Memoria de trabajo: Modelo de Baddeley

Proposition1.3.10allows one to generalize the definition of a polar s-hedron. A polar s-hedron can be viewed as a reduced closed subschemeZ of_P(E) = |E|∨_{consisting ofspoints. Obviously,}

h0(OZ) = dimH0(P(E),OZ) =s.

More generally, we may consider non-reduced closed subschemesZ ofP(E)

of dimension0 satisfying h0(OZ) = s. The set of such subschemes is pa- rameterized by a projective algebraic variety Hilbs(_P(E))called thepunctual Hilbert schemeofP(E)of0-cycles of lengths.

AnyZ∈Hilbs(_P(E))defines the subspace

IZ(d) =P(H0(P(E),IZ(d)))⊂H0(P(E),OP(E)(d)) =S d_(E).

The exact sequence

0→H0(_P(E),IZ(d))→H0(P(E),OP(E)(d))→H0(P(E),OZ) (1.56) →H1(P(E),IZ(d))→0

shows that the dimension of the subspace

hZid=P(H0(P(E),IZ(d))⊥)⊂ |Sd(E∨)| (1.57) is equal toh0(OZ)−h1(IZ(d))−1 =s−1−h1(IZ(d)).IfZ is reduced and consists of pointsp1, . . . , ps, thenhZid = hvd(p1), . . . , vd(ps)i, where

vd:P(E)→P(Sd(E))is the Veronese map. Hence,dimhZid =s−1if the pointsvd(p1), . . . , vd(ps)are linearly independent. We say thatZ islinearly

d-independentifdimhZid=s−1.

Definition 1.3.11 A generalizeds-hedron off ∈ Sd_(E∨_{)is a linearlyd-}

independent subschemeZ∈Hilbs(_P(E))which is apolar tof. Recall thatZis apolar tofif, for eachk≥0,

IZ(k) =H0(P(E),IZ(k))⊂APk(f). (1.58) According to this definition, a polar s-hedron is a reduced generalized s-hedron. The following is a generalization of Proposition1.3.9.

Proposition 1.3.12 A linearlyd-independent subschemeZ ∈ Hilbs(P(E))

is a generalized polars-hedron off ∈Sd_(E∨_{)if and only if} IZ(d)⊂APd(f).

Proof We have to show that the inclusion in the assertion impliesIZ(d) ⊂ APk(f)for anyk≤d. For anyψ0∈Sd−k(E)and anyψ∈IZ(k), the product

ψψ0 belongs toIZ(k). ThusDψψ0(f) = 0. By the duality,Dψ(f) = 0, i.e.

ψ∈APk(f).

Example1.3.13 LetZ∈Hilbs(P(E))be the union ofkfat pointspk, i.e. at eachpi ∈Zthe idealIZ,piis equal to themi-th power of the maximal ideal.

Obviously, s= k X i=1 n+mi−1 mi−1 .

Then the linear system |IZ(d)| consists of hypersurfaces of degree d with pointspi of multiplicity≥ mi. One can show (see [314], Theorem 5.3) that

Zis apolar tof if and only if

f =ld−m1+1

1 g1+· · ·+l_kd−mk+1gk,

wherepi = V(li)andgiis a homogeneous polynomial of degreemi−1or the zero polynomial.

Remark 1.3.14 It is not known whether the set of generalizeds-hedra of f

is a closed subset of Hilbs(_P(E)). It is known to be true fors≤d+ 1since in this casedimIZ(d) = t:= dimSd(E)−sfor allZ ∈Hilbs(P(E))(see

[314], p.48). This defines a regular map of Hilbs(P(E))to the Grassmannian

Gt−1(|Sd(E)|)and the set of generalizeds-hedra equal to the preimage of a closed subset consisting of subspaces contained in APd(f). Also we see that

h1_(I

Z(d)) = 0, henceZis always linearlyd-independent.

1.3.4 Secant varieties and sums of powers

Consider theVeronese mapof degreed

νd:|E| → |Sd(E)|, [v]7→[vd],

defined by the complete linear system|Sd_E∨_{|. The image of this map is the}

Veronese varietyV_dnof dimensionnand degreedn. It is isomorphic toPn. By

choosing a monomial basisti_{in the linear space of homogeneous polynomials}

of degreedwe obtain that the Veronese variety is isomorphic to the subvariety ofP n+d d −1 given by equations Ai·Aj−AkAm = 0, i+j=k+m,

whereAiare dual coordinates in the space of polynomials of degree d. The

linear system of hypersurfaces of degreedis called an-dimensional Veronese variety of degreedn_.

One can combine the Veronese mapping and the Segre mapping to define aSegre-Veronese varietyVn1,...,nk(d1, . . . , dk). It is equal to the image of the

mapPn1_{× · · · ×}

Pnk_{defined by the complete linear system|O}

Pn1(d1)· · ·

O_Pnk(dk)|.

The notion of a polars-hedron acquires a simple geometric interpretation in terms of the secant varieties of the Veronese varietyVn

d. If a set of points [l1], . . . ,[ls] in|E|is a polar s-hedron of f, then [f] ∈ h[ld1], . . . ,[lds]i, and hence[f]belongs to the(s−1)-secant subspace ofVn

d. Conversely, a general point in this subspace admits a polar s-hedron. Recall that for any irreducible nondegenerate projective varietyX ⊂PN of dimensionritst-secant variety

Sect(X)is defined to be the Zariski closure of the set of points inPN which

lie in the linear span of dimensiontof some set oft+ 1linearly independent points inX.

The counting constants easily gives

dimSect(X)≤min(r(t+ 1) +t, N).

The subvarietyX ⊂ PN is calledt-defective if the inequality is strict. An

example of a1-defective variety is a Veronese surface inP5.

A fundamental result about secant varieties is the following Lemma whose modern proof can be found in [608], Chapter II, and in [151]

Lemma 1.3.15(A. Terracini) Letp1, . . . , pt+1be generalt+ 1points inX andpbe a general point in their span. Then

Tp(Sect(X)) =Tp1(X), . . . ,Tpt+1(X).

The inclusion part

Tp1(X), . . . ,Tpt+1(X)⊂Tp(Sect(X))

is easy to prove. We assume for simplicity thatt= 1. Then Sec1(X)contains the coneC(p1, X)which is swept out by the lines p1q, q ∈ X. Therefore,

Tp(C(p1, X))⊂Tp(Sec1(X)). However, it is easy to see thatTp(C(p1, X)) contains_Tp1(X).

Corollary 1.3.16 Sect(X)6=PN if and only if, for anyt+ 1general points

ofX, there exists a hyperplane section ofX singular at these points. In par- ticular, ifN ≤r(t+ 1) +t, the varietyXist-defective if and only if for any

t+ 1general points ofX there exists a hyperplane section ofX singular at these points.

Example1.3.17 LetX = Vn_d ⊂ P

d+n n

−1

be a Veronese variety. Assume

n(t+ 1) +t > d+n_n

−1. A hyperplane section ofX is isomorphic to a hypersurface of degreedinPn. Thus Sect(Vdn)6=|S

d_(E∨_{)|if and only if, for}

anyt+ 1general points inPn, there exists a hypersurface of degreedsingular

at these points.

Consider a Veronese curveV1

d⊂Pd. Assume2t+ 1≥d. Sinced <2t+ 2, there are no homogeneous forms of degreedwhich havet+ 1multiple roots. Thus the Veronese curve Rd = vd(P1) ⊂ Pd is nott-degenerate fort ≥

(d−1)/2.

Taken= 2andd= 2. For any two points in_P2_{there exists a conic singular} at these points, namely the double line through the points. This explains why a Veronese surfaceV2

2 is1-defective. Another example is V2

4 ⊂ P14 and t = 4. The expected dimension of Sec4(X)is equal to14. For any 5 points inP2 there exists a conic passing

through these points. Taking it with multiplicity 2, we obtain a quartic which is singular at these points. This shows thatV2

4is4-defective.

The following Corollary of Terracini’s Lemma is called theFirst Main Theo- rem on apolarityin [207]. The authors gave an algebraic proof of this Theorem without using Terracini’s Lemma.

Corollary 1.3.18 A general homogeneous form inSd(E∨)admits a polar

s-hedron if and only if there exist linear forms l1, . . . , ls ∈ E∨ such that, for any nonzero ψ ∈ Sd(E), the idealAP(ψ) ⊂ S(E∨)does not contain {ld₁−1, . . . , ld−1

s }.

Proof A general form inSd_(E∨_{)admits a polar s-hedron if and only if the}

secant variety Secs−1(Vnd)is equal to the whole space. This means that the span of the tangent spaces at some pointsqi=V(lid), i= 1, . . . , s,is equal to the whole space. By Terracini’s Lemma, this is equivalent to that the tangent spaces of the Veronese variety at the pointsqiare not contained in a hyperplane defined by someψ ∈ Sd_{(E) = S}d_(E∨₎∨_{. It remains for us to use that the}

tangent space of the Veronese variety atqiis equal to the projective space of all homogeneous formsl_id−1l, l∈E∨\ {0}(see Exercise 1.18). Thus, for any nonzeroψ ∈ Sd_{(E), it is impossible thatP}

ld−1 i l

(ψ) = 0for allland for all

i. But P_ld−1 i l

(ψ) = 0for all lif and only if P_ld−1 i

(ψ) = 0. This proves the assertion.

The following fundamental result is due to J. Alexander and A. Hirschowitz [5]. A simplified proof can be found in [53] or [91].

Theorem 1.3.19 Ifd >2, the Veronese varietyVn_d ist-defective if and only if

(n, d, t) = (2,4,4),(3,4,8),(4,3,6),(4,4,13).

In all these cases the secant varietySect(Vn_d)is a hypersurface. The Veronese varietyVn

2 ist-defective only if1≤t≤n. Itst-secant variety is of dimension

n(t+ 1)−1

2(t−2)(t+ 1)−1.

For the sufficiency of the condition, only the case(4,3,6)is not trivial. It asserts that for7general points in_P3_{there exists a cubic hypersurface which} is singular at these points. To see this, we use a well-known fact that anyn+ 3 general points inPn lie on a Veronese curve of degreen(see, for example,

[279], Theorem 1.18). So, we find such a curveRthrough 7 general points in

P4and consider the 1-secant variety Sec1(R). It is a cubic hypersurface given

by the catalecticant invariant of a binary quartic form. It contains the curveR

as it singular locus.

Other cases are easy. We have seen already the first two cases. The third case follows from the existence of a quadric through nine general points inP3. The

square of its equation defines a quartic with 9 points. The last case is similar. For any 14 general points there exists a quadric inP4containing these points.

In the case of quadrics we use that the variety of quadrics of corankris of codimensionr(r+ 1)/2in the variety of all quadrics.

Obviously, ifdimSecs−1(Vdn) < dim|SdE∨)| = n+d

n

−1, a general formf ∈ Sd_(E∨_{)cannot be written as a sum ofspowers of linear forms.}

SincedimSecs−1(Vnd)≤min{(n+ 1)s−1, n+d

n

−1}, the minimal number

s(n, d)of powers needed to writefas a sum of powers of linear forms satisfies

s(n, d)≥l 1 n+ 1 _n+d n _m . (1.59) If Vn

d is not (s−1)-defective, then the equality holds. Applying Theorem 1.3.19, we obtain the following.

Corollary 1.3.20 s(n, d) =l 1 n+ 1 _n+d n _m

unless(n, d) = (n,2),(2,4),(3,4),(4,3),(4,4). In these exceptional cases

s(n, d) =n+ 1,6,10,8,15instead of expecteddn−1

2 e,5,9,8,14.

Remark1.3.21 Ifd > 2, in all the exceptional cases listed in the previous corollary,s(n, d)is larger by one than the expected number. The variety of forms of degreedthat can be written as the sum of the expected number of

powers of linear forms is a hypersurface in|O_Pn(d)|. In the case(n, d, t) =

(2,4,5), the hypersurface is of degree 6 and is given by the catalecticant matrix which we will discuss later in this chapter. The curves parameterized by this hypersurface are Clebsch quartics which we will discuss in Chapter 6. The case(n, d) = (4,3)was studied only recently in [422]. The hypersurface is of degree 15. In the other two cases, the equation expresses that the second partials of the quartic are linearly dependent (see [241], pp. 58-59.)

One can also consider the problem of a representation of several forms

f1, . . . , fk ∈Sd(E∨)as a sum of powers of the same set (up to proportional- ity) of linear formsl1, . . . , ls. This means that the forms share a common polar s-hedron. For example, a well-known result from linear algebra states that two general quadratic formsq1, q2inkvariables can be simultaneously diagonal- ized. In our terminology this means that they have a common polar k-hedron. More precisely, this is possible if thedet(q1+λq2)hasn+ 1distinct roots (we will discuss this later in Chapter 8 while studying del Pezzo surfaces of degree 4). Suppose fj = s X i=1 a(j)_i ld_i, j= 1, . . . , k. (1.60) We view this as an elementφ∈U∨_⊗Sd_(E∨_{), whereU =}

Ck. The mapφ

is the sum ofslinear mapsφof rank 1 with the images spanned byld i. So, we can view eachφas a vector inU∨⊗Sd_(E∨_{)equal to the image of a vector in} U∨⊗E∨embedded inU∨⊗E∨byu⊗l7→u⊗ld. Now, everything becomes clear. We consider the Segre-Veronese embedding

|U∨| × |E∨|,→ |U∨| × |Sd(E∨)|,→ |U∨⊗Sd(E∨)|

defined by the linear system of divisors of type(1, d)and view[φ]as a point in the projective space|U∨⊗Sd_(E∨_{)|which lies on the(s−1)-secant variety}

ofVk−1,n(1, d).

For any linear mapφ∈Hom(U, Sd_(E∨_{)), consider the linear map}

Tφ:Hom(U, E)→Hom( 2 ^ U, Sd−1(E∨)), defined by Tφ(α) :u∧v7→Dα(u)(φ(v))−Dα(v)(φ(u)).

We call this map theToeplitz map. Suppose thatφis of rank 1 with the image spanned byld_{, thenT}

φis of rank equal todimV2U−1 = (k−2)(k+ 1)/2. If we choose a basisu1, . . . , ukinUand coordinatest0, . . . , tninE, then the

image is spanned byld−1(aiui−ajuj), wherel=Paiti. This shows that, if

φbelongs to Secs−1(|U∨| × |E∨|),

rankTφ≤s(k−2)(k+ 1)/2. (1.61) The expected dimension of Secs−1(|U∨| × |E∨|)is equal tos(k+n)−1.

Thus, we expect that Secs−1(|U∨| × |E∨|)coincides with |U∨⊗Sd(E∨)| when s≥l k k+n _n+d n _m . (1.62)

If this happens, we obtain that a general set ofkforms admits a common polar s-hedron. Of course, as in the casek = 1, there could be exceptions if the secant variety is(s−1)-defective.

Example1.3.22 Assume d = 2andk = 3. In this case the matrix ofTφ is a square matrix of size3(n+ 1). Let us identify the spacesU∨andV2_U

by means of the volume formu1∧u2∧u3 ∈ V 3

U ∼= C. Also identify

φ(ui) ∈ S2_(E∨_{) with a square symmetric matrixA}

i of size n+ 1. Then, an easy computation shows that one can represent the linear mapTφ by the skew-symmetric matrix   0 A1 A2 −A1 0 A3 −A2 −A3 0  . (1.63)

Now condition (1.61) for

s=lk n+d n k+n m =l3(n+ 2)(n+ 1) 2(n+ 3) m =        1 2(3n+ 2) ifnis even, 1 2(3n+ 1) ifnis odd≥3, 3 ifn= 1

becomes equivalent to the condition

Λ=Pf   0 A1 A2 −A1 0 A3 −A2 −A3 0  = 0. (1.64)

It is known that the secant variety Secs−1(|U| × |E|)of the Segre-Veronese variety is a hypersurface ifn≥3is odd and the whole space ifnis even (see [549], Lemma 4.4). It implies that, in the odd case, the hypersurface is equal toV(Λ). Its degree is equal to 3(n+ 1)/2. Of course, in the even case, the pfaffian vanishes identically.

is an invariant of the net2 of quadrics in P3 that vanishes on the nets with

common polar pentahedron. Following [248], we callΛtheToeplitz invariant. Let us write its generatorsf1, f2, f3 in the form (1.60) withn = 3ands =

1

2(3n+ 1) = 5. Since the four linear formsliare linearly dependent, we can normalize them by assuming thatl1+· · ·+l5= 0and assume thatl1, . . . , l5 span a 4-dimensional subspace. Consider a cubic form

F =1 3 5 X i=1 l3i,

and find three vectorsviinC4such that

(l1(vj), . . . , l5(vj)) = (a(j)₁ , . . . , a(j)₅ ), j = 1,2,3.

Now we check that fj = Dvj(F) for j = 1,2,3. This shows that the net

spanned byf1, f2, f3is a net of polar quadrics of the cubicF. Conversely, we will see later that any general cubic form in 4 variables admits a polar pentahe- dron. Thus any net of polars of a general cubic surface admits a common polar pentahedron. So, the Toeplitz invariant vanishes on a general net of quadrics in

P3if and only if the net is realized as a net of polar quadrics of a cubic.

Remark1.3.23 Let(n, d, k, s)denote the numbers such that we have the strict inequality in (1.62). We call such4-tuples exceptional. Examples of excep- tional 4-tuples are(n,2,3,1₂(3n+ 1))with oddn≥2. The secant hypersur- faces in these cases are given by the Toeplitz invariantΛ. The case(3,2,3,5) was first discovered by G. Darboux [154].3 _{It has been rediscovered and ex-}

tended to any oddn by G. Ottaviani [421]. There are other two known ex- amples. The case(2,3,2,5)was discovered by F. London [367]. The secant variety is a hypersurface given by the determinant of order6of the linear map Tφ(see Exercise 1.30). The examples(3,2,5,6)and(5,2,3,8)were discov- ered recently by E. Carlini and J. Chipalkatti [64]. The secant hypersurface in the second case is a hypersurface of degree 18 given by the determinant ofTφ. There are no exceptional 4-tuples(n,2,2, s)[64] and no exceptional 4-tuples (n, d, k, s)for largen(with some explicit bound)[1]. We refer to [96], where the varieties of common polar s-hedra are studied.

Remark 1.3.24 Assume that one of the matricesA1, A2, A3in (1.63) is in- 2 _{We employ classical terminology calling a 1-dimensional (resp. 2-dimensional, resp.}

3-dimensional) linear system apencil(resp. anet, resp. aweb).

3 _{Darboux also wrongly claimed that the case(3,2,4,6)is exceptional, the mistake was} pointed out by Terracini [560] without proof, a proof is in [64].

vertible, say let it beA2. Then   I 0 0 0 I −A1A−21 0 0 I     0 A1 A2 −A1 0 A3 −A2 −A3 0     I 0 0 0 I 0 0 −A−₂1A1 I   =   0 0 A2 0 B A3 −A2 −A3 0  , where B =A1A−21A3−A3A−21A1. This shows that

rank   0 A1 A2 −A1 0 A3 −A2 −A3 0  =rankB+ 2n+ 2.

The condition that rankB ≤2is known in the theory of vector bundles over the projective plane asBarth’s conditionon the net of quadrics inPn. It does

not depend on the choice of a basis of the net of quadrics spanned by the quadrics with matricesA1, A2, A3. Under Barth’s condition, the discriminant curvedet(z0A1+z1A2+2A3) = 0of singular quadrics in the net is aDarboux curveof degreen+ 1(see [24]).

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