vectors (1, 0, . . . , 0), etc. Now, it is obvious that f does not depend
on the variables t0 , . . . , tn−r−1 .
It follows from Theorem 1.1.5 that the first polar Pa (X) of a point a with respect
to a hypersurface X passes through all singular points of X. One can say more.
Proposition 1.1.7 Let a be a singular point of X of multiplicity m. For each
r ≤ deg X − m, Par (X) has a singular point at a of multiplicity m and the tangent
cone of Par (X) at a coincides with the tangent cone TCa (X) of X at a. For any
point b a, the rth polar Pbr (X) has multiplicity ≥ m − r at a and its tangent cone
at a is equal to the rth polar of TCa (X) with respect to b.
Proof Let us prove the first assertion. Without loss of generality, we may assume
that a = [1, 0, . . . , 0]. Then X = V( f ), where
f = t0d−m fm (t1 , . . . , tn ) + t0d−m−1 fm+1 (t1 , . . . , tn ) + · · · + fd (t1 , . . . , tn ).
(1.15)
The equation fm (t1 , . . . , tn ) = 0 defines the tangent cone of X at b. The equation
of Par (X) is
d−m−r
∂r f
d−m−i d−m−r−i
=
r!
fm+i (t1 , . . . , tn ) = 0.
t0
r
r
∂t0
i=0
It is clear that [1, 0, . . . , 0] is a singular point of Par (X) of multiplicity m with the
tangent cone V( fm (t1 , . . . , tn )).
Now we prove the second assertion. Without loss of generality, we may assume
that a = [1, 0, . . . , 0] and b = [0, 1, 0, . . . , 0]. Then the equation of Pbr (X) is
r
∂ r fd
∂r f
d−m ∂ fm
= 0.
r = t0
r + ··· +
∂t1
∂t1
∂t1r
The point a is a singular point of multiplicity ≥ m − r. The tangent cone of Pbr (X)
r
at the point a is equal to V( ∂∂tfrm ) and this coincides with the rth polar of TCa (X) =
1
V( fm ) with respect to b.
We leave it to the reader to see what happens if r > d − m.
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Keeping the notation from the previous proposition, consider a line through
the point a such that it intersects X at some point x a with multiplicity larger
than one. The closure ECa (X) of the union of such lines is called the enveloping
cone of X at the point a. If X is not a cone with vertex at a, the branch divisor
of the projection p : X \ {a} → Pn−1 from a is equal to the projection of the
enveloping cone. Let us find the equation of the enveloping cone.
As above, we assume that a = [1, 0, . . . , 0]. Let H be the hyperplane t0 = 0.
Write in a parametric form ua + vx for some x ∈ H. Plugging in equation (1.15),
we get
P(t) = td−m fm (x1 , . . . , xn ) + td−m−1 fm+1 (x1 , . . . , xm ) + · · · + fd (x1 , . . . , xn ) = 0,
where t = u/v.
We assume that X TCa (X), i.e. X is not a cone with a vertex at a (otherwise, by definition, ECa (X) = TCa (X)). The image of the tangent cone under
the projection p : X \ {a} → H Pn−1 is a proper closed subset of H. If
fm (x1 , . . . , xn ) 0, then a multiple root of P(t) defines a line in the enveloping cone. Let Dk (A0 , . . . , Ak ) be the discriminant of a general polynomial P =
A0 T k + · · · + Ak of degree k. Recall that
A0 Dk (A0 , . . . , Ak ) = (−1)k(k−1)/2 Res(P, P )(A0 , . . . , Ak ),
where Res(P, P ) is the resultant of P and its derivative P . It follows from the
known determinant expression of the resultant that
Dk (0, A1 , . . . , Ak ) = (−1)
k2 −k+2
2
A20 Dk−1 (A1 , . . . , Ak ).
The equation P(t) = 0 has a multiple zero with t 0 if and only if
Dd−m ( fm (x), . . . , fd (x)) = 0.
So, we see that
ECa (X) ⊂ V(Dd−m ( fm (x), . . . , fd (x))),
(1.16)
ECa (X) ∩ TCa (X) ⊂ V(Dd−m−1 ( fm+1 (x), . . . , fd (x))).
It follows from the computation of ∂∂trf in the proof of the previous Proposition that
0
the hypersurface V(Dd−m ( fm (x), . . . , fd (x))) is equal to the projection of Pa (X) ∩ X
to H.
Suppose V(Dd−m−1 ( fm+1 (x), . . . , fd (x))) and TCa (X) do not share an irreducible
component. Then
r
V(Dd−m ( fm (x), . . . , fd (x))) \ TCa (X) ∩ V(Dd−m ( fm (x), . . . , fd (x)))
= V(Dd−m ( fm (x), . . . , fd (x))) \ V(Dd−m−1 ( fm+1 (x), . . . , fd (x))) ⊂ ECa (X),
gives the opposite inclusion of (1.16), and we get
ECa (X) = V(Dd−m ( fm (x), . . . , fd (x))).
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978-1-107-01765-8 - Classical Algebraic Geometry: A Modern View
Igor V. Dolgachev
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10
Polarity
Note that the discriminant Dd−m (A0 , . . . , Ak ) is an invariant of the group SL(2) in
its natural representation on degree k binary forms. Taking the diagonal subtorus,
we immediately infer that any monomial Ai00 · · · Aikk entering in the discriminant
polynomial satisfies
k
k
is = 2
si s .
k
s=0
s=0
It is also known that the discriminant is a homogeneous polynomial of degree
2k − 2 . Thus, we get
k
k(k − 1) =
si s .
s=0
In our case k = d − m, we obtain that
deg V(Dd−m ( fm (x), . . . , fd (x))) =
d−m
(m + s)i s
s=0
= m(2d − 2m − 2) + (d − m)(d − m − 1) = (d + m)(d − m − 1).
This is the expected degree of the enveloping cone.
Example 1.1.8
Assume m = d − 2, then
D2 ( fd−2 (x), fd−1 (x), fd (x)) = fd−1 (x)2 − 4 fd−2 (x) fd (x),
D2 (0, fd−1 (x), fd (x)) = fd−2 (x) = 0.
Suppose fd−2 (x) and fd−1 are coprime. Then our assumption is satisfied, and we
obtain
ECa (X) = V( fd−1 (x)2 − 4 fd−2 (x) fd (x)).
Observe that the hypersurfaces V( fd−2 (x)) and V( fd (x)) are everywhere tangent to the enveloping cone. In particular, the quadric tangent cone TCa (X) is
everywhere tangent to the enveloping cone along the intersection of V( fd−2 (x))
with V( fd−1 (x)).
For any nonsingular quadric Q, the map x → P x (Q) defines a projective isomorphism from the projective space to the dual projective space. This is a special
case of a correlation.
According to classical terminology, a projective automorphism of Pn is called
a collineation. An isomorphism from |E| to its dual space P(E) is called a correlation. A correlation c : |E| → P(E) is given by an invertible linear map
φ : E → E ∨ defined uniquely up to proportionality. A correlation transforms
points in |E| to hyperplanes in |E|. A point x ∈ |E| is called conjugate to a point
y ∈ |E| with respect to the correlation c if y ∈ c(x). The transpose of the inverse
map t φ−1 : E ∨ → E transforms hyperplanes in |E| to points in |E|. It can be considered as a correlation between the dual spaces P(E) and |E|. It is denoted by c∨
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