Unstable hyperplanes for Steiner bundles and multidimensional matrices

(de Gruyter 2001

Unstable hyperplanes for Steiner bundles and multidimensional matrices

Vincenzo Ancona and Giorgio Ottaviani*

(Communicated by G. Gentili)

Abstract. We study some properties of the natural action of SL…V0† SL…Vp†on non- degenerate multidimensional complex matricesAAP…V0n nVp†of boundary format (in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize the non-stable ones as the matrices which are in the orbit of a ``triangular'' matrix, and the matrices with a stabilizer containingCas those which are in the orbit of a ``diagonal'' matrix. For pˆ2 it turns out that a non-degenerate matrixAAP…V0nV1nV2†detects a Steiner bundleSA(in the sense of Dolgachev and Kapranov) on the projective space Pⁿ, nˆdim…V2† ÿ1. As a consequence we prove that the symmetry group of a Steiner bundle is contained in SL…2†and that the SL…2†-invariant Steiner bundles are exactly the bundles introduced by Schwarzen- berger [Schw], which correspond to ``identity'' matrices. We can characterize the points of the moduli space of Steiner bundles which are stable for the action of Aut…Pⁿ†, answering in the

®rst nontrivial case a question posed by Simpson. In the opposite direction we obtain some results about Steiner bundles which imply properties of matrices. For example the number of unstable hyperplanes ofSA(counting multiplicities) produces an interesting discrete invariant ofA, which can take the values 0;1;2;. . .;dimV0‡1 ory; they case occurs if and only ifSAis Schwarzenberger (andAis an identity). Finally, the Gale transform for Steiner bun- dles introduced by Dolgachev and Kapranov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices.

1 Introduction

A multidimensional matrix of boundary format is an element AAV0n nVp

whereViis a complex vector space of dimensionki‡1 foriˆ0;. . .;pand

k0 ˆX^p

iˆ1

ki:

* Both authors were supported by MURST and GNSAGA-INDAM

We denote by DetA the hyperdeterminant of A(see [GKZ]). Lete₀^j†;. . .;e_kj^j† be a basis inVjso that everyAAV0n nVphas a coordinate form

AˆX

ai0;...;ipe^0†_i0 n ne^p†_ip :

Let x₀^j†;. . .;x_kj^j† be the coordinates in V_j. Then A has the following di¨erent

descriptions:

1) A multilinear form

X

…i0;...;ip†

ai0;...;ipx^0†_i0 n nx_ip^p†:

2) An ordinary matrix M_Aˆ …m_i1i0† of size …k₁‡1† …k₀‡1† whose entries are multilinear forms

m_i1i0ˆ X

…i2;...;ip†

a_i0;...;ipx^0†_i2 n nx^p†_ip : …1:1†

3) A sheaf morphism f_Aon the productXˆP^k2 P^kp:

O_X^k0‡1!^fA O_X…1;. . .;1†^k1‡1: …1:2†

Theorem 3.1 of chapter 14 of [GKZ] easily translates into:

Theorem.The following properties are equivalent:

i) DetA00;

ii) the matrix M_Ahas constant rank k₁‡1on X ˆP^k2 P^kp;

iii) the morphism f_A is surjective so that S_A ˆkerf_A is a vector bundle of rank k₀ÿk₁.

The above remarks set up a basic link between non-degenerate multidimensional matrices of boundary format and vector bundles on a product of projective spaces. In the particular casepˆ2 the (dual) vector bundleSAlives on the projective spacePⁿ, nˆk2, and is a Steiner bundle as de®ned by Dolgachev and Kapranov in [DK]. We can keep forSAthe name Steiner also in the case pX3.

The action of SL…V0† SL…Vp† on V0n nVp translates to an action on the corresponding bundle in two steps: ®rst the action of SL…V0† SL…V1†leaves the bundle in the same isomorphism class; then SL…V2† SL…Vp† acts on the classes, i.e. on the moduli space of Steiner bundles. It follows that the invariants of matrices for the action of SL…V₀† SL…V_p†coincide with the invariants of the action of SL…V₂† SL…V_p†on the moduli space of the corresponding bundles.

Moreover the stable points of both actions correspond to each other.

The aim of this paper is to investigate the properties and the invariants of both the above actions. When we look at the vector bundles, we restrict ourselves to the case pˆ2, that is Steiner bundles on projective spaces. This is probably the ®rst case where Simpson's question ([Simp], p. 11) about the natural SL…n‡1†-action on the moduli spaces of bundles onPⁿ has been investigated.

Section 2 is devoted to the study of multidimensional matrices. We denote by the same letter matrices inV₀n nV_p and their projections inP…V₀n nV_p†. In Theorem 2.4 we prove that a matrixAAP…V₀n nV_p†of boundary format with DetA00 is not stable for the action of SL…V₀† SL…V_p†if and only if there is a coordinate system such that a_i0...ip ˆ0 for i₀>P_p

tˆ1i_t. A matrix satisfying this condition is called triangulable. The other main results of this section are Theorems 2.5 and 2.6 which describe the behaviour of the stabilizer subgroup Stab…A†. In Remark 5.14 we introduce a discrete SL…V0† SL…V1† SL…V2†-invariant of non- degenerate matrices in P…V0nV1nV2† and we show that it can assume only the values 0;. . .;k0‡2;y.

The second part of the paper, consisting of Sections 3 to 6, can be read independently of Section 2, except that we will use Theorem 2.4 in two crucial points (Theorem 5.9 and Section 6). In this part we study the Steiner bundles on Pⁿˆ P…V†. As we mentioned above, they are rank-n vector bundles S whose dual S appears in an exact sequence

0!S!WnO!^fA InO…1† !0 …1:3†

whereWandIare complex vector spaces of dimensionn‡kandk, respectively. The map f_A corresponds to AAWnVnI (which is of boundary format) and f_A is surjective if and only if DetA00. We denote by Sn;k the family of Steiner bundles described by a sequence as (1.3).Sn;1 contains only the quotient bundle. Important examples of Steiner bundles are the Schwarzenberger bundles, whose construction goes back to the pioneering work of Schwarzenberger [Schw]. Other examples are the logarithmic bundles W…logH† of meromorphic forms on Pⁿ having at most loga- rithmic poles on a ®nite union H of hyperplanes with normal crossing; Dolgachev and Kapranov showed in [DK] that they are Steiner. The Schwarzenberger bundles are a special case of logarithmic bundles, when all the hyperplanes osculate the same rational normal curve. Dolgachev and Kapranov proved a Torelli type theorem, namely that the logarithmic bundles are uniquely determined up to isomorphism by the above union of hyperplanes, with a weak additional assumption. This assumption was recently removed by ValleÁs [V2], who shares with us the idea of looking at the schemeW…S† ˆ fHAPⁿ⁴jh⁰…S_H†00gHPⁿ⁴of unstable hyperplanes of a Steiner bundle S. ValleÁs proves that any SASn;k with at least n‡k‡2 unstable hyper- planes with normal crossing is a Schwarzenberger bundle and W…S† is a rational normal curve. We strengthen this result by showing the following: for anySASn;k

any subset of closed points inW…S†has always normal crossing (see Theorem 3.10).

Moreover SASn;k is logarithmic if and only if W…S† contains at least n‡k‡1 closed points (Corollaries 5.11 and 5.10). In particular ifW…S†contains exactlyn‡ k‡1 closed points thenSFW…logW…S††. The Torelli Theorem follows.

It turns out that the length ofW…S†de®nes an interesting ®ltration into irreducible subschemes of Sn;k which gives also the discrete invariant of multidimensional matrices of boundary format mentioned above. This ®ltration is well behaved with respect to the PGL…n‡1†-action onPⁿ and also with respect to the classical notion of association reviewed in [DK]. Eisenbud and Popescu realized in [EP] that the association is exactly what nowadays is called Gale transform. For Steiner bundles corresponding toAAWnVnI this operation amounts to exchanging the role of V withI, so that it corresponds to the transposition operator on multidimensional matrices.

The Gale transform for Steiner bundles can be decribed by the natural isomorphism Sn;k=SL…n‡1† !Skÿ1;n‡1=SL…k†:

Both quotients in the previous formula are isomorphic to the GIT-quotient P…WnVnI†=SL…W† SL…V† SL…I†

which is a basic object in linear algebra.

As an application of the tools developed in the ®rst section we show that all the points ofSn;k are semistable for the action of SL…n‡1†and we compute the stable points. Moreover we characterize the Steiner bundles SASn;k whose symmetry group (i.e. the group of linear projective transformations preserving S) contains SL…2†or containsC.

Finally we mention that W…S† has a geometrical construction by means of the Segre variety. From this construction W…S† can be easily computed by means of current software systems.

We thank J. ValleÁs for the useful discussions we had on the subject of this paper.

2 Multidimensional matrices of boundary format and geometric invariant theory It is well known that all one dimensional subgroups of the complex Lie group SL…2†

either are conjugated to the maximal torus consisting of diagonal matrices (which is isomorphic toC) or are conjugated to the subgroupCF 1 b

0 1

bAC

. De®nition 2.1.A…p‡1†-dimensional matrix of boundary formatAAV0n nVp

is called triangulable if one of the following equivalent conditions holds:

i) there exist bases inVj such thatai0;...;ipˆ0 fori0>P_p

tˆ1it;

ii) there exist a vector space U of dimension 2, a subgroup CHSL…U† and iso- morphismsVjFS^kjUsuch that ifV0n nVpˆ0_nAZWnis the decomposi- tion into a direct sum of eigenspaces of the induced representation then we have AA0_nX0W_n.

Proof of the equivalence between i)andii). Let x;y be a basis ofUsuch thattAC acts onxandyastxandt^ÿ1y. Sete_k^j†:ˆx^ky^kjÿk kj

k AS^kjU for j>0 ande^0†_k :ˆ

x^k0ÿky^k k₀

k AS^k0Uso thate^0†_i0 n ne_ip^p†is a basis ofS^k0Un nS^kpUwhich diagonalizes the action ofC. The weight ofe^0†_i0 n ne^p†_ip is 2…P_p

tˆ1itÿi0†, hence ii) implies i). The converse is trivial.

The following de®nition agrees with the one in [WZ], p. 639.

De®nition 2.2.A…p‡1†-dimensional matrix of boundary formatAAV0n nVp

is called diagonalizable if one of the following equivalent conditions holds:

i) there exist bases inV_j such thata_i0;...;ipˆ0 fori₀0P_p

tˆ1i_t;

ii) there exist a vector space U of dimension 2, a subgroup CHSL…U† and iso- morphismsV_jFS^kjUsuch thatAis a ®xed point of the induced action ofC. The following de®nition agrees with the one in [WZ], p. 639.

De®nition 2.3.A…p‡1†-dimensional matrix of boundary formatAAV0n nVp

is an identity if one of the following equivalent conditions holds:

i) there exist bases inVj such that

ai0;...;ipˆ 0 fori₀0P_p

tˆ1i_t 1 fori0ˆP_p

tˆ1it;

ii) there exist a vector space Uof dimension 2 and isomorphisms VjFS^kjU such thatAbelongs to the unique one-dimensional SL…U†-invariant subspace ofS^k0U nS^k1Un nS^kpU.

The equivalence between i) and ii) follows easily from the following remark: the matrixAsatis®es the condition ii) if and only if it corresponds to the natural multi- plication map S^k1Un nS^kpU!S^k0U (after a suitable isomorphism UFU has been ®xed).

From now on, we consider the natural action of SL…V0† SL…Vp† on P…V0n nVp†. We may suppose pX2. The de®nitions of triangulable, diago- nalizable and identity apply to elements ofP…V0n nVp†as well. In particular all identity matrices ®ll a distinguished orbit in P…V0n nVp†. The hyper- determinant of elements of V₀n nV_p was introduced by Gelfand, Kapranov and Zelevinsky in [GKZ]. They proved that the dual variety of the Segre product P…V₀† P…V_p† is a hypersurface if and only if k_jWP

i0jk_i for jˆ0;. . .;p (which is obviously true for a matrix of boundary format). When the dual variety is a hypersurface, its equation is called the hyperdeterminant of format …k₀‡1†

…k_p‡1† and denoted by Det. The hyperdeterminant is a homogeneous polynomial function over V0n nVp so that the condition DetA00 is meaningful for AAP…V0n nVp†. The function Det is SL…V0† SL…Vp†-invariant, in

particular if DetA00 thenAis semistable for the action of SL…V0† SL…Vp†.

We denote by Stab…A†HSL…V0† SL…Vp†the stabilizer subgroup ofAand by Stab…A†⁰ its connected component containing the identity. The main results of this section are the following.

Theorem 2.4. Let AAP…V₀n nV_p† of boundary format such that DetA00.

Then

A is triangulable , A is not stable for the action of SL…V₀† SL…V_p†:

Theorem 2.5.Let AAP…V0n nVp†be of boundary format such thatDetA00.

Then

A is diagonalizable ,Stab…A†contains a subgroup isomorphic toC: We state the following theorem only in the case pˆ2, although we believe that it is true for all pX2. We point out that in particular dim Stab…A†W3 which is a bound independent ofk₀,k₁,k₂.

Theorem 2.6. Let AAP…V₀nV₁nV₂† of boundary format such that DetA00.

Then there exists a2-dimensional vector space U such thatSL…U†acts over V_iFS^kiU and according to this action on V₀nV₁nV₂ we have Stab…A†⁰HSL…U†.Moreover the following cases are possible:

Stab…A†⁰F

0 (trivial subgroup) C

C

SL…2† (this case occurs if and only if A is an identity).

8>

>>

<

>>

>:

Remark.WhenAis an identity then Stab…A†FSL…2†.

Let X_j be the ®nite set f0;. . .;jg. We setB:ˆX_k1 X_kp. Aslice (in theq- direction) is the subsetf…a₁;. . .;a_p†AB:a_qˆkgfor somekAX_q. Two slices in the same direction are calledparallel. Anadmissible pathis a ®nite sequence of elements …a₁;. . .;a_p†AB starting from …0;. . .;0†, ending with …k₁;. . .;k_p†, such that at each step exactly one a_i increases by 1 and all other remain unchanged. Note that each admissible path consists exactly ofk0‡1 elements.

Tom Thumb's Lemma 2.7.Put a mark(or a piece of bread)on every element of every admissible path.Then two parallel slices contain the same number of marks.

Proof.Any admissible pathPcorresponds to a sequence ofk0integers between 1 and p such that the integer i occurs exactlyki times. We call this sequence the code of the path P. More precisely the j-th element of the code is the integerisuch that ai

increases by 1 from the j-th element of the path to the …j‡1†-th element. The occurrences of the integer i in the code divide all other integers di¨erent from i appearing in the code into ki‡1 strings (possibly empty); each string encodes the part of the path contained in one of thek_i‡1 parallel slices. The symmetric group S_ki‡1acts on the setAof all the admissible paths by permuting the strings. LetP_jⁱthe number of elements (marks) of the pathPAAon the slicea_iˆ j. In particular for all sAS_ki‡1 we have

X

PAA

P_jⁱˆ X

PAA

…sP†_jⁱˆ X

PAA

P_sⁱÿ1…j†;

which proves our lemma.

We will often use the following well-known lemma.

Lemma 2.8. If O_X^k !^f F is a morphism of vector bundles on a variety X with kW rankF ˆ f and cj…F†00for some jXf ÿk‡1,then the degeneracy locus Dk…f† ˆ fxAXjrank…f_x†Wkÿ1gis nonempty of codimensionWf ÿk‡1.

Proof.Suppose thatDk…f† ˆq. Then consider the projectionXP^kÿ1!^p Xand let Hbe the pullback of the hyperplane divisor according to the second projection. The natural composition

O!pO^knH !pFnH

gives a section of pFnH without zeroes, hence pFnH has a trivial line sub- bundle. It follows

0ˆcf…pFnH† ˆpcf…F† ‡ ‡pcfÿk‡1…F† H^kÿ1;

which is a contradiction because 1;. . .;H^kÿ1 are independent modulopH…X;C†.

We getD_k…f†0 qand the result follows from the Theorem 14.4 (b) of [Fu].

A square matrix with a zero left-lower submatrix with the NE-corner on the diag- onal has zero determinant. The following lemma generalizes this remark to multi- dimensional matrices of boundary format.

Lemma 2.9. Let AAV0n nVp. Suppose that in a suitable coordinate system there is…b₁;. . .;b_p†ABsuch that ai0...ipˆ0for ikWb_k …kX1†and i0Xb₀:ˆP_p

tˆ1b_t. ThenDetAˆ0.

Proof.The submatrix ofAgiven by elementsai0...ipsatisfyingikWb_k (kX1) gives on X ˆP^b2 P^bp the sheaf morphism

O_X^b1‡1!O_X…1;. . .;1†^b0

whose rank by Lemma 2.8 drops on a subvariety of codimensionWb₀ÿb₁ˆ P_p

tˆ2b_tˆdimP^b2 P^bp; hence there are nonzero vectorsviAV_ifor 1WiWp such thatA…v1n nvp† ˆ0 and then DetAˆ0 by Theorem 3.1 of Chapter 14 of [GKZ].

Lemma 2.10. Let pX2 and a_jⁱ be integers with 0WiWp, 0WjWk_i satisfying the inequalities a_j⁰Xa⁰_j‡1 for 0WjWk₀ÿ1,a_jⁱWa_j‡1ⁱ for i>0, 0WjWk_iÿ1 and the linear equations

X^ki

jˆ0

a_jⁱˆ0 for0WiWp a⁰_T^p

tˆ1b_i‡a_b¹₁‡ ‡a_b^p_pˆ0 for all…b₁;. . .;b_p†AB:

Then there is N AQsuch that

a_i⁰ˆN…k0ÿ2i†; a_i^jˆN…ÿkj‡2i† j>0:

Moreover NAZif at least one k_jis not even,and2NAZif all the k_j are even.

Proof.If 1WsWpandb_sX1 we have the two equations a⁰_T^p

tˆ1bt‡a_b¹₁‡ ‡a_b^s_s‡ ‡a_b^p

p ˆ0;

a⁰_T^p

tˆ1b_tÿ1‡a_b¹₁‡ ‡a_b^s_sÿ1‡ ‡a_b^p_pˆ0:

Subtracting we obtain a⁰_T^p

tˆ1b_tÿa⁰_T^p

tˆ1b_tÿ1 ˆ ÿ…a_b^s_sÿa_b^s_sÿ1†;

so that the right-hand side does not depend ons.

Moreover for pX2 from the equations a⁰_T^p

tˆ1b_t ‡a_b¹₁‡ ‡a_b^q_q‡1‡ ‡a_b^s_sÿ1‡ ‡a_b^p_pˆ0;

a⁰_T^p

tˆ1btÿ1‡a_b¹₁‡ ‡a_b^q

q‡ ‡a_b^s_s‡ ‡a_b^p

pˆ0

we get

a_b^q_q‡1ÿa_b^q_qˆa_b^s_sÿa_b^s_sÿ1;

which implies that the right-hand side does not depend onb_seither. Leta_b^s_sÿa_b^s_sÿ1ˆ 2NAZ. Thena_t^sˆa₀^s‡2Ntfort>0,s>0. By the assumptionP_ks

tˆ0a_t^sˆ0 we get …k_s‡1†a₀^s‡2NX^ks

tˆ1

tˆ0;

that is

a₀^sˆ ÿk_sN:

The formulas for a_i^s anda⁰_i follow immediately. If some k_s is odd we have 2NAZ andk_sNAZso thatNAZ.

Proof of Theorem2.4. IfAis triangulable it is not stable. Conversely suppose Anot stable and denote byAagain a representative ofAinV₀n nV_p. By the Hilbert±

Mumford criterion there exists a 1-parameter subgroup l:C!SL…V₀† SL…V_p†such that lim_t!0l…t†Aexists. Let

a₀^sW Wa_k^s_s; 0WsWp

be the weights of the 1-parameter subgroup of SL…Vs†induced byl; with respect to a basis consisting of eigenvectors the coordinate ai0...ip describes the eigenspace of l whose weight isa_i⁰₀‡a¹_i1‡ ‡a_i^p_p. Recall that

X^ki

jˆ0

a_sⁱˆ0; 0WsWp:

We note that for all…b₁;. . .;b_k†ABwe have a⁰_T^p

tˆ1bt‡a¹_b1‡ ‡a_b^p_pX0; …2:1†

otherwise the coe½cient a_i0...ip is zero for i_kWb_k, 1WkWp and i₀XP_p

tˆ1b_t and Lemma 2.9 implies DetAˆ0. The sum on all …b₁;. . .;b_k†AB for any admissible path of the left-hand side of (2.1) is nonnegative. The contribution ofa^t's in this sum is zero by Lemma 2.7. Also the contribution ofa⁰'s is zero because it is zero on any admissible path. It follows that

a⁰_T^p

tˆ1b_t‡a_b¹₁‡ ‡a_b^p_pˆ0 for all…b₁;. . .;b_k†AB;

and by Lemma 2.10 we get explicit expressions for the weights which imply thatAis triangulable.

Proof of Theorem2.5. Again we denote byAany representative ofAinV0n n Vp. IfAis diagonal in a suitable basise^0†_i0 n ne_ip^p†, we construct a 1-parameter subgroup l:C!SL…V0† SL…Vp† by the equation l…t†e^0†_i0 n ne^p†_ip :ˆ

t^{i0ÿTtˆ1p it}e^0†_i0 n ne^p†_ip , so that CHStab…A†. Conversely let CHStab…A†. By Theorem 2.4, A is triangulable and by Lemma 2.9 all diagonal elements ai0...ip

with i0ˆP_p

tˆ1it are nonzero. We can arrange the action on the representative in order that the diagonal corresponds to the zero eigenspace. Then the assumption

CHStab…A†and the explicit expressions of the weights as in the proof of Theorem 2.4 show thatAis diagonal.

We will prove Theorem 2.6 by geometric arguments at the end of Section 6.

3 Preliminaries about Steiner bundles

De®nition 3.1.A Steiner bundle overPⁿˆP…V†is a vector bundleSwhose dualS appears in an exact sequence

0!S!WnO!^fA InO…1† !0 …3:1†

whereWandIare complex vector spaces of dimensionn‡kandkrespectively.

A Steiner bundle is stable ([BS], Theorem 2.7 or [AO], Theorem 2.8) and is invariant by small deformations ([DK], Corollary 3.3). Hence the moduli spaceS_n;k of Steiner bundles de®ned by (3.1) is isomorphic to an open subset of the Maruyama moduli scheme of stable bundles. On the other handS_n;k is also isomorphic to the GIT-quotient of a suitable open subset of P…Hom…W;InV†† for the action of SL…W† SL…I†(see Section 6). It is interesting to remark that these two approaches give two di¨erent compacti®cations of Sn;k, but we do not pursue this direction in this paper. For other results aboutP…Hom…W;InV††, see [EH] and [C].

De®nition 3.2. Let SASn;k be a Steiner bundle. A hyperplane H AP…V† is an unstable hyperplane ofSifh⁰…S_jH †00. The setW…S†of the unstable hyperplanes is the degeneracy locus overP…V†of the natural mapH¹…S…ÿ1††nO!H¹…S†n O…1†, hence it has a natural structure of scheme. W…S† is called the scheme of the unstable hyperplanes ofS. Note that sinceh⁰…S_jH †W1 ([V2]) the rank of the previous map drops at most by one.

3.3. Let us describe more explicitly the map H¹…S…ÿ1††nO!H¹…S†nO…1†.

From (3.1) it follows thatH¹…S…ÿ1††FI andH¹…S†F…VnI†=W. The projec- tionVnI !^B …VnI†=W can be interpreted as a mapVnH¹…S…ÿ1†† !H¹…S† which induces onP…V†the required morphismH¹…S…ÿ1††nO!H¹…S†nO…1†.

For a generic S, W…S† ˆq. Examples show that W…S†can have a nonreduced structure.

We recall that ifDis a divisor with normal crossing then W…logD†is the bundle of meromorphic forms having at most logarithmic poles overD. IfHis the union of m hyperplanes Hi with normal crossing, it is shown in [DK] that for mWn‡1, W…logH† splits while for mXn‡2 we haveSˆW…logH†ASn;k where kˆmÿ nÿ1.

The following is a simple consequence of [BS], Theorem 2.5.

Proposition 3.4.Let SASn;k,then

h⁰…S…t†† ˆ0,tWkÿ1:

Proof. S…t†F 5^nÿ1S…ÿk‡t†. The5^nÿ1-power of the sequence dual to (3.1) is 0!S^nÿ1InO…ÿn‡1ÿk‡t† !S^nÿ2InWnO…ÿn‡2ÿk‡t† !

!5^nÿ1WnO…ÿk‡t† !5^nÿ1S…ÿk‡t† !0;

and from this sequence the result follows.

Let us ®x a basis in each of the vector spaces W and I. Then the morphism f_A in (3.1) can be represented by a k …n‡k† matrix A (it was called MA in the introduction, see (1.1)) with entries inV. In order to simplify the notations we will use the same letterAto denote also its class inP…Hom…W;InV††.Ahas rankkat every point ofP…V†. Two such matrices represent isomorphic bundles if and only if they lie in the same orbit of the action of GL…W† GL…I†.

3.5.In particularH⁰…S…t††identi®es with the space of…n‡k† 1-column vectorsv with entries inS^tVsuch that

Avˆ0: …3:2†

MoreoverHAW…S†(as closed point) if and only if there are nonzero vectorsw₁of size…n‡k† 1 andi₁ of sizek1 both with constant coe½cients such that

Aw₁ˆi₁H: …3:3†

This means thatw1is in the kernel of the mapWFH⁰…WnOH† !H⁰…InOH…1††:

3.6. According to the theorem stated in the introduction AAHom…W;VnI† has nonzero hyperdeterminant if and only if it corresponds to a vector bundle. The locus inP…Hom…W;VnI††where the hyperdeterminant vanishes is an irreducible hyper- surface of degreek n‡k

k

([GKZ], Chapter 14, Corollary 2.6). It is interesting to remark that Proposition 3.4 can be proved also as a consequence of [GKZ], Chapter 14, Theorem 3.3.

3.7.The above description has a geometrical counterpart. HereP…V†is the projective space of lines inV, dual to the usual projective spacePof hyperplanes inV. Consider inP…VnI†the varietyX_r corresponding to elements ofVnI of rankWr. In par- ticularX₁ is the Segre varietyP…V† P…I†. Letmˆmin…n;kÿ1†so thatX_m is the variety of non maximum rank elements. ThenAAHom…W;VnI†de®nes a vector bundle if and only if it induces an embeddingP…W†HP…VnI†such that at every smooth point of XmVP…W†, P…W† and Xm meet transversally. This follows from [GKZ], Chapter 14, Propostion 3.14 and Chapter 1, Proposition 4.11.

3.8.W…S†has the following geometrical description. Let p_V be the projection of the Segre varietyP…V† P…I†on theP…V†. Then

W…S†_redˆ p_V‰P…W†V…P…V† P…I††Š_red

(according to the natural isomorphismP…V† ˆP…V†). In fact i₁H in formula (3.3) is a decomposable tensor inVnI.

3.9.About the scheme structure we remark thatW…S†is the degeneration locus of the morphismInO_P…V†!VnI

W nO_P…V†…1†. The following construction is standard.

The projective bundle PˆP…InOP…V†† !^p P…V† is isomorphic to the Segre varietyT ˆP…V† P…I† ˆP…V† P…I†andOP…1†FOT…0;1†. The morphism

C!VnI

W nVnI de®nes a section ofO_T…1;1†nVnI

W with zero locusZˆTVP…W†. Now assume that dimW…S† ˆ0, hence dimTˆ0. By applyingp to the exact sequence

OTn VnI W

!OT…1;1† !OZ!0

we get that the structure sheaf ofW…S†is contained inpOZ. We do not know if the equality always holds. In particular if Z is reduced also W…S†is reduced. We will show in Proposition 6.5 that a multiple point occurs inZi¨ it occurs inW…S†.

Theorem 3.10. Let SASn;k be a Steiner bundle. Then any set of distinct unstable hyperplanes of S has normal crossing.

Proof. We ®x a coordinate systemx₀;. . .;x_n on Pⁿ and a basise¹;. . .;e^n‡k ofW.

Let Abe a matrix representing S. If the assertion is not true, we may suppose that W…S† contains the hyperplanes x₀ˆ0;. . .;x_jˆ0, P_j

iˆ0x_iˆ0 for some j such that 1WjWnÿ1. By (3.3) there are c⁰AW,b⁰ AI such that Ac⁰ˆb⁰x₀. We may suppose that the ®rst coordinate of c⁰ is nonzero, hence A ‰c⁰;e²;. . .;e^n‡kŠ ˆ

‰b⁰x₀;. . .Š ˆA⁰.

The matrix A⁰still representsS, hence by (3.3) there arec¹AW,b¹AI such that A⁰c¹ˆb¹x1. At least one coordinate ofc¹after the ®rst is nonzero, say the second. It follows thatA⁰ ‰e¹;c¹;e³;. . .;e^n‡kŠ ˆ ‰b⁰x0;b¹x1;. . .Š ˆA⁰⁰and againA⁰⁰represents S. Proceeding in this way we get in the end that

‰b⁰x₀;. . .;b^jx_j;. . .Š

is a matrix representingS, which we denote again byA.

By (3.3) there arecˆ…c1;. . .;cn‡k†^t AW,bAIsuch thatAcˆ ‰b⁰x0;. . .b^jxj;. . .Š

cˆbP_j

iˆ0xi.

Now we distinguish two cases. If ciˆ0 for iXj‡2 we get bˆc1b⁰ˆc2b¹

ˆ ˆc_j‡1b^j, that is the submatrix of A given by the ®rst j‡1 columns has generically rank one. If we take the k …n‡kÿj† matrix which has b^j as ®rst column and the lastn‡kÿjÿ1 columns ofAin the remaining places, we obtain a morphism

O^k!OlO…1†^n‡kÿjÿ1;

which by Lemma 2.8 has rankWkÿ1 on a nonempty subschemeZofPⁿ. It follows that alsoAhas rankWkÿ1 onZ, contradicting the assumption thatSis a bundle.

So this case cannot occur.

In the second case there exists a nonzero c_i for someiXj‡2, we may suppose c_j‡200. Then the matrix

A⁰ˆA ‰e¹;. . .;e^j‡1;c;e^j‡3;. . .;e^n‡kŠ ˆ b⁰x₀;. . .b^jx_j;bX^j

iˆ0

x_i. . .

" #

representsS.

The lastn‡kÿjÿ2 columns ofA⁰ de®ne a sheaf morphismO^k!O…1†^n‡kÿjÿ2 on the subspace P^nÿjÿ1ˆ fx0 ˆ ˆxj ˆ0g and again by Lemma 2.8 we ®nd a point where the rank ofAisWkÿ1. So neither case can occur.

Proposition 3.11. Let SASn;k and let x₁;. . .;x_sAW…S†, sWn‡k. There exists a matrix representing S whose ®rst s columns are ‰b¹x₁;. . .;b^sx_sŠ, where the bⁱ are vectors with constant coe½cients of size k1. Moreover any p columns among

b¹;. . .;b^s with pWk are independent. Conversely if the ®rst s columns of a matrix

representing S have the form‰b¹x₁;. . .;b^sx_sŠthenx₁;. . .;x_sAW…S†.

Proof. The last assertion is obvious. The proof of the existence of a matrixArepre- sentingShaving the required form is analogous to that of Theorem 3.10. Then it is su½cient to prove that b¹;. . .;b^p are independent. Suppose P_p

iˆ1bⁱl_iˆ0. Letxˆ Q_p

iˆ1x_i. Letcbe the…n‡k† 1 vector (whith coe½cients inS^pÿ1V) whosei-th entry

is lix=x_i for iˆ1;. . .;p and zero otherwise. It follows that AcˆxP_p

iˆ1bⁱliˆ0 and by (3.2) we get a nonzero section of S…pÿ1†, which contradicts Proposition 3.4.

3.12 Elementary transformations.ConsiderH ˆ fxˆ0gAW…S†. The mapOH!S_jH induces a surjective mapS!OHand an exact sequence

0!S⁰!S!OH!0 …3:4†

(see also [V2], Theorem 2.1); it is easy to check (e.g. by Beilinson's theorem) that S⁰AS_n;kÿ1. According to [M] we say that S⁰ has been obtained from S by an

elementary transformation. By Proposition 3.11 there exists a matrixArepresenting Sof the following form

Aˆ

x

0

... A⁰

0 2 66 64

3 77

75 …3:5†

whereA⁰is a matrix representingS⁰. Sinceh⁰…S_jH †W1,S⁰is uniquely determined by SandH.

Theorem 3.13. With the above notations we have the inclusion of schemes W…S†H W…S⁰†UH.In particular we have:

i) lengthW…S⁰†XlengthW…S† ÿ1;

ii) ifdimW…S⁰† ˆ0thenmult_HW…S⁰†Xmult_HW…S† ÿ1,so that if H is a multiple point of W…S†,then H AW…S⁰†;

iii) ifdimW…S⁰† ˆ0then for any hyperplane K0H, multKW…S⁰†XmultKW…S†.

Proof.The sequence dual to (3.4)

0!S!S⁰!O_H…1† !0 gives the commutative diagram onP…V†:

0 ƒƒ! O???y ƒƒ! H¹…S…ÿ1††nO ƒƒ! H¹…S⁰…ÿ1††nO ƒƒ! 0

??

?y

??

?y

0 ƒƒ! H⁰…O_H…1††nO…1† ƒƒ! H¹…S†nO…1† ƒƒ! H¹…S⁰†nO…1† ƒƒ! 0 It follows that the matrixB⁰of the map

H¹…S⁰…ÿ1††nO!H¹…S⁰†nO…1†

can be seen as a submatrix of the matrixBof the map H¹…S…ÿ1††nO!H¹…S†nO…1†:

In a suitable system of coordinates:

Bˆ

y₁

...

y_n 0 B⁰ 2 66 66 4

3 77 77

5 …3:6†

where…y₁;. . .;y_n†is the ideal ofH(in the dual space). It follows that I…W…S⁰†† …y₁;. . .;y_n†HI…W…S††;

which concludes the proof.

4 The Schwarzenberger bundles

Let U be a complex vector space of dimension 2. The natural multiplication map S^kÿ1UnSⁿU!S^n‡kÿ1U induces the SL…U†-equivariant injective map S^n‡kÿ1U!S^kÿ1UnSⁿUand de®nes a Steiner bundle onP…SⁿU†FPⁿas the dual of the kernel of the surjective morphism

O_P…Sⁿ_U†nS^n‡kÿ1U!O_P…Sⁿ_U†…1†nS^kÿ1U:

It is called a Schwarzenberger bundle (see [ST], [Schw]). Let us remark that in the correspondence between Steiner bundles and multidimensional matrices mentioned in the introduction, the Schwarzenberger bundles correspond exactly to the identity matrices (see De®nition 2.3).

By interchanging the role of S^kÿ1U andSⁿU we obtain also a Schwarzenberger bundle onP…S^kÿ1U†FP^kÿ1as the dual of the kernel of the surjective morphism

O_P…S^kÿ1_U†nS^n‡kÿ1U!O_P…S^kÿ1_U†…1†nSⁿU:

Both the above bundles are SL…U†-invariant. We sketch the original Schwarzen- berger construction for the ®rst one. The diagonal mapu7!uⁿand the isomorphism P…SⁿU†FPⁿ detect a rational normal curve P…U† ˆCnHPⁿ. In the same way a second rational normal curve P…U† ˆCn‡kÿ1 arises in P…S^n‡kÿ1U†. We de®ne a morphism

P…SⁿU† ˆSⁿP…U† !Gr…P^nÿ1;P…S^n‡kÿ1U††

npoints inP…U† 7!Span of npoints inC_n‡kÿ1:

The pullback of the dual of the universal bundle on the Grassmannian is a Schwarzenberger bundle.

It is easy to check that if S is a Schwarzenberger bundle then W…S† ˆC_nH P…SⁿU†(the dual rational normal curve). See e.g. [ST], [V1].

This can be explicitly seen from the matrix form given by [Schw], Proposition 2

M_Aˆ

x0 . . . xn

... ...

x₀ . . . x_n

2 64

3

75: …4:1†

Let t1;. . .;tn‡k be any distinct complex numbers. Let wbe the …n‡k† …n‡k†

Vandermonde matrix whose…i;j†entry ist^iÿ1†_j ; the…i;j†-entry of the productMAwis t^iÿ1†_j …P_n

kˆ0xkt_j^k†; hencefP_n

kˆ0xkt^k ˆ0gAW…S†for alltACby Proposition 3.11.

On the other handW…S†is SL…U†-invariant; if it were strictly bigger thanC_n then it would contain the hyperplane Hˆ fx₀‡x₁ˆ0g, which lies in the next SL…U†- orbit; now equation (3.3) implies immediately thatw₁ˆi₁ˆ0.

In Theorem 5.13 we will need the following result.

Lemma 4.1. Let S be a Schwarzenberger bundle and let …x₀;. . .;x_n† be coordinates inP…V†such that S is represented(with respect to suitable basis of I and W)by the matrix M_Ain…4:1†.Let…y₀;. . .;y_n†be dual coordinates inP…V†.Then the morphism H¹…S…ÿ1††nO!H¹…S†nO…1†(with respect to the obvious basis)is represented by the matrix

Bˆ

y₁ ÿy₀ y₁ ÿy₀

... ...

y₁ ÿy₀ y₂ 0 ÿy₀

... ... ...

y₂ 0 ÿy₀ y₂ ÿy₁

y₃ 0 0 ÿy₀

... ... ... ...

2 66 66 66 66 66 66 66 66 66 66 4

3 77 77 77 77 77 77 77 77 77 77 5 :

Proof.By (3.3) it is enough to check that the composition W !^A VnI!^B …VnI†=W is zero, which is straightforward.

Theorem 4.2([Schw], Theorem 1, see also [DK], Proposition 6.6).The moduli space of Schwarzenberger bundles is PGL…n‡1†=SL…2†, which is the open subscheme of the Hilbert scheme parametrizing rational normal curves.

In particularW…S†uniquely determinesSin the class of Schwarzenberger bundles.

5 A ®ltration ofSn;kand the Gale transform of Steiner bundles De®nition 5.1.

S_n;kⁱ :ˆ fSASn;kjlengthW…S†Xig:

In particular

Sn;kˆS_n;⁰_kIS_n;k¹ I :

We will see (Corollary 5.5) thatS_n;^y_k corresponds to Schwarzenberger bundles.

Each S_n;kⁱ is invariant for the action of SL…V†on Sn;k. We will see in Section 6 that all the points of Sn;k are semistable (in the sense of Mumford's GIT) for the action of SL…V†.

LetSbe the open subset ofP…Hom…W;VnI††representing Steiner bundles. The quotientSn;k=SL…V†is isomorphic toS=SL…W† SL…I† SL…V†.

By interchanging the role of V and I, also Skÿ1;n‡1=SL…I† turns out to be iso- morphic toS=SL…W† SL…I† SL…V†, so that we obtain an isomorphism

S_n;k=SL…n‡1†FS_kÿ1;n‡1=SL…k†:

For anyEASn;k=SL…n‡1†we will call theGale transformofEthe corresponding class inSkÿ1;n‡1=SL…k†and we denote it byE^G. In [DK] the above construction is called association. Here we follow [EP]. Our Gale transform is a generalization of the one in [EP]. In fact in the caseiˆn‡k‡1 Eisenbud and Popescu in [EP] review the classical association between PGL…n‡1†-classes of n‡k‡1 points of Pⁿ in general position and PGL…k†-classes ofn‡k‡1 points ofP^kÿ1 in general position and call it Gale transform. If we take the union H of n‡k‡1 hyperplanes with normal crossing in Pⁿ (as points in the dual projective space) the Gale transform (as points in the dual projective space)H^Gconsists of a PGL…k†-class of n‡k‡1 hyperplanes with normal crossing in P^kÿ1. As remarked in [DK], ‰W…logH†Š^GF

‰W…logH^G†Š. That is, the Gale transform in our sense reduces to that in [EP] when the Steiner bundles are logarithmic. It is also clear that the PGL-class of Schwar- zenberger bundles overP…V†corresponds under the Gale transform to the PGL-class of Schwarzenberger bundles overP…I†.

We point out that one can de®ne the Gale transform of a PGL-class of Steiner bundles but it is not possible to de®ne the Gale transform of a single Steiner bundle.

This was implicit (but not properly written) in [DK]. Nevertheless by a slight abuse we will also speak about the Gale transform of a Steiner bundleS, which will be any Steiner bundle in the class of the Gale transform ofSmod SL…n‡1†.

The following elegant theorem due to Dolgachev and Kapranov is a ®rst beautiful application of the Gale transform.

Theorem 5.2([DK], Theorem 6.8).Any SAS_n;2is a Schwarzenberger bundle.

Proof.

S_n;2=SL…n‡1†FS_1;n‡1=SL…2†;

and it is obvious that a Steiner bundle on the lineP¹is Schwarzenberger.

Theorem 5.3. Two Steiner bundles having in common n‡k‡1 distinct unstable hyperplanes are isomorphic.

Proof. We prove that ifSis a Steiner bundle such that the hyperplanesfx_iˆ0g for

iˆ1;. . .;n‡k‡1 belong toW…S†, thenSis uniquely determined. By Proposition

3.11 there exist column vectors a_iAC^k such that S is represented by the matrix

‰a¹x₁;. . .;a^n‡kx_n‡kŠ. Moreover by (3.3) there arebAC^n‡k andcAC^k such that

‰a¹x₁;. . .;a^n‡kx_n‡kŠbˆcx_n‡k‡1:

We claim thatallthe components ofbare nonzero. The last formula can be written

‰a¹b¹;. . .;a^n‡kb^n‡k;ÿcŠ ‰x₁;. . .;x_n‡k‡1Š^t ˆ0

where in the right matrix we identify x_i with the …n‡1† 1 vector given by the coordinates of the corresponding hyperplane. We may suppose that there exists s with 1WsWn‡kÿ1 such that b_iˆ0 for 1WiWs and b_i00 for s‡1WiW n‡k. IfsXk, it follows thatn‡1 hyperplanes among thex_ihave a nonzero syzygy, which contradicts Proposition 3.11. HencesWkÿ1 and we have

‰a^s‡1b^s‡1;. . .;a^n‡kb^n‡k;ÿcŠ ‰x_s‡1;. . .;x_n‡k‡1Š^tˆ0:

The rank of the right matrix isn‡1, hence the rank of the left matrix isWkÿs, in particular the ®rstkÿs‡1 columns are dependent and this contradicts Proposition 3.11. This proves the claim.

In particular‰a¹;. . .;a^n‡k;ÿcŠ Bˆ0 where

BˆDiag…b1;. . .;bn‡k;1† ‰x₁;. . .;x_n‡k‡1Š^t

is a…n‡k‡1† …n‡1†matrix with constant entries of rank…n‡1†. Therefore the matrix‰a¹;. . .a^n‡k;ÿcŠis uniquely determined up to the (left) GL…k†-action, which implies thatSis uniquely determined up to isomorphism.

Corollary 5.4. A Steiner bundle is logarithmic if and only if it admits at least …n‡k‡1†unstable hyperplanes.

Proof.In factHHW…W…logH††by formula (3.5) of [DK] and Proposition 3.11.

Corollary 5.5([V2], Theorem 3.1]).A Steiner bundle is Schwarzenberger if and only if it admits at least…n‡k‡2†unstable hyperplanes.In particularS_n;k^y coincides with the moduli space of Schwarzenberger bundles.

Proof. Let S be a Steiner bundle, and H AW…S†. Let us consider the elementary transformation (3.12)

0!S⁰!S!O_H!0

where S⁰ASn;kÿ1; by Theorem 3.13, S⁰ has n‡k‡1 unstable hyperplanes. Pick- ing H⁰AW…S⁰† and repeating the above procedure after …kÿ2† steps we reach a S^kÿ2†ASn;2; by Theorem 5.2,S^kÿ2†is a Schwarzenberger bundle. In particular the remainingn‡4 unstable hyperplanes lie on a rational normal curve. It is then clear that any subset ofn‡4 hyperplanes inW…S†lies on a rational normal curve. Since there is a unique rational normal curve through n‡3 points in general position, it follows that W…S† is contained in a rational normal curve, so that S is a Schwar- zenberger bundle by Theorem 5.3.

Theorem 5.6.Let nX2,kX3.

i) S_n;kⁱ for0WiWn‡k‡1is an irreducible unirational closed subvariety ofSn;k of dimension…kÿ1†…nÿ1†…k‡n‡1† ÿi‰…nÿ1†…kÿ2† ÿ1Š.

ii) S_n;k^n‡k‡1contains as an open dense subset the variety of Steiner logarithmic bundles which coincides with the open subvariety of Sym^n‡k‡1Pⁿ⁴ consisting of hyper- planes inPⁿwith normal crossing.

Proof.(ii) follows from Theorem 5.3.

The irreducibility in (i) follows from the geometric construction 3.8. The numerical computation in (i) is performed (for iWn‡k) by adding i…n‡kÿ1†(moduli of i points inP…V†nP…I†) ton…kÿ1†…n‡kÿi†(dimension of Grassmannian of linear P^n‡kÿ1inP…VnI†containing the span of the aboveipoints) and subtractingk²ÿ1 (dim SL…I†).

Remark 5.7.In the case…n;k† ˆ …2;3†the generic Steiner bundle is logarithmic (this was remarked in [DK], 3.18). In fact the genericP⁴ linearly embedded in P⁸ meets the Segre varietyP²P² in degP²P²ˆ6ˆn‡k‡1 points.

Remark.The dimension ofS_n;kⁱ =SL…n‡1†is equal to…n‡k‡1ÿi†‰…kÿ2†…nÿ1†

ÿ1Š ‡n…kÿ1†forkX3,nX2, 0WiWn‡k‡1 and it is 0 foriXn‡k‡2.

5.8. Corollary 5.5 implies the following property of the Segre variety: if a generic linearP…W†meetsP…V† P…I†inn‡k‡2 points, thenP…W†meets it in in®nitely many points.

Theorem 5.9. Consider a nontrivial (linear) action of SL…2† ˆSL…U† over Pⁿ. If a Steiner bundle isSL…2†-invariant then it is a Schwarzenberger bundle andSL…U†acts overPⁿˆP…SⁿU†.HenceS_n;k^y is the subset of the ®xed points of the action ofSL…2†

onS_n;k.

Proof. By Theorem 2.4 there exists a coordinate system such that all the entries (except the ®rst) of the ®rst column of the matrix representing the Steiner bundle S are zero. By Proposition 3.11, W…S† is nonempty. By the assumption W…S† is

SL…2†-invariant and closed; it follows that W…S† is a union of rational curves and of simple points. If W…S† is in®nite we can apply Corollary 5.5. If W…S† is ®nite we argue by induction onk. We pick upHAW…S†and we consider the elementary transformation 0!S⁰!S!O_H!0. We get for allgASL…U†the diagram

S ƒƒƒ!^f O_H

??

?yⁱ

gS ƒƒƒ!^gf O_H:

Since h⁰…S_jH †W1 we get that f and gfi coincide up to a scalar multiple. We obtain a commutative diagram

0 ƒƒƒ! S???y⁰ ƒƒƒ! S ƒƒƒ! OH ƒƒƒ! 0

??

?y^F

??

?y^F

0 ƒƒƒ! gS⁰ ƒƒƒ! gS ƒƒƒ! OH ƒƒƒ! 0:

It follows thatS⁰FgS⁰, hence SL…U†HSym…S⁰†and by the inductive assumption S⁰ is Schwarzenberger and SL…U† acts over Pⁿ ˆP…SⁿU†. Hence W…S† is in®nite and we apply again Corollary 5.5.

Corollary 5.10.IfHis the union of n‡k‡1hyperplanes with normal crossing then

W…W…logH†† ˆ

H whenHdoes not osculate a rational normal curve, C_n whenHosculates the rational normal curve C_n,

(this case occurs iff W…logH†is Schwarzenberger).

8<

:

Proof. HHW…logH†by Proposition 3.11. The result follows by Theorem 5.3 and Corollary 5.5.

Corollary 5.11. Let SAS_n;k be a Steiner bundle.If W…S†contains at least n‡k‡1 hyperplanes then for every subset HHW…S† consisting of n‡k‡1 hyperplanes SFW…logH†,in particular S is logarithmic.

Corollary 5.12 (Torelli theorem, see [DK] for kXn‡2 or [V2] in general). Let H and H⁰ be two ®nite unions of n‡k‡1 hyperplanes with normal crossing in P…V†

with kX3not osculating any rational normal curve.Then HˆH⁰,W…logH†FW…logH⁰†: