1.5.1 Binary forms
LetU be a 2-dimensional linear space andf ∈Sd(U∨)\ {0}. The hypersur-
faceX = V(f)can be identified with a positive divisor div(f) = Pm
ixi of degreedon|U| ∼= P1. SinceV2U ∼= C, we have a natural isomorphism
U →U∨of linear representations of SL(U). It defines a natural isomorphism between the projective line|U| and its dual projective lineP(U). In coordi-
nates, a pointa= [a0, a1]is mapped to the hyperplaneV(a1t0−a0t1)whose zero set is equal to the pointa. IfX is reduced (i.e.f has no multiple roots), then, under the identification of|U|andP(U),Xcoincides with its dualX∨.
In general,X∨ consists of simple roots off. Note that this is consistent with the Pl¨uckeri-Teissier formula. The degrees of the Hessian and the Steinerian coincide, although they are different ifd >3. Assume thatXis reduced. The partial derivatives off define the polar mapg : |U| → |U|of degreed−1. The ramification divisor He(X)consists of 2d−4 points and it is mapped bijectively onto the branch divisor St(X).
Example 1.5.1 We leave the cased = 2to the reader. Consider the case
d= 3. In coordinates
f =a0t30+ 3a1t20t1+ 3a2t0t21+a3t31. All invariants are powers of thediscriminant invariant
∆ =a20a23+ 4a0a32+ 4a 3
1a3−6a0a1a2a3−3a21a 2
2. (1.81)
whose symbolic expression is(12)2(13)(24)(34)2 (see [581], p. 244). The Hessian covariant H= (a0a2−a21)t 2 0+ (a0a3−a1a2)t0t1+ (a1a3−a22)t 2 1. Its symbolic expression is(ab)axby. There is also a cubic covariant
J =J(f, H) = det t30 3t20t1 3t0t21 t31 a2 −2a1 a0 0 a3 −a2 −a1 a0 0 −a3 −2a2 a1
with symbolic expression(ab)2(ac)2bxc2x. The covariantsf, HandJ form a complete system of covariants, i.e. generate the module of covariants over the algebra of invariants.
Example1.5.2 Consider the cased= 4. In coordinates,
f =a0t40+ 4a1t30t1+ 6a2t20t 2
1+ 4a3t0t31+a4t41.
There are two basic invariantsS andT on the space of quartic binary forms. Their symbolic expression areS = (12)4andT = (12)2(13)2(23)2. Explic- itly,
S=a0a4−4a1a3+ 3a22, (1.82)
T =a0a2a4+ 2a1a2a3−a0a23−a 2
1a4−a32.
Note thatT coincides with the determinant of the catalecticant matrix off. Each invariant is a polynomial inSandT. For example, the discriminant in- variant is equal to
∆ =S3−27T2.
The Hessian He(X) =V(H)and the SteinerianS(X) = V(K)are both of degree 4. We have H= (a0a2−a21)t 4 0+ 2(a0a3−a1a2)t30t1+ (a0a4+ 2a1a3−3a22)t 2 0t 2 1 +2(a1a4−a2a3)t0t31+ (a2a4−a23)t 4 1.
and
K= ∆((a0t0+a1t1)x3+3(a1t0+a2t1)x2y+3(a2t0+a3t1)xy2+(a3t0+a4t1)y3).
Observe that the coefficients of H (resp. K) are of degree 2 (resp. 4) in coefficients of f. There is also a covariant J = J(f, H) of degree 6 and the module of covariants is generated byf, H, J overC[S, T]. In particular,
K=αT f+βSH,for some constantsαandβ. By takingf in the form
f =t40+ 6mt20t21+t41, (1.83) and comparing the coefficients we find
2K=−3T f+ 2SH. (1.84)
Under identification|U|=P(U), a generalizedk-hedronZoff ∈Sd(U∨)
is the zero divisor of a formg∈Sk(U)which is apolar tof. Since
H1(|E|,IZ(d))∼=H1(P1,OP1(d−k)) = 0, k≥d+ 1,
anyZis automatically linearly independent. Identifying a point[g]∈ |Sk(U)| with the zero divisor div(g), we obtain
Theorem 1.5.3 Assumen= 1. Then
VSP(f;k) =|APk(f)|.
Note that the kernel of the map
Sk(U)→Sd−k(U∨), ψ7→Dψ(f)
is of dimension≥dimSk(U)−dimSd−k(U∨) =k+1−(d−k+1) = 2k−d. ThusDψ(f) = 0for some nonzeroψ∈Sk(U), whenever2k > d. This shows thatfhas always generalized polark-hedron fork > d/2. Ifdis even, a binary form has an apolard/2-form if and only ifdetCatd/2(f) = 0. This is a divisor in the space of all binaryd-forms.
Example1.5.4 Taked= 3. Assume thatf admits a polar 2-hedron . Then
f = (a1t0+b1t1)3+ (a2t0+b2t1)3.
It is clear thatf has 3 distinct roots. Thus, iff = (a1t0+b1t1)2(a2t0+b2t1) has a double root, it does not admit a polar2-hedron. However, it admits a generalized2-hedron defined by the divisor2p, wherep = (b1,−a1). In the secant variety interpretation, we know that any point in|S3(E∨)|either lies
on a unique secant or on a unique tangent line of the rational cubic curve. The space AP2(f) is always one-dimensional. It is generated either by a binary quadric(−b1ξ0+a1ξ1)(−b2ξ0+a2ξ1), or by(−b1ξ0+a1ξ1)2.
Thus VSP(f,2)oconsists of one point or empty but VSP(f,2)always con- sists of one point. This example shows that VSP(f,2)6= VSP(f,2)oin gen- eral.
1.5.2 Quadrics
It follows from Example1.3.17that Sect(Vn2)6=|S2(E∨)|if and only if there exists a quadric witht+1singular points in general position. Since the singular locus of a quadricV(q)is a linear subspace of dimension equal to corank(q)− 1, we obtain that Secn(Vn
2) =|S2(E∨)|, hence any general quadratic form can be written as a sum ofn+1squares of linear formsl0, . . . , ln. Of course, linear algebra gives more. Any quadratic form of rankn+ 1can be reduced to sum of squares of the coordinate functions. Assume thatq=t20+· · ·+t2n. Suppose we also haveq=l2
0+· · ·+ln2. Then the linear transformationti 7→lipreserves
qand hence is an orthogonal transformation. Since polar polyhedra ofqand
λqare the same, we see that the projective orthogonal group PO(n+ 1)acts transitively on the set VSP(f, n+1)oof polar(n+1)-hedra ofq. The stabilizer groupGof the coordinate polar polyhedron is generated by permutations of coordinates and diagonal orthogonal matrices. It is isomorphic to the semi- direct product2noSn+1(the Weyl group of root systems of typesBn, Dn), where we use the notation2n for the 2-elementary abelian group (
Z/2Z)n.
Thus we obtain
Theorem 1.5.5 Letqbe a quadratic form inn+ 1variables of rankn+ 1. Then
VSP(q, n+ 1)o∼=PO(n+ 1)/2noSn+1. The dimension ofVSP(q, n+ 1)ois equal to 1
2n(n+ 1).
Example1.5.6 Taken = 1. Using the Veronese mapν2 : P1 → P2, we
consider a nonsingular quadricQ=V(q)as a pointpinP2not lying on the conicC =V(t0t2−t21). A polar 2-gon ofqis a pair of distinct pointsp1, p2 onC such thatp ∈ hp1, p2i. The set of polar 2-gons can be identified with the pencil of lines throughpwith the two tangent lines toC deleted. Thus
W(q,2)o = P1\ {0,∞} = C∗. There are two generalized 2-gons2p0and 2p∞defined by the tangent lines. Each of them gives the representation ofqas l1l2, whereV(li)are the tangents. We have VSP(f,2) =VSP(f,2)
o ∼ =P1.
Letq ∈ S2(E∨)be a nondegenerate quadratic form. We have an injective map (1.77) VSP(q, n+ 1)o→G(n,H2q(E))∼=G(n, n+2 2 −1). (1.85)
Its image is contained in the subvarietyG(n,H2
q(E))σof subspaces isotropic with respect to the Mukai skew forms.
Recall that the Grassmann varietyG(m, W)of linearm-dimensional sub- spaces of a linear spaceW of dimensionN carries the natural ranknvector bundleS, theuniversal subbundle. Its fiber over a pointL∈G(m, W)is equal toL. It is a subbundle of the trivial bundleWG(m,W)associated to the vector spaceW. We have a natural exact sequence
0→ S →WG(m,W)→ Q →0,
whereQis theuniversal quotient bundle, whose fiber overLis equal toW/L. By restriction, we can view the Mukai formσq : V
2
E → V2
H2
q(E∨)as a section of the vector bundleV2
S∨⊗V2
E∨. The image of VSP(q, n+ 1)is contained in the zero locus of a section of this bundle defined byσq. Since the rank of the vector bundle is equal to n2 n+12 , we expect that the dimension of its zero locus is equal to
dimG(n, n+22 −1)− n 2 n+ 1 2 =n( n+22 −1−n)− n 2 n+ 1 2 .
Unfortunately, this number is ≤ 0 forn > 2, so the expected dimension is wrong. However, whenn= 2, we obtain that the expected dimension is equal to3 = dimVSP(q,3). We can viewσω,qas a hyperplane in the Pl¨ucker embed- ding ofG(2,H2
q(E))∼= G(2,5). So, VSP(q,3)embeds into the intersection of 3 hyperplane sections ofG(2,5).
Theorem 1.5.7 Letqbe a nondegenerate quadratic form on a 3-dimensional vector space E. Then the image ofVSP(q,3)inG(2,H2
q(E)), embedded in the Pl¨ucker space, is a smooth irreducible 3-fold equal to the intersection of
G(2,H2
q(E))with a linear space of codimension 3. Proof We havedimH2
q(E) = 5, soG(2,H2q(E))∼=G(2,5)is of dimension 6. Hyperplanes in the Pl¨ucker space are elements of the space|V2
H2 q(E)∨|. Note that the functionssq,ωare linearly independent. In fact, a basisξ0, ξ1, ξ2 inE gives a basisω01 = ξ0∧ξ1, ω02 = ξ0∧ξ2, ω12 = ξ1∧ξ2inV
2
E. Thus the space of sectionssq,ω is spanned by 3 sectionss01, s02, s12 corre- sponding to the forms ωij. Without loss of generality, we may assume that
q = t20+t21 +t22. If we take a = t0t1 +t22, b = −t 2 0+t 2 1+t 2 2, we see that s01(a, b) 6= 0, s12(a, b) = 0, s02(a, b) = 0. Thus a linear dependence between the functionssij implies the linear dependence between two func- tions. It is easy to see that no two functions are proportional. So our 3 func- tions sij,0 ≤ i < j ≤ 2 span a 3-dimensional subspace of V
2 H2
and hence define a codimension 3 projective subspaceLin the Pl¨ucker space |V2H2
q(E)|. The image of VSP(q,3)under the map (1.85) is contained in the intersectionG(2, E)∩L. This is a 3-dimensional subvariety ofG(2,H2
q(E)), and hence containsµ(VSP(q,3))as an irreducible component. We skip an ar- gument, based on counting constants, which proves that the subspaceL be- longs to an open Zariski subset of codimension 3 subspaces ofV2
H2 q(E) for which the intersectionL∩G(2,H2
q(E))is smooth and irreducible (see [184]).
It follows from the adjunction formula and the known degree ofG(2,5)that the closure of VSP(q,3)oinG(2,H2
q(E))is a smooth Fano variety of degree 5. We will discuss it again in the next chapter.
Remark1.5.8 One can also consider the varieties VSP(q, s)fors > n+ 1. For example, we have
t20−t22= 12(t0+t1)2+12(t0−t1)2−12(t1+t2)2−12(t21−t2) 2,
t20+t21+t22= (t0+t2)2+ (t0+t1)2+ (t1+t2)2−(t0+t1+t2)2. This shows that VSP(q, n+ 2),VSP(q, n+ 3)are not empty for any nonde- generate quadricQinPn, n≥2.