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ARRANGEMENTS OF HYPERPLANES AND VECTOR BUNDLES ON P n.

arXiv:alg-geom/9210001v1 6 Oct 1992ARRANGEMENTS OF HYPERPLANES AND VECTOR BUNDLES ON P n.I.Dolgachev and M.Kapranov*Introduction.Let X be a smooth algebraic variety and D be a divisor with normal crossing on X. Thepair (X, D) gives rise to a naturalsheaf Ω1X(log D) of differential 1-forms on X with logarithmic poles on D. For eachpoint x ∈X the space of sections of this sheaf in a small neighborhood of x is generatedover OX,x by regular 1-forms and by forms d log fi where fi = 0 is a local equation ofan irreducible component of D containing X. This sheaf (and its exterior powers) wasoriginally introduced by Deligne [De] to define a mixed Hodge structure on the openvariety X −D.

An important feature of the sheaf Ω1X(log D) is that it is locally free i.e.can be regarded as a vector bundle on X.In this paper we concentrate on a very special case when X = P n is a projective spaceand D = H1 ∪... ∪Hm is a union of hyperplanes in general position. It turns out that thecorresponding vector bundles are quite interesting from the geometric point of view.

Itwas shown in an earlier paper [K] of the second author that in this case Ω1P n(log D) definesan embedding of P n into the Grassmann variety G(n, m −1) whose image becomes, afterthe Pl¨ucker embedding, a Veronese variety V m−3ni.e. a variety projectively isomorphic tothe image of P n under the map given by the linear system of all hypersurfaces of degreem −3.If the case when the hyperplanes osculate a rational normal curve in P n thebundle Ω1P n(log D) coincides with the secant bundle Emn of Schwarzenberger [Schw1-2].The corresponding Veronese variety consists in this case of chordal n −1 -dimensionalsubspaces to a rational normal curve in P m−2.The main result of this paper (Theorem 7.2) asserts that in the case m ≥2n + 3 thearrangement of m hyperplanes H = {H1, ..., Hm} can be uniquely reconstructed from thebundle E(H) = Ω1P n(log S Hi) unless all its hyperplanes osculate the same rational normalcurve of degree n. To prove this we study the variety C(H) of jumping lines for E(H).The consideration of this variety is traditional in the theory of vector bundles on P n, see[Bar, Hu].

In our case this variety is of some geometric interest. For example, if n = 2i.e.

we deal with m lines in P 2 then (in the case of odd m) C(H) is a curve in the dualP 2 containing the points corresponding to lines from H. The whole construction therefore* Research of both authors was partially supported by the National Science Foundation.1

gives a canonical way to draw an algebraic curve through a collection of points in (thedual) P 2.For 5 points p1, ..., p5 in P 2 this construction gives the unique conic through pi.For 7 points p1, ..., p7 the construction gives a plane sextic curve with for which pi aredouble points. A non-singular model of this curve is isomorphic to the plane quartic curveof genus 3 which is classically associated to 7 points via Del Pezzo surfaces of degree 2, see[C2][DO].For even number of lines in P 2 the set of jumping lines is typically finite.

In this casemore interesting is the curve of jumping lines of second kind introduced by Hulek [Hu].The study of this curve will be carried out in the subsequent paper [DK].Let us only formulate the answer for 6 lines (considered as points p1, ..., p6 in thedual plane). In this case Hulek’s curve will be a sextic of genus 4 of which pi are nodes.It is described as follows.

Blow up the points pi. The result is isomorphic to a cubicsurface S in P 3.

The inverse images if the points pi and the strict preimages of quadricsthrough various 5-tuples of pi form a Schl¨afli double sixer of lines on the cubic surface. Toeach such double sixer there is classically associated a quadric Q in P 3 called the Schurquadric [Schur][R] (see also [B], p.162).

It is uniquely characterized by the property thatthe corresponding pairs of lines of the double sixer are orthogonal with respect to Q. Oursextic curve in P 2 lifted to the surface S becomes the intersection S ∩Q.In fact, much of Hulek’s general theory of stable bundles on P 2 with odd first Chernclass can be neatly reformulated in terms of (suitably generalized) Schur quadrics. Thiswill be done in [DK].Thus our approach gives a unified treatment of many classical constructions associat-ing a curve to a configuration of points in a projective space.

It appears that a systematicstudy of logarithmic bundles in other situations (like surfaces other than P 2) will provide arich supply of concrete examples and and give additional insight into the geometry relatedto vector bundles.We would like to thank L. Ein and H. Terao for useful discussions and correspondencerelated to this work.2

§1. Arrangements of hyperplanes.1.1.

Let V be a complex vector space of dimension n+1 and P n = P(V ) be the projectivespace of lines in V .Let H = (H1, ..., Hm) be a set (arrangement) of hyperplanes inP n. Dually it defines a set (a configuration) of m points in the dual vector space ˇP n =P(V ∗). We say that H is in (linearly) general position if the intersection of any k ≤n + 1hyperplanes from H is of codimension exactly k. Throughout this paper we shall mostlydeal with arrangements in general position.1.2.

We choose a linear equation fi ∈V ∗for each hyperplane Hi of H. This system ofchoices defines a linear mapαH : Cm →V ∗;(λ1, ..., λm) 7→Xλifi.The kernel of this map will be denoted by IH. It consists of linear relations between thelinear forms fi.

By transposing, we obtain a linear mapα∗H : (Cm)∗−→I∗H. (1.1)Assume that m ≥n + 2 so that IH ̸= 0.

After making a natural identification betweenthe space Cm and its dual space (Cm)∗(defined by the bilinear form P xiyi) we obtainfrom the map (1.1) m linear forms on the space IH i.e. an arrangement of hyperplanes inP(H).

We denote the arrangement thus obtained by Has and refer to it as the associatedarrangement. It is clear that H is in general position if and only if the restriction of themap αH to any coordinate subspace Ck in Cm with k ≤n + 1 is of maximal rank.

Thisimplies that Has is in general position if and only if H is. In the latter case, the dimensionof P(IH) is equal to m −n −2.

From now on whenever we speak about the associationwe assume that the arrangements are in general position. Note that to define the mapαH we need a choice of order on the set of hyperplanes from H. Making this choice weautomatically make a choice on the set Has.The notion of association was introduced by A.Coble [C1].

For modern treatment see[DO]. This notion has been rediscovered, under the names ”duality” or ”orthogonality”several times later, notably in the context of combinatorial geometries (see [CR], §11) andhypergeometric functions (see [GG]).Obviously the association is a self-dual operation, so (Has)as = H where we make acanonical identification between the spaces V and V ∗∗.Let V and I be two vector spaces of dimensions n + 1 and m −n −1 respectively andlet H and H′ be two arrangements of m hyperplanes in P(V ) and P(I) respectively.

We3

shall say that H and H′ are associated if there is a projective isomorphism P(I) →P(IH)taking H′ to the associated configuration Has (matching the ordering, if it was made). Inparticular, when m = 2n + 2, we can speak about self-associated configurations.By duality we can speak about associated configurations of points in projective spaces.The following proposition (equivalent to a result by A.Coble) gives a criterion of beingassociated (resp.

self-associated) in terms of the Segre (resp. Veronese) embedding.

Westate it in terms of configurations of points.1.3. Proposition.

a) Let V, I be vector spaces of dimensions n + 1 and m −n −1 re-spectively. Let pi ∈P(V ), qi ∈P(I), i = 1, ..., m, be two configurations of m points .

Lets(pi, qi) ∈P(V ⊗I) be the image of the pair (pi, qi) with respect to the Segre embeddings : P(V ) × P(I) →P(V ⊗I). The configurations of points (p1, ..., pm) and (q1, ..., qm) areassociated to each other if and only if the points s(pi, qi) are projectively dependent butany proper subset of them is projectively independent.b) Let V be a vector space of dimension n + 1 and pi ∈P(V ), i = 1, ..., 2n+ 2 be a config-uration of points.

Let v(pi) ∈P(S2V ) be the image of pi under the Veronese embeddingv : P(V ) →P(S2V ).The configuration (p1, ..., p2n+2) is self-associated if and only ifthe points v(pi) are projectively dependent but any proper subset of them is projectivelyindependent.Proof: b) follows from a). For the proof of a), see, e.g., [K].1.4.Let H be an arrangement of m hyperplanes in P(V ) and Has be the associatedarrangement in the space P(IH).

LetW = {(λ1, ..., λm) ∈Cm :Xλi = 0}.Define a linear maptH : IH ⊗V →Wby the formulatH( (a1, ..., am), v) = (a1f1(v), ..., amfm(v)).This map considered as an element of the tensor product I∗H⊗V ∗⊗W will be of considerableimportance in the sequel. We shall refer to it as the fundamental tensor of the configurationH.It is clear that the fundamental tensor tHas ∈(IHas)∗⊗I∗H ⊗W = V ∗⊗I∗H ⊗W of theassociated configuration Has is obtained from the fundamental tensor TH ∈I∗H ⊗V ∗⊗Wby interchanging of factors in the tensor product.In coordinates, fixing a basis e1, ..., en+1 in V and its dual basis in V ∗, let A =∥aij∥1≤i≤n+1, 1≤j≤m be the matrix whose columns are the coordinates of the linear func-tions fi and B = ∥bij∥1≤i≤m−n−1, 1≤j≤m be similar matrix for the associated arrangement.4

We can choose B in such a way that B ◦A = 0. Then the coordinates of the tensor tH aregiven by the formula(tH)ijk = bikakj.1.5.

Proposition. Suppose that H is in general position.

Then for any non-zero vectorv ∈V the linear operator tH(v) : IH →W defined by the fundamental tensor tH, isinjective.Proof: If (a1, ..., am) ∈Ker(tH(v)) then aifi(v) = 0 for all i = 1, ..., m. Let J = {i :fi(v) = 0}. Then for any i /∈J we have ai = 0.

Since H is in general position, |J| ≤n.Hence P aifi = 0 is a non-trivial linear relation between ≤n linear functions among fi.This contradicts the assumption of general position for H.5

§2. Logarithmic bundles.2.1.

Let H = (H1, ..., Hm) be an arrangement of m hyperplanes in P n = P(V ) in generalposition. We shall define the divisor S Hi also by H. This divisor has normal crossing.This means that for any point x ∈P n its local equation can be given by t1...tk = 0 wheret1, ..., tk is a part of a system of local parameters at x.

In this situation one can definethe sheaf Ω1P n(log H) of differential 1-forms with logarithmic poles along H, see [De]. Itis a subsheaf of the sheaf j∗Ω1U where U = P n −H and j : U ֒→P n is the embedding.If x ∈P n and t1...tk = 0 is a local equation of the divisor H near x, as above, then thesection of Ω1P n(log H) near x are meromorphic differential forms which can be expressedas ω + P uid log ti where ω is a 1-form and ui are functions , all regular near x.

It is notdifficult to see that the sheaf Ω1P n(log H) is locally free of rank n, see [De].We shall denote the sheaf Ω1P n(log H) by E(H) and call it the logarithmic bundleassociated to H. It will be the main object of study in this paper. We will not make adistinction between vector bundles and locally free coherent sheaves of their sections.2.2.

The sheaf E(H)∗dual to E(H) has a nice interpretation in terms of vector fields. Wesay that a regular vector field ∂defined in some open subset U ⊂P n is tangent to H if forany x ∈U the vector ∂(x) lies in the intersection of the tangent hyperplanes at x to all Hicontaining x (in particular, ∂(x) = 0 if x is a point of n -tuple intersection).

Such fieldsform a coherent subsheaf in the tangent sheaf TP n. It is easy to see by local calculationsthat this sheaf is isomorphic to the dual sheaf E(H)∗.2.3. Proposition.

Let ǫi : Hi ֒→P n be the embedding map. We have the canonicalexact sequence of sheaves on P n0 →Ω1P n →E(H)res→mMi=1ǫi∗OHi →0(2.1)where res is the Poincar´e residue morphism defined locally by the formulaa1d log t1 + ... + akd log tk + bk+1dtk+1 + ... + bndtn 7−→(a1(x), ..., ak(x), 0, ..., 0)where (t1, ..., tn) is a system of local coordinates at x such that t1...tk = 0 is a local equationof the divisor H at x.Proof: See [De].The next two propositions follow simply from the above exact sequence.6

2.4. Proposition.

The Chern polynomial c(E(H)) = P ci(E(H))ti of the bundle E(H)is given byc(E(H)) = (1 −ht)−m+n+1where h is the class of a hyperplane in P n. In particular, the determinant Vn E(H) isisomorphic to the line bundle O(m −n −1) on P n.2.5. Proposition.

a) The space H0(P n, E(H)) has dimension m−1 and consists of formsmXi=1αid log fi = d log(mYi=1f αii ),αi ∈C,Xαi = 0.b) More generally, dim H0(E(H)(k)) = (n + 1)k+n−1n−k+nn+ mk+n−1n−1.c) Hi(E(H)(k)) = 0 for 1 ≤i ≤n −2 and any k ∈Z.Note that we can now identify the space H0(P n, E(H)) with the space W introducedin n. 1.4.2.6. The logarithmic bundles can be obtained from the bundle Ω1P n by applying elementarytransformations of vector bundles.These transformations were introduced first in thecase of vector bundles over curves by A.Tyurin [T] and their general definition is due toMaruyama [M1-2].

Let us recall this concept.Let E be a rank r vector bundle over a smooth algebraic variety X and Z ⊂X -a hypersurface.Denote by i : Z →X the embedding.Suppose that we have chosensome quotient bundle F of the restriction i∗E. Then we have a surjective map of sheavesE →i∗F on X.

We define the coherent sheaf Elm−Z,F as the kernel of this surjection. Itis easy to see that it is locally free of rank r i.e., can also be regarded as a vector bundle.This bundle is called the elementary transformation of E along (Z, F).Note than when E is a line bundle and F = i∗E then Elm−Z,F is just the twisted sheafE(−Z).2.7.

The bundle E can be reconstructed from its elementary transformation by applyingthe ”inverse” elementary transformation Elm+Z,F. In the situation of n.2.6, the definitionof Elm+ is as follows.

Let E(Z) be the sheaf whose sections are sections of E with simplepoles along Z. Then Elm+Z,F (E) is a subsheaf of E(Z) whose sections after multiplyingby the local equation of Z belong to Elm−Z,F (E) = Ker{E →i∗F}.

It is easy to see thatElm+ and Elm−are mutually inverse operations.2.8. For any 1 ≤i ≤m let H≤i be the truncated arrangement (H1, ..., Hi).

By definition,H≤0 = ∅and E(H≤0) = Ω1P n. The residue exact sequence from n.2.4 induces the followingexact sequence0 →E(H≤i−1) →E(H≤i) →ǫi∗OHi →0.7

Passing to the dual exact sequence and using the adjunction formula we find the followingexact sequence0 →E(H≤i)∗→E(H≤i−1)∗→ǫi∗OHi(1) →0. (2.2)Thus, by definition, we obtain2.9.

Proposition. For each i ≤m we have isomorphismsE(H≤i−1) ∼= Elm−Hi,ǫi∗OHi (E(H≤i)),E(H≤i)∗∼= Elm−Hi,ǫi∗OHi(1)(E(H≤i−1)∗).It is the second isomorphism which will be useful for us in §5 later.2.10.

Proposition. Assume 1 ≤m ≤n + 1.

ThenE(H) ∼= (OP n)⊕(m−1) ⊕OP n(−1)n+1−m.Proof: Since m ≤n+1, we can choose homogeneous coordinates x1, ..., xn+1 in P n = P(V )such that the hyperplane Hi, 1 ≤i ≤m, is given by the equality xi = 0. By Serre’stheorem [H] coherent sheaves on P n correspond to graded C[x1, ..., xn+1] -modules (modulofinite-dimensional ones), the correspondence being given by F 7→L H0(P n, F(i)).

Weshall describe the module corresponding to E(H). Denote this module by M ′.

The ringC[x1, ..., xn+1] will be denoted shortly by A.Denote by ξ = P xi∂/∂xi the Euler vector field on V . By Lieξ and iξ we shall denotethe Lie derivative along ξ and the contraction of 1-forms with ξ.Let ˜H ⊂V be the configuration of coordinate hyperplanes {xi = 0}, i = 1, ..., m. LetM be the space of all global sections of the sheaf Ω1V (log ˜H) on V .

It is a graded A -module; the graded component Mr consists of forms ω such that Lieξω = r · ω.It is clear that the space of sections H0(P n, E(H)(p)) can be identified with the sub-space in Mr consisting of forms ω such that iξω = 0. Hence our module M ′ correspondingto E(H) is the kernel of the homomorphism M →A given by iξ.The graded A -module M is free: it is isomorphic to Am ⊕An−m(−1) with the basisd log x1, ..., d log xm, dxm+1, ..., dxn+1, the first m elements being in degree 0, the remainingones in degree 1.

Since iξ(d log xi) = 1, iξ(dxi) = xi, we find that M ′ is the kernel of thehomomorphismAm ⊕An−m(−1) →A,(a1, ..., an) 7→mXj=1aj +n+1Xj=m+1ajxj. (2.3)However, an element (a1, ..., an) from the kernel of (2.3) is uniquely determined by thecomponents (a2, ..., an) which may be arbitrary: we just define a1 to be equal −(Pmj=2 aj +8

Pn+1j=m+1 ajxj). This means that M is isomorphic to Am−1 ⊕A(−1)n−m+1 as a graded A- module.

Hence, by Serre’s theorem, E(H) ∼= (OP n)⊕(m−1) ⊕OP n(−1)n+1−m.2.11. The logarithmic bundles can be used to define a map from the projective space toa Grassmannian with the image isomorphic to a Veronese variety.

Let us explain this inmore detail.By a Veronese variety we mean a subvariety in a projective space P(n+dn )−1 which isprojectively isomorphic to the image of the Veronese mappingvn,d : P n = P(V ) →P(n+dn )−1 = P(SdV ). (2.4)Let E be a vector space of dimension n + d and G(n, E) – the Grassmannian of n di-mensional linear subspaces in E. We shall often identify it with the Grassmannian D(d, E∗)of d - dimensional subspaces in the dual spaces E∗.

Consider its Pl¨ucker embeddingG(n, E) ֒→P(n^E) = P(n+dn )−1. (2.5)Note that the dimensions of the ambient spaces for the Pl¨ucker embedding and the Veroneseembedding coincide.

Therefore it makes sense to speak about n -dimensional Veronesevarieties in the Grassmannian G(n, E). The following result, proven in [K], shows that thelogarithmic bundle E(H) defines an embedding of P n into a Grassmannian whose imageis a Veronese variety.2.12.

Theorem. Let H be an arrangement of m ≥n + 2 hyperplanes in P n in generalposition.

Denote by W the space H0(P n, E(H)) ∼= Cm−1. For any point x ∈P n considerthe subspace of W consisting of all sections vanishing at x, and let φH(x) be the dualsubspace of W ∗.

Then:a) The dimension of φH(x) equals n for all x ∈P n;b) The correspondence x 7→φH(x) is a regular embedding φH : P n ֒→G(n, W)∗.c) The image φH(P n) in G(n, W ∗) becomes, after the Pl¨ucker embedding G(m−n−1, W) ⊂P(Vn W ∗), a Veronese variety.In particular, E(H) is the inverse image of the bundle S∗on G(n, W ∗) where S is thetautological subbundle over G(n, W ∗).2.13. Corollary.

Assume that m = n + 2. Then E(H) ∼= TP n(−1) where TP n is thetangent bundle of P n.Proof: Since dim(W) = n + 1, the map φH defines an isomorphism P n = P(V ) →G(n, W ∗) = G(1, W) = P(W).

In this case the tautological subbundle S on G(n, W ∗) isisomorphic to Ω1P (W )(1). Hence E(H) is isomorphic to TP n(−1).9

2.14. Let us call a rank n vector bundle E on P n normalized if c1(E) ∈{0, −1, ..., −n +1}.

If E is any rank n vector bundle E on P n and c1(E) = na + b where a ∈Z, b ∈{0, −1, ..., −n + 1} then we denote by Enorm the normalized bundle E(−a).In our case c1(E(H)) = m −n −1 so the normalized bundle Enorm(H) has the formE(H)(−d+1), where m = 1+nd+r, 0 ≤r ≤n−1. Its first Chern class equals r −n.

Thecase when the first Chern class of the normalized bundle is zero i.e. when m = nd +1, willplay a special role for us since many results below rely on a good theory of jumping linesfor bundles with c1 = 0, see [Bar].10

§3. Steiner bundles.Vector bundles of logarithmic forms turn out to belong to a more general class ofbundles remarkable for the existence of a very simple resolution.3.1.

Definition. A vector bundle E on P n = P(V ) is called a Steiner bundle if E admitsa resolution of the form0 →I ⊗OP n(−1)τ−→W ⊗OP n →E →0(3.1)where I and W are vector spaces identified with the corresponding trivial vector bundles.The bundles of this type were considered earlier by several people, see [E][BS].

Thename ”Steiner bundles” will be explained later in this section.Note that applying the exact sequence of cohomology, we immediately obtainW ∼= H0(P n, E),I ∼= H0(P n, E ⊗Ω1P n(1)).(3.2)3.2. Proposition.

A vector bundle E be Steiner bundle if and only if the cohomologygroups Hq(P n, E⊗Ωp(p)) vanish for all q > 0 and also for q = 0, p > 1. The resolution (3.1)is defined functorially in E. More precisely, the tensor t is the only non-trivial differentiald−1,01: E−1,01→E0,01of the Beilinson spectral sequence with the first termEpq1 = Hq(P n, F ⊗Ω−p(−p)) ⊗O(p) eqno(2.3)converging to E in degree 0 and to 0 in degrees ̸= 0.The proposition follows easily from considering the Beilinson spectral sequence, see[E], Proposition 2.2.3.3.

Corollary. The property of being a Steiner bundle is an open property.3.4.

A map τ between sheaves I ⊗OP n and W ⊗OP n, an in (2.1), is uniquely determinedby a tensort ∈Hom (V, Hom (I, W)) = V ∗⊗I∗⊗W. (3.3)This tensor should be such that the map τ is fiberwise injective.Thus we see that the fundamental tensor tH of an arrangement of hyperplanes in P n(see n.1.4) allows one to define a coherent sheaf as the cokernel of the mapτH : IH ⊗OP n(−1) →W ⊗OP n.(3.4).Here the spaces I = IH and W are defined in nn.

1.2 and 1.4 respectively. It turns outthat this sheaf is isomorphic to our logarithmic bundle E(H).11

3.5. Theorem.

Let H be an arrangement of m hyperplanes in general position in P(V ).Suppose that m ≥n + 2. Then the logarithmic bundle E(H) is a Steiner bundle.

Thecorresponding tensor is the fundamental tensor tH of the configuration H.Proof: Let v ∈V be a non-zero vector. The fiber of the map (3.4) over the point Cv ∈P(V )has, in the notation of §1 the formtH(v) : IH →W,(a1, ..., am) 7→(a1f1(v), ..., amfm(v)).

(3.5)To prove our theorem, we shall construct, for any v, an explicit isomorphism betweenCoker tH(v) and the fiber at Cv of the bundle E(H). Consider the map of vector spacesπv : W →E(H)Cv,(a1, ..., am) 7→Xai(d log fi)|Cv,(3.6)where E(H)Cv is the fiber of E(H) at Cv.

It follows from Theorem 2.12 a) that πv is asurjection. Thus our theorem is a consequence of the following lemma.3.6.

Lemma. We have Ker πv = Im tH(v).

In other words, a section P λid log fi, P λi =0 of the bundle E(H) vanishes at Cv if and only if λi = aifi(v) for some (a1, ..., am) ∈IH.Proof: It suffices to show that Im tH(v) ⊂Ker πv since the spaces in question have thesame dimension.Let J = {i : Cv ∈Hi} and HJ = Ti∈J Hi. A section ω of E(H) vanishes at Cv if andonly if ω is regular near Cv as a 1-form and, moreover, vanishes on the tangent subspaceto HJ.Suppose that λi = aifi(v) where (a1, ..., am) ∈IH.

Then for i ∈J we have λi = 0since fi = 0 on Hi. Hence the form ω = P λid log fi is regular at Cv.

Let ξ ∈V be suchthat fi(ξ) = 0 for i ∈J i.e. ξ represents a vector tangent to HJ at Cv.

Then the value ofω on this tangent vector equalsXi/∈Jλifi(ξ)fi(v) =Xi/∈Jaifi(v)fi(ξ)fi(v)=Xi/∈Jaifi(ξ) =mXi=1aifi(ξ) = 0.This proves the lemma and hence Theorem 3.5.Let us mention that is possible to give an alternative proof of Theorem 3.5 by usingProposition 3.2.3.7. Let E be a rank r Steiner bundle on P n given by the resolution (3.1).

We have noticedalready that W = H0(P n, E). It is obvious that E is generated by its global sections.Hence we obtain a regular map γ : P n →G(r, W ∗) that takes a point x ∈P n into thedual of the subspace of sections vanishing at x.

This map can be defined ”synthetically”by means of the following ”Grassmannian Steiner construction” [K].12

Let m = dim(W)+1 so that dim(I) = m−1−r. Take m−r −1 projective subspacesL1, ..., Lm−r−1 in the projective space P(W ∗), each of codimension n + 1.Denote by]Li[ the ”star” of Li i.e.

the projective space of dimension n formed by hyperplanes inP(W ∗) containing Li. Identify all the stars ]Li[ with each other by choosing projectiveisomorphisms φi : P n →]Li[.

Suppose that for any x ∈P n the corresponding hyperplanesφi(x) are independent. Consider the locus of subspaces in P(W ∗) of codimension m−r−1(i.e.

of dimension r −1) which are intersections of the corresponding hyperplanes fromstars ]Li[ i.e. the subspaces of the formφ1(x) ∩... ∩φm−r−1(x), x ∈P n.(3.7)This locus lies in G(n, W ∗).

It is called the Grassmannian Steiner construction.Thisis s straightforward generalization of the classical Steiner construction of rational normalcurves, see [GH], Ch.4, §3.The following proposition shows that this construction isequivalent to that of Steiner bundle. This explains the name.3.8.

Proposition. Let X be a projective space of dimension n embedded in some wayinto the Grassmannian Gn(W) of codimension n subspaces in W, dim(W) = m −1.

LetQ be the rank n bundle on Gn(W), whose fiber over a subspace L ⊂W is W/L. Let E bethe restriction of Q to X.

The possibility of representing X by the Grassmannian Steinerconstruction is equivalent to the fact that E is a Steiner bundle.Proof: The choice of m−n−1 parametrized star (]Li[, φi : P n = P(V ) →]Li[) is equivalentto the choice of m−n−1 surjective linear operators ai : W ∗→V ∗. Namely, given such ai,we associate to any hyperplane in V ∗i.e.

to any point of P(V ) its inverse image under ai.Thus a point π ∈Gn(W) corresponding to x ∈P(V ) is T Ker(ai(x)). Define a linear mapA : Cm−n−1 →Hom (W ∗, V ∗) by setting ai = A(ei) where e1, ..., em−n−1 is the standardbasis of Cm−n−1.

Denote the space Cm−n−1 by I. This defines a tensor t ∈I∗⊗W ⊗V ∗which, in its turn, defines a morphism of sheaves I ⊗OP (V )(−1) →W ⊗OP (V ).

Ourbundle E must be the cokernel of this morphism. Indeed, dualizing, we have to show thatE∗is the kernel of the dual map W ∗⊗OP (V ) →I∗⊗OP (V )(1).

This is defined by alinear map A† : V →Hom (W ∗, I∗) associated to t. The fiber of this bundle over a pointCv of P(V ) is equal to the kernel of the linear map A†(v) : L∗→I∗. The latter is dualto the point of X ⊂Gn(W) corresponding to x.

This identifies the fibres. The conversereasoning is obvious.3.9.

Proposition [E]. The rank of a non-trivial Steiner bundle on P n is greater or equalto n.Proof: Let r be the rank.

A Steiner bundle is given by a linear map V →Hom (I, W)where dim(I) = dim(W) −r. Let D be the subvariety of Hom (I, W) consisting of linear13

maps of not maximal rank. It is well known that its codimension equals r+1 (see [ACGH],p.67).

Therefore if V is of dimension > r + 1 every linear map t : V →Hom (I, W) willmap some non - zero vector v ∈V to a matrix of not maximal rank. This does not occurfor Steiner bundles.Thus logarithmic bundles provide examples of Steiner bundles of maximal possiblerank.

In the rest of this section we shall consider only rank n Steiner bundles on P n.3.10.Recall that a vector bundle E is called stable if for any torsion - free coherentsubsheaf F ⊂E we havedeg(F)/rk(F) < deg(E)/rk(E).It is well known that the property of stability is preserved under tensoring with invertiblesheaves.The following fact is a particular case of results of Bohnhorst and Spindler ([BH],Theorem 2.7).3.11.Theorem. Any non-trivial Steiner bundle on P n is stable.3.12.

Proposition. Let E be a non-trivial rank 3 Steiner bundle on P 3 with c1(E) = 3k(i.e.

dim(W) = 3k + 3). Then the normalized bundle Enorm = E(−k) is an instantonbundle on P 3 i.e.

c1(Enorm) = 0 and H1(P 3, Enorm(−2)) = 0.Proof: Follows at once from the resolution (3.1).3.15. Corollary.

Let H be a configuration of m hyperplanes in P n in general positionwith m > n + 2. Then:a) The logarithmic bundle E(H) is stable.b) If n = 3 and m = 3d + 1 then the normalized bundle Enorm(H) = E(H)(−d + 1) is aninstanton bundle on P 3.Denote by MP 2(a, b) the moduli space of stable rank 2 vector bundles on P 2 withc1 = a, c2 = b.

It is known to be an irreducible algebraic variety of dimension 4b −a2 −3,see [OSS], Ch.2, §4. Note that MP 2(a, b) is isomorphic to the moduli space of normalizedbundles namely to MP 2(0, (4b −a2)/4) for a even and to MP 2(−1, (4b −a2 + 1)/4) if a isodd.3.16.

Corollary. If a = m −3, b =m−22for some m then the moduli space MP 2(a, b)contains a dense Zariski open subset consisting of Steiner bundles.In other words, a generic stable bundle with these Chern classes is a Steiner bundle.14

Proof: The property of being a Steiner bundle is open (Corollary 3.3). The said modulispace indeed contains Steiner bundles - the logarithmic bundles corresponding to configu-rations of m lines: they are stable by Theorem 3.11 and have the required Chern classesby Proposition 2.2.

Since the moduli space is irreducible, we are done.Let us reformulate the above corollary in terms of normalized bundles.3.17. Corollary.

For any d > 0 each of the moduli space MP 2(0, d(d −1)), MP 2(−1, (d −1)2) has an open dense subset consisting of twisted Steiner bundles.3.18. Notice that dimMP 2(3, 6) = dimMP 2(−1, 4) = 12.

On the other hand, arrangementsof 6 lines in P 2 also depend on 12 parameters. We will show later that the map H 7→E(H)from the space of arrangements of 6 lines to the moduli space dimMP 2(3, 6) is genericallyinjective.

This will show that a generic bundle from MP 2(3, 6) is a logarithmic bundleassociated to an arrangement of 6 lines in P 2.3.19.The operation of association discussed in section 1 can be extended to Steinerbundles. Namely, we can view the defining tensor (3.3) as a linear map V ∗⊗I∗→W andconsider the corresponding mapτ ′ : V ⊗OP (I)(−1) →W ⊗OP (I)of vector bundles on the projective space P(I).3.20.

Proposition - Definition. The map τ ′ is injective on all the fibers if and only if τis.

In this case the Steiner bundle Coker(τ ′) is said to be associated to the Steiner bundleE = Coker(τ) and denoted by Eas.Proof: The condition that τ is not fiberwise injective means that there are non-zero v ∈V, i ∈I such that t(v ⊗i) = 0. The same condition is equivalent to the fact that τ ′ is notfiberwise injective.

This proves the ”proposition” part.The next proposition follows immediately from definitions of n.1.4 and its proof is leftto the reader.3.21. Proposition.

Let H be an arrangement of hyperplanes in P(V ) in general positionand Has be its associated arrangement in P(I). Then there is a natural isomorphism ofvector bundlesE(Has) ∼= (E(H))as.15

§4. Monoids, codependence and monoidal complexes.In this section we describe some constructions of projective geometry which will beused in the study of logarithmic bundles, more precisely, in the description of jumpinglines for such bundles.4.1.

We shall work in projective space P n with homogeneous coordinates x0, ..., xn. Pro-jective subspaces in P n will be shortly called flats.

For a subset S ⊂P n let < S > denotethe flat (projective subspace) spanned by S. In particular, for two points p ̸= q ∈P n thenotation < p, q > means the line through p and q.Let X be a hypersurface in a smooth algebraic variety Y and x ∈X be a point. Wesay that x is a k -tuple point of Y i f the whole (k −1) -st infinitesimal neighborhoodx(k−1) ⊂Y is contained in X.As usual, if L is a line bundle on a projective variety X, we shall denote by |L|the complete linear system of divisors on X formed by zero loci of sections of L i.e.|L| = P(H0(X, L)).4.2.

Let Z ⊂P n be an irreducible variety. A hypersurface X ⊂P n of degree d is called aZ -monoid if each point of Z is a (d −1) -tuple point of X.

For example, a Z -monoid ofdegree 2 is just a quadric containing Z.We denote by Md(Z) the projective subspace of |OP n(d)| formed by all Z -monoidsof degree d.We shall be mostly interested in the case when Z ⊂P n is a flat. In this case, denotingc = codim Z, we find by easy calculation, thatdim Md(Z) =c + d −2d −1(n −c −1) +c + d −1d−1.

(4.1)In particular, if codim Z = 2 then dim Md(Z) = nd.4.3. Proposition.

Let Z ⊂P n be a flat of dimension k ≤n −2. Any Z -monoid is arational variety ruled in P k’s.Proof: Projecting X from the subspace Z we find a rational map to P n−k−1 whose fibersare flats of dimension k. Indeed, take any (k+1) -dimensional flat L containing Z.

Then Zis a hyperplane in L. The intersection L ∩X is a hypersurface of degree d in L containingd −1 times the hyperplane Z. This means that L ∩X = (d −1)Z + H(L) where H(L) issome hyperplane in L. So H(L) is the fiber of the said rational map over L, as claimed.4.4.

For any flat Z ⊂P n we denote by ]Z[ the star of Z i.e. the projective space ofhyperplanes containing Z, cf.

n.3.8. Obviously dim ]Z[= codim Z −1.16

Assume that codimZ = 2. There is a simple way to construct irreducible Z -monoidsof degree d by means of the classical Steiner construction.

Take any point x ∈P n −Z andany regular mapψ : ]Z[ ∼= P 1 →]x[ ∼= P n−1of degree d −1 ,i.e. a map given by a linear subsystem of |OP 1(d −1)|.

Denote by H theunique hyperplane containing Z and x and assume that ψ(H) ̸= H. Consider the varietyX(Z, x, ψ) which is the union of codimension 2 flats L ∩ψ(L), L ∈]Z[. We claim that thisis a Z -monoid of degree d containing the point x.In fact, take a line l which has no common points with Z ∪{x}.

Then ]Z[ is identifiedwith l by the correspondence taking H ∈]Z[ to the intersection point xH = H ∩l. Themap ψdefines a degree d −1 map f : l →l defined as follows: xH 7→ψ(H) ∩l.

Thegraph of this map Γf ⊂l × l ∼= P 1 × P 1 intersects the diagonal in d points. Thereforel intersects X(Z, x, ψ) at d points so deg X(Z, x, ψ) = d. To see that X(Z, x, ψ) is a Z-monoid, we take any general hyperplane P containing Z.

The intersection X(Z, x, ψ) ∩Pis a hypersurface Y in P of degree d which set-theoretically is the union of Z and anotherhyperplane in P,the latter entering with multiplicity 1. This implies that Z enters withmultiplicity (d −1) into Y so Z consists of (d −1) -tuple points of X(Z, x, ψ).Conversely, every Z -monoid containing a point x outside Z is equal to Z(Z, x, ψ) forsome regular map ψ of degree d from ]Z[ to ]x[.

This follows from the proof of Proposition4.3. So we have proving the following fact.4.5.

Proposition. Let Z ⊂P n be a codimension 2 flat and x ∈P n −Z.

Then there is abijection between the set of irreducible Z -monoids of degree d containing x and the set ofregular maps ψ : ]Z[ →]x[ of degree d −1 with the property that ψ(< Z, x >) ̸=< Z, x >.4.6. We are going to relate monoids to a property of point sets in projective spaces whichwe call codependence.Let p = (p1, ..., pr) be an ordered r -tuple of points in P n−1 and q = (q1, ..., qr) bean ordered r -tuple of points i n P 1.

We say that p and q are d -codependent if there is ahypersurface Y ⊂P n−1 × P 1 of bi-degree (1, d) which contains the points (xi, yi). We saythat p and q are strongly d -codependent if there is an irreducible such hypersurface.4.7.

Proposition. Let (Λ1, ..., Λr) be an ordered r - tuple of hyperplanes in P n−1 and(q1, ..., qr) be an ordered r -tuple of points in P 1.

Let us regard each Λi as a point pi in thedual projective space ˇP n−1. Then p = (p1, ..., pr) and q = (q1, ..., qr) are d -codependent(resp.

strongly d -codependent) if and only if there is a regular map ψ : P 1 →P n−1 ofdegree ≤d (resp. of degree exactly d) such that ψ(qi) ∈Λi.Proof: Suppose that p and q are s trongly d -codependent.Let Y be an irreduciblehypersurface of bi-degree (1, d) in ˇP n−1 × P 1 containing all (pi, qi).

Denote by F(u, v) =17

F(u0, ..., un−1; v0, v1) the bi-homogeneous equation of Y . We obtain, for any v = (v0, v1)a linear form u 7→F(u, v) on P n−1.

Since Y is irreducible, this form is non-zero and so itskernel is a hyperplane, denoted ψ(v), in ˇP n−1 i.e. a point in the initial P n−1.

This givesthe desired map map ψ from P 1 to P n−1. The rest of the proof is obvious.4.8.

Corollary. Let Z ⊂P n be a codimension 2 flat and q, p1, ..., pr be points in P n −Z.The following conditions are equivalent:(i) There exists a Z -monoid of degree d (resp.

an irreducible Z -monoid of degree d)containing q, p1, ..., pr. (ii) There exists a regular map ψ : ]Z[ →]q[ of degree ≤d −1 (resp.

of degree exactlyd −1) such that ψ(< Z, pi >) contains the line < q, pi >. (iii) The collection of points < q, pi >∈]q[ ∼= P n−1 and < Z, pi >∈]Z[ ∼= P 1 are (d −1)-codependent (resp.

strongly (d −1) -codependent).The proof is immediate.4.9.Proposition. For r = nd all (d −1) - codependent pairs of nd - tuples(x1, ..., xnd, y1, ..., ynd) form a hypersurface Ξ in (P n−1)nd × (P 1)nd.

Let xi0, ..., xi,n−1 bethe homegeneous coordinates of the point xi ∈P n−1 and let yi0, yi1 be the homogeneouscoordinates of the point yi ∈P 1. The equation of Ξ is the determinant of size nd × ndwhose i -th row is the following vector of length nd:(xi0yd−1i0, xi0yd−2i0yi1, ..., xi0yd−2i1; xi1yd−1i0, xi1yd−2i0yi1, ..., xi1yd−2i1; ......; xi,n−1yd−1i0, xi,n−1yd−2i0yi1, ..., xi,n−1yd−1i1).Proof: Let V be the space of polynomials in (x0, ..., xn−1, y1, y2) homogeneous of degree1 in xi and of degree d −1 in yj.

Then dim(V ) = nd. The entries of the determinant inquestion are the values of nd monomials forming a basis of V , on our nd points.

So thevanishing of the determinant is equivalent to the linear dependence of the vectors given bythese values i.e. to the (d −1) -codependence of (xi) and (yi).4.10.

Let p1, ..., pnd+1 be nd + 1 points in P n in general position. The monoidal complexC(p1, ..., pnd+1) is, by definition the locus of all the codimension 2 flats Z ⊂P n for whichthere exists a Z -monoid of degree d containing p1, ..., pnd+1.According to classical terminology of Pl¨ucker, by complexes one meant 3-parametricfamilies of lines in P 3 i.e.

a hypersurface in the Grassmannian G(2, 4). As we shall see,our C(p1, ..., pnd+1) is a hypersurface in the Grassmannian G(n −1, n + 1).

This explainsthe word complex.Monoidal complexes will be used in the next section to describe jumping lines oflogarithmic vector bundles.18

4.11. Theorem.

Let G = G(n −1, n + 1) be the Grassmannian of codimension 2 flats inP n. Then:a) For any points p1, ..., pnd+1 ∈P n in general position the variety C(p1, ..., pnd+1) iseither the whole G or a hypersurface in G. In the latter case its degree (i.e. the degreeof any equation in Pl¨ucker coordinates defining this hypersurface) equals nd(d −1)/2.b) Any codimension 2 flat Z containing one of the points pi belongs to C(p1, ..., pnd+1)and, moreover, is a (d −1) - tuple point of this variety.The proof of this theorem will be organized as follows.

As a first step we shall analyzethe equation of C(p1, ..., pnd+1) and find its degree. Second step will be to prove part b)of the theorem, again by using the equation.

These three steps will be done in nn. 4.12and 4.13 respectively.4.12.

We shall define codimension 2 flats by pairs of linear forms whose coefficients areput into rows of a by 2 × (n + 1) matrixA =a10a11...a1na20a21...a2n.The flat corresponding to A will be denoted Z(A). Its Pl¨ucker coordinates are just 2 by2 minors of A.

A representation of a flat Z as Z(A) gives a parametrization of the pencil]Z[ of hyperplanes through Z i.e. an explicit identification ]Z[= P 1.

Explicitly, to a point(t1, t2) ∈P 1 we associate the hyperplane given by the equationnXj=0(t1a1j + t2a2j)xj = 0.Denote the last point pnd+1 by q. We can choose a coordinate system in P n in such a waythat q has coordinates (1 : 0 : ... : 0).

The projective space ]pnd+1[ of lines through q isidentified with P n−1. Explicitly, if p = (b0 : ... : bn) ∈P n is another point then the line< q, p > has homogeneous coordinates (b1, ..., bn).Let bij, j = 0, ..., n, be the homogeneous coordinates of the point pi ∈P n, i = 1, ..., nd.The hyperplane < Z(A), pi >∈]Z(A)[ has, under the above identification ]Z(A)[ = P 1,the homogeneous coordinates (Pj a1jbij, Pj a2jbij).Applying Corollary 4.8, we find that a flat Z(A) belongs to the variety C(p1, ..., pnd+1)if and only if the two nd -tuples of points((bi1, ..., bin) ∈P n−1, i = 1, ..., nd)and((Xja1jbij,Xja2jbij) ∈P 1, i = 1, ..., nd)are (d −1) -codependent.

Substituting them into the determinant of Proposition 4.9, wefind an equation on matrix elements aij whose degree in these elements equals nd(d −1)19

(since each entry of the determinant will have degree d−1 in aij). The Pl¨ucker coordinates,being 2 by 2 minors of A, have degree 2 in aij.

Hence the degree of equation in Pl¨uckercoordinates equals nd(d −1)/2 as required.4.13. Let us prove part b) of Theorem 4.11.

By symmetry it suffices to prove that ifpnd+1 ∈Z then Z is a (d −1) -tuple point of C(p1, ..., pnd+1). Let us keep the notationsand conventions introduced in the proof of part a), in particular, assume that pnd+1 = (1 :0; ... : 0).

A flat Z(A) contains pnd+1 if and only if a10 = a20 = 0.First, let us prove that such Z(A) lies in C(p1, ..., pnd+1). By Corollary 4.8, this meansthat the collections of points(bi1 : ... : bin) ∈P n−1and(ti, si) = (nXj=1a1jbij,nXj=1a2jbij) ∈P 1, i = 1, ..., ndare always (d −1) -codependent.

To show this, we construct a polynomial F(b1, ..., bn, t, s)homogeneous of degree 1 in b1, ..., bn and of degree (d−1) in t, s such that for all i we haveF(bi1, ..., bin, ti, si) = 0 . In fact, we can construct at least d −1 linearly independent suchpolynomials, namelyFm(b1, ..., bn, t, s) = (nXj=1a2jbj)tmsd−1−m −(nXj=1a1jbj)tm−1sd−m,m = 1, 2, ..., d −1.The possibility of finding d −1 such polynomials means that the kernel of nd × nd -matrixwhose determinant, by Proposition 4.9, defines C(p1, ..., pnd+1), has dimension ≥d −1.Since matrices with such properties are (d −1) -tuple points of the variety of degeneratematrices, we find that Z(A) is a (d −1) - tuple point of C(p1, ..., pnd+1).So we have proven all the assertion of Theorem 4.11 except the fact that the equationof the monoidal complex cannot be identically zero.

This will be proven in §5.Unfortunately, we do not know whether the case C(p1, ..., pnd+1) = G occurs. It doesnot occur if n = 2 (see Corollary 5.4).

Another case when this does not occur is as follows.Proposition 4.14. If d ≤3 and n is arbitrary then for any nd+1 points p1, ..., pnd+1 ∈P nin linearly general position the monoidal complex C(p1, ..., pnd+1) does not coincide withthe whole Grassmannian G.Proof: Suppose the contrary i.e., that for any codimension 2 flat Z there is a Z -monoid Xof degree d through p1, ..., pnd+1.

If we take Z to lie in the hyperplane H =< p1, ..., pn >then we find that any such monoid X should be the union of H and some Z -monoid ofdegree d −1 through pn+1, ..., pnd+1. Let H′ =< pn+1, ..., p2n >.

We take Z = H ∩H′.By the above the corresponding monoid should be the union of H, H′ and a Z -monoid of20

degree d−2 through the remaining points p2n+1, ..., pnd+1. If d = 2 this means that 2n+1generic points p1, ..., p2n+1 lie onthe union of two hyperplanes H ∪H′, which is impossible.If d = 3, the above means that n + 1 points p2n+1, ..., p3n+1 lie on a monoid of degree 1i.e.

on a hyperplane, which is also impossible.4.15. Examples in P 2.

Consider first the case n = 2. Then each 2d + 1 points in P 2in general position define the monoidal complex C(p1, ..., p2d+1).

It is a curve of degreed(d −1) with (d −1) -tuple point at each pi. Let us consider some particular cases.a) Let d = 2.

Then the curve C(p1, ..., p5) is just the unique conic through the pointspi.b) Let d = 3. Then C(p1, ..., p7) is a curve of degree 6 and genus 3 with double pointsat pi.

By definition, it is the locus of all possible singular points of cubics through p1, ..., p7.This curve has the following (classical, see [DO]) description.Let Sσ→P 2 be the blow-up of P 2 in p1, ..., p7.This is a Del Pezzo surface.Itsanticanonical linear system has dimension 2 and defines a double cover Sτ→˜P 2 (this ˜P 2 isdifferent from the first one) ramified along a plane quartic curve C′ ⊂˜P 2. We claim thatC′ is birationally isomorphic to C(p1, ..., p7).Indeed, the anticanonical linear system of P 2 consists of cubic curves.

The curves ofthe anticanonical linear system of the blown-up surface S can be viewed, after projectionSσ→P 2, as plane cubics through p1, ..., p7. Denote this linear system by L ∼= P 2.

Thesecond projective plane ˜P 2 is the space of lines in L. The projection τ associates to a pointp ∈S with projection z = σ(p) ∈P 2 the set of all plane cubics through p1, ..., p7 whichalso meet p (so this set is a line in L i.e. a pencil of cubics).

All the cubics from this pencilalso contain some ninth point p′ which is conjugate to p with respect to the double coverτ. The map τ ramifies at p when p′ = p i.e.

the cubics from L have a node at z = τ(p).c) Let d = 4. Then C(p1, ..., p9) is a curve of degree 12 with triple points at p1, ..., p9.Its genus equals 28.

We de not know any special geometric significance of this curve.4.16.Examples in P 3.Let us consider the case n = 3.The monoidal complexC(p1, ..., p3d+1) is a line complex of degree 3d(d −1)/2.a) For d = 2 i.e. for 7 points in P 2 we get the so-called Montesano complex.

It consistsof lines in P 3 which lie on a quadric passing through points p1, ..., p7. It is not difficult tosee that this complex (which is a threefold in G(2, 4)) is isomorphic to a P 1 -bundle overa Del Pezzo surface of degree 2.

This latter surface is obtained by blowing up the sevenpoints of P 2 corresponding to p1, ..., p7 ∈P 3 by association (see n.1.2 above). We refer to[Mo] for more details about the geometry of this complex.b) Let d = 3.

The complex C(p1, ..., p9) is a complex of degree 9 consisting of lineswhich appear as double lines of cubic surfaces through p1, ..., p9.21

4.17. Examples in P n. We consider only the case d = 2 i.e.

of 2n+1 points in P n. Thiscase gives a complex of codimension 2 flats which can be called the generalized Montesanocomplex. We consider all quadrics in P n through p1, ..., p2n+1 and pick those among themwhich contain a P n−2 i.e.

quadrics of rank ≤4. The union of all (n −2) -flats on all thequadrics of rank ≤4 through pi gives a hypersurface in the Grassmannian G(n −1, n + 1)whose degree equals n. This is our complex.Note the case when all pi lie on a rational normal curve C in P n of degree n. In thiscase the generalized Montesano complex will be the locus of all (n−2) -flats intersecting thecurve C. Its equation will be the Chow form of C i.e.

the resultant of two indeterminatepolynomials of degree n. This is a consequence of the following easy fact.4.18. Lemma.

Let C ⊂P n be a rational normal curve and Z a codimension 2 flat in P n.Then the two conditions are equivalent:(i) There exists a quadric through C and Z. (ii) C ∩Z ̸= ∅.Proof: (ii)⇒(i): Let x ∈C ∩Z.

Let |O(2)| be the linear system of all quadrics in P n. Weconsider three projective subspaces L, M, N ⊂|O(2)| consisting respectively of quadricscontaining C, containing Z and containing x. Then L, M ⊂N.

The codimension of Lin the whole |O(2)| is 2n + 1 and hence its codimension in N is 2n. The space M hasdimension 2n.

Hence L ∩M ̸= ∅so there exists a quadric with required properties. (i)⇒(ii): Let Q be a quadric containing C ∪Z.

Then C and Z are subvarieties in Qof complementary dimensions. In the case n > 3 (as well as in the case when n = 3 and Qis singular) this alone implies that the intersection is non-empty.

If n = 3 and Q is smooththe non-emptiness follows from the fact that C regarded as a curve on Q = P 1 × P 1 hasbidegree (1,2) or (2,1). The case n = 2 is trivial.22

§5. Monoidal complexes and splitting of logarithmic bundles.5.1.

One of the main tools for the study of vector bundles on P n is the restriction ofbundles to projective subspaces to P n especially to lines. By Grothendieck’s theorem anyvector bundle on P 1 splits into a direct sum of line bundles L OP 1(ai).In this section we use this approach for logarithmic bundles E(H) where H = (H1, ...,Hm) is an arrangement of m hyperplanes in P n in general position.Let us write thenumber m in the form m = nd + 1 + r where d, r are integers and 0 ≤r ≤n −1.

Calla line l in P n a jumping line for E(H) (or for H, if no confusion arises), if the restrictionE(H)|l is not isomorphic to Ol(d)⊕r ⊕Ol(d −1)⊕(n−r).Of special interest for us will be the case m = nd + 1. In this case the normalizedbundle E(H)norm = E(H)(−d + 1) has first Chern class 0.

A line l will be in this casea jumping line for H if the restriction E(H)norm|l is non - trivial i.e. not isomorphic toO⊕nl.The main result of this section is as follows.5.2.

Theorem. Suppose m = nd + 1.

Let p1, ..., pnd+1 be points in ˇP n correspondingto hyperplanes H1, ..., Hnd+1 ∈H.For any line l ⊂P n let ]l[ be the correspondingcodimension 2 flat in ˇP n. Then a line l ⊂P n is a jumping line for the bundle E(H) ifand only if the flat ]l[ belongs to the monoidal complex C(p1, ..., pnd+1). In particular, thelocus of jumping lines of E(H) is the support of a divisor in the Grassmannian G(2, n + 1)of degree nd(d −1)/2.5.3.

Corollary. Assume n = 2.

Then the monoidal complex C(p1, ..., p2d+1) (which is inthis case a subvariety in the dual plane ˇP 2), does not coincide with the whole ˇP 2.Proof: This is a consequence of Theorems 5.2, 3.11 and of the Grauert - M¨ulich theorem[OSS] which implies that the locus of jumping lines oif a stable rank 2 bundle on P 2 is infact a curve.5.4. Corollary.

Assume that d ≤3. Then for any configuration H of nd + 1 hyperplanesH1, ..., Hnd+1 ⊂P n in general position the locus of jumping lines of the bundle Enorm(H)does not coincide with the whole Grassmannian.Proof: This follows from Proposition 4.14.Applying Corollary 4.8, we can give an equivalent, more geometric description of theproperty of a line to be jumping.5.5.

Corollary. Suppose m = nd + 1.

Let l ⊂P n be a line intersecting the Hi in distinctpoints. Then l is a jumping line for H if and only if there is a regular map ψ : l →Hnd+123

of degree ≤d −1 such that ψ(l ∩Hi) ∈Hnd+1 ∩Hi for i = 1, ..., nd.This reformulation is asymmetric: one of the hyperplanes, namely Hnd+1, acts as a“screen”. Of course, any other Hi can be chosen for this role.5.6.

Let E be a vector bundle on P n. We say that E is projectively trivial if E ∼= OP 1(a)⊕bfor some a ∈Z, b ∈Z+. In this case the projective bundle P(E) is trivial and, moreover,canonically trivialized.

To get the trivialization we note that P(E) = P(E(−a)). If W isthe space of sections of E(−a) then E(−a) is canonically identified with W ⊗OP n. HenceP(E) is canonically identified with P n × P(W).

For any two points x, x′ ∈P n we get theidentification of fibresΨE,x,x′ : P(Ex) →P(W) →P(Ex′).We shall call this system of identifications the canonical canonical projective connection ofthe projectively trivial bundle E.In our situation of logarithmic bundles it follows that whenever m = nd + 1 and l isnot a jumping line for H, we get a canonical projective connection in the restricted bundleE(H)|l. We are going to describe this connection explicitly.Note that the fiber of the bundle E(H)∗= TP n(log H) at any point x ∈P n not lyingon any Hi is identified with the tangent space TxP n. Therefore the fibre P(E(H∗)x) iscanonically identified with the projective space P n−1xof all lines through x.

This meansthat for any non-jumping line l the projective connection gives us isomorphisms which wedenoteΦH,l,x,x′ : P n−1x→P n−1x′,x, x′ ∈l −H.Our next result describes this identification.5.7. Proposition.

Let m = nd + 1 and let l be a non-jumping line for H. Let x ∈l −Hbe any point and λ ∈P n−1xbe a line in P n through x. Then there is a unique regularmap ψx,λ : l →Hnd+1 of degree d such that ψ(l ∩Hi) ⊂Hnd+1 ∩Hi for each i = 1, ..., ndand ψ(x) = λ ∩Hnd+1.

For any other point x ∈l −H the value at λ of the projectiveconnection map ΦH,l,x,x′ : P n−1x→P n−1x′equals the line < x′, ψx,λ(x′) >∈P n−1x′.Now we start to prove our results. We shall begin with Theorem 5.2.

We need twolemmas.5.8. Lemma.

A vector bundle E∗on P 1 of rank n and first Chern class (−n(d −1)) doesnot have the form O(−d + 1)⊕n if and only if H0(P 1, E∗(d −2)) ̸= 0.Proof: As any bundle on P 1, our E∗has the form Lni=0 Ol(ai) where P ai = −n(d −1).The condition (a1, ..., an) ̸= (−(d−1), ..., −(d−1)) is equivalent, under the above constaint24

on the sum, to the condition ”∃i : ai ≥−d+2” which is tantamount to H0(P 1, E∗(d−2)) ̸=0.The next lemma concerns the case when H consists of just one hyperplane H. In thiscase, as we have seen in Proposition 2.10, the logarithmic bundles E(H) is itself projectivelytrivial. So the canonical projective connection on E(H) gives identifications:ΦH,x,x′ : P n−1x→P n−1x′,x, x′ ∈P n −H.5.9.

Lemma. The identification ΦH,x,x′ takes a line λ through x to the line < λ∩H, x′ >through x′.Proof: Let H ⊂Cn+1 be the linear hyperplane corresponding to the projective hyperplaneH ⊂P n. By Proposition 2.10, we have an isomorphism E(H) ∼= OP n(−1)⊕n.

We canmake this statement more precise by showing the existence of a natural isomorphismE(H) ∼= H∗⊗OP n(−1).Denote the space Cn+1 shortly by V . Let x be any point of P n = P(V ) and let x be the 1-dimensional subspace in V representing x.

The tangent space TxP n is canonically identifiedwith x ⊗V/x. Denote by U the open set P n −H.

If x ∈U then the map H →V →V/xis an isomorphism so we get identification TxP n = x∗⊗H. Correspondingly, the fiber atx of Ω1P n becomes identified with x ⊗H∗i.e., with the fiber at x of H∗⊗OP n(−1).

Weget an isomorphism of restricted bundles φ : E(H)|U ∼= H∗⊗OP n(−1). Using the factthat E(H) is isomorphic to OP n(−1)⊕n, we can extend the isomorphism φ to the wholeP n. In this model for E(H) the fiber P n−1xof E(H)∗at x is canonically identified withH by assigning to the line λ through x the point λ ∩H.

Our lemma follows from thisimmediately.5.10. Now we are ready to prove Theorem 5.2.

Let us consider the bundle TP n(log H) asthe result of sucessive elementary transformations starting with the bundle25

TP n(log Hnd+1), as in Proposition 2.9. The latter bundle is projectively trivial.

Considera line l ⊂P n. We can assume that l ∩Hi are distinct points of l. Then the restriction to lof the bundle TP n(log H) is the elementary transformation of TP n(log Hnd+1) = Ol(1)⊕nwith respect to points yi = l ∩Hi and subspaces TyiHi ⊂TyiP n.5.11. Lemma.

Consider on the projective line P 1 the vector bundle O⊕nP 1 = OP 1 ⊗Ewhere E is an n -dimensional vector space. Let y1, ..., ynd be distinct points of P 1 andΛ1, ..., Λnd be hyperplanes in E. We regard Λi as a hyperplane in the fiber of our bundleover yi.

Then the following conditions are equivalent:(i) The elementary transformation Elm{y1,...,ynd}, {Λ1,...,Λnd}(E ⊗OP 1(1)) is not isomor-phic to OP 1(−d + 1)⊕n;(ii) The two nd -tuples (Λ1, ..., Λnd) ∈P(E)nd and (y1, ..., ynd) ∈(P 1)nd are d -codependent in the sense of n. 4.6.Proof: Denote the elementary transformation in condition (i) by Elm. Then, By Lemma5.8,, condition (i) is equivalent to nonvanishing of H0(Elm(E ⊗O(1))(d −2)).

Let x0, x1be homogeneous coordinates in P 1 A section of Elm(E ⊗O(−1))(d −2) is, by definition, ahomogeneous polynomial s(x) = (x0, x1) of degree d−1 with values in E such that s(yi) ⊂Λi. This is exactly the characterization of (d −1) -codependence given in Proposition 4.7.5.12.

Corollary. Let H = (H1, ..., Hnd+1) be a configuration of hyperplanes in P n ingeneral position.

A line l ⊂P n not lying in any Hi, is a jumping line for H if and only ifthe nd -tuples (H1 ∩Hnd+1, ..., Hnd ∩Hnd+1) ∈ˇHnd+1 and (l ∩H1, ..., l ∩Hnd+1) ∈lnd are(d −1) -codependent.Proof: Let l be given and suppose that l does not lie in Hnd+1. Let Hnd+1 be the linearhyperplane in Cn+1 corresponding to Hnd+1.By Proposition 5.7 the restriction of TP n(log Hnd+1) to l is isomorphic to Hnd+1 ⊗Ol(1).The restriction of the bundle E∗(H) to l is the elementary transformation ofthis projectively trivial bundle with respect to points yi = l ∩Hi and hyperplanes Λi =TyiHi.

So we can apply Lemma 5.11. An explicit projective trivialization of the bundleTP n(log Hnd+1) given in Proposition 5.7, identifies the projectivizations of fiber at everypoint y ∈l −Hnd+1 with Hnd+1.

Under this identification our hyperplanes Λi correspondto hyperplanes Hnd+1 ∩Hi ⊂Hnd+1. So the assertion follows from Lemma 5.85.13.

To finish the proof of Theorem 5.2, let us reformulate Corollary 5.12 in terms of thedual space ˇP n. Hyperplanes Hi correspond to points pi of ˇP n, the line l corresponds toa flat Z of codimension 2. The projective space ˇHnd+1 of hyperplanes in Hnd+1 becomesthe space P n−1pnd+1 of lines through pnd+1 and l itself becomes identified with the pencil]Z[ of lines through Z.

Under these identifications the hyperplane Hnd+1 ∩Hi in Hnd+126

corresponds to the line < pnd+1, pi >. Now Theorem 5.2 follows from the definition of themonoidal complex and Corollary 4.8.5.14.

Proposition 5.7 now becomes just a reformulation of the fact that the projectiveconnection in the projectively trivial bundle E = OP 1(a)⊕b is induced by global sectionsof E(−a). So it is proven.5.15.

For any stable rank r bundle E on P n there exists a Zariski open set U ⊂G(2, n+1)such that for l ∈U the splitting type (a1 ≥... ≥ar) of E|U is constant. A splitting type(a1, ..., ar) is called generic (or rigid) if a1 −ar ≤1.

By Grauert - M¨ulich theorem it isalways generic if r = 2. We conjecture that the splitting type of E(H) is always generic.27

§6. Schwarzenberger bundles.In this section we shall show that our logarithmic bundles generalize the constructionof Schwarzenberger [Schw1-2] of vector bunldes of rank n on P n.6.1.

Note that the following choices are equivalent:(i) An isomorphism P n ∼= |OP 1(n)| of P n with the n -fold symmetric product of P 1. (ii) A dual isomorphism ˇP n ∼= |OP 1(n)|∗.

(iii) A map ν : P 1 →ˇP n = |OP 1(n)|∗given by a complete linear system. (iv) A rational normal curve of degree n (Veronese curve, for short) ˇC in P n, the imageof the map in (iii).

(v) A map ˇν : P 1 →P n ∼= |OP 1(n)| given by a complete linear system. (vi) A Veronese curve C in P n, the image of this map.Fix any of them.

Then every point x ∈P n is identified with a positive divisor Dx of degreen on P 1. Choose some m ≥n+2.

Let V (x) be the subspace of sections s ∈H0(P 1, O(m−2)whose divisor of zeroes div(s) satisfies the condition div(s) ≥Dx. Denote by V (x)⊥⊂H0(P 1, O(m −2))∗the orthogonal subspace.

Its dimension is equal to n. In this way weobtain a mapP n →G(n, H0(P 1, O(m −2))∗) = G(n, m −1),x 7→V (x)⊥. (6.1)Let S be the tautological bundle on G(n, m−1) whose fiber over a point represented by ann -dimensional linear subspace is this subspace.

The pull-back, with respect to (6.1), of Sis a rank n vector bundle on P n. It is defined by any of the above six choices, in particular,by a choice of the Veronese curve C ⊂P n. We denote the dual bundle by E(C, m) andcall it the Schwarzenberger bundle of degree m associated to C. Thus fibers of E(C, m)have the formE(C, m)x =H0(P 1, O(m −2)){s ∈H0(P 1, O(m −2)) : div(s) ≥Dx}. (6.2)If we fix another isomorphism P m−2 ∼= |OP 1(m −2)|, this times by means of anotherVeronese curve R of degree m −2 in ˇP m−2, then we can view each point x ∈P n as apositive divisor Dx on R and the space P(V (x)⊥) as the projective subspace < Dx >spanned by Dx i.e.

as an (n −1) - secant flat of R. For this reason the projective bundleP(E(C, m)) is called the n -secant bundle of R, see [Schw2].One can easily show that E(C, m) is generated by its space of global sections whichis canonically isomorphic to H0(P 1, O(m −2)) = Cm−1.6.2. The bundle E(C, m) is in fact a Steiner bundle.

This fact is well known, see e.g.,[BS], Example 2.2. Let us give a precise statement.28

Fix an isomorphism P n = P(V ), dim V = n + 1. Then the choice of a Veronese curveC is given by an isomorphism V ∼= SnA, where A is a 2-dimensional vector space andthe points of the curve C are represented by the n -th powers ln, l ∈A.

Consider themultiplication mapt : V ⊗Sm−n−2A = SnA ⊗Sm−n−2A →Sm−2A.(6.3)6.3. Proposition.

The Schwarzenberger bundle E(C, m) is a Steiner bundle on P n =P(SnA) defined by vector spaces I = Sm−n−2A, W = Sm−2A and the tensor t : V ⊗I →Wgiven by (6.3).Proof: This follows from formula (6.2) and the fact that for two sections f ∈H0(P 1, O(a)),g ∈H0(P 1, O(b)) we have div(f) ≥div(g) if and only if f is divisible by g.6.4. Theorem.

Let H = (H1, ..., Hm) be an arrangement of m hyperplanes in P n ingeneral position. Suppose that all Hi considered as points of ˇP n, lies on a Veronese curveˇC (equivalently, all Hi osculate the dual Veronese curve C ⊂P n).Then there is anisomorphismE(H) ∼= E(C, m).Proof: This is equivalent to Theorem 3.8.5 from [K] which describes the Veronese varietyin the Grassmannian corresponding to E(H).

We prefer to give a direct proof here.Let IH be the space defined in n.1.4 and W ⊂Cm be the space of vectors with sumof coordinates zero, see formula (1.2). We shall construct explicit isomorphismsα : Sm−n−2(A) →IH,β : Sm−2A →Wwhich take the multiplication tensor (6.3) into the fundamental tensor tH which definesE(H) as a Steiner bundle.Let fi = 0 be the equation of the hyperplane Hi from H. The condition that Hiosculates C means that fi considered as an element of V ∗= SnA∗can be written in theform uni where ui ∈A∗.

Let pi ∈P 1 = P(A) be the point corresponding to ui. Let usidentify the space Sm−2A = H0(P 1, O(m −2)) with the space H0(P 1, Ω1P 1(p1 + ... + pm)of forms with simple poles at (p1, ..., pm).

After that the map β is defined by the formulaβ(ω) = (resp1(ω), ..., respm(ω)),ω ∈H0(P 1, Ω1P 1(p1 + ... + pm)). (6.4)By the residue theorem the sum of components of β(ω) equals 0 i.e.

β(ω) ∈W. Now,if we fix a point q different from the pi’s then we can identify the space H0(P 1, O(n)) =SnA with the space of rational functions on P 1 with poles of order ≤n at q.

Let usdenote this latter space by L(nq). Let us also regard the space Sm−n−2A as the space29

H0(P 1, Ω(p1 + ... + pm −nq)) of forms with at most simple poles at pi and with zero oforder ≤n at q. This is a subspace of H0(P 1, Ω1P 1(p1 + ... + pm) and we define α to be therestriction of β to this subspace.Let us see that this is correct i.e.,the image of α indeed lies in the space IH.

Bydefinition (see n.1.4), IH ⊂Cm consists of (λ1, ..., λm) such that P λifi(v) = 0 for everyv ∈V . In our case λi = respi(ω), where ω ∈H0(P 1, Ω(p1 + ... + pm −nq)).

Since we haveidentified V = SnA = L(nq), we have, for any v ∈V :Xλifi(v) =Xλi(uni , v) =Xrespi(ω) · (uni , v) =Xrespi(ω · v) = 0since the form ω · v has poles only at pi.This shows the correctness of the definition of maps α and β. After the identificationgiven by these maps it is obvious that the multiplication tensor becomes identified withthe fundamental tensor tH.6.5.

Corollary. Suppose that H, H′ are two arrangements of m hyperplanes in generalposition in P n such that all the hyperplanes from H and H′ osculate the some fixedVeronese curve C ⊂P n. Then the logarithmic bundles E(H) and E(H′) are isomorphic.6.6.

Proposition. Let C, C′ be two Veronese curves in P n such that for some m ≥n + 2the Schwarzenberger bundles E(C, m) and E(C′, m) are isomorphic.

Then C = C′.Proof: We shall show that C can be recovered intrinsically from E(C, m). Since E(C, m)is a Steiner bundle, its defining tensor t : V ⊗I →W is determined by E(C, m) itself (seeProposition 3.2).

We know that there are isomorphisms V ∼= SnA, I ∼= Sm−n−2A, W ∼=Sm−2A, with dimA = 2 which take the tensor t into the multiplication tensor (6.3). Weshall see that as soon as such isomorphisms exist, the Veronese curves in P(V ), P(I), P(W)consisting of perfect powers of elements of A, are defined by t alone.

Indeed, t gives amorphism T : P(V ) × P(I) →P(W). The Veronese curve R in P(W) is recovered as thelocus of w ∈P(W) such that T −1(w) consists of just one point, say (v, i).

The loci of v(resp. i) corresponding to various w ∈R constitute the Veronese curves of perfect powersin P(V ) and P(I) respectively.

Proposition 6.6 is proven.6.7. Consider as an example the case when m = n+3.

It is well known that any n+3 pointsin general position in P n lie on a unique Veronese curve,see [GH], p.530. So Corollary 6.5and Proposition 6.6 lead to the following conclusion.For two arrangements H and H′ of n + 3 hyperplanes in general position in P n thelogarithmic bundles E(H) and E(H′) are isomorphic if and only if the Veronese curvesosculated by H and H′ coincide.Moreover, any deformation of a bundle E(H) is again of this type, as the following propo-sition shows.30

6.8. Proposition.

Any Steiner bundle E on P n of rank n with dim I = 2, dim W = n+2is a Schwarzenberger bundle E(C, n + 3) for some Veronese curve C ⊂P n.Proof: Let Eas be the associated Steiner bundle on P 1 (see n. 3.20). By Proposition3.21, it suffices to show that Eas = E(H) for some arrangement of points on P 1.

But thisis obvious since Eas is of rank 1 and hence is determined by its first Chern class, whichis equal to n + 1 in our case. Thus taking any n + 3 points on P 1, we realize Eas as alogarithmic bundle and hence realize E as a Schwarzenberger bundle.31

.§7. A Torelli theorem for logarithmic bundles.7.1.

Let Agen(m, n) be the variety of all arrangements of m unordered hyperplanes in P nin general position. So Agen(m, n) is an open subset in the symmetric product Symm( ˇP n)(Note that we do not factorize modulo projective transformations).

The correspondenceH 7→E(H) defines a mapAgen(m, n) 7−→M(n, (1 −t)n+1−m),(7.1)where M(n, (1 −t)n+1−m) is the moduli space of stable rank n bundles on P n with Chernpolynomial (1−t)n+1−m. We are interested in the question whether this map is an embed-ding.

The statements that some moduli space is embedded into another are traditionallycalled ”Torelli theorems” after the classical Torelli theorem about the embeddine of themoduli space of curves into the moduli space of Abelian varieties. The following theoremwhich is the main result of this section shows that Ψ is very close to an embedding, atleast for large m.7.2.

Theorem. Let m ≥2n + 3 and let H, H′ be two arrangements of m hyperplanesin P n in general position.

Suppose that the corresponding logarithmic bundles E(H) andE(H′) are isomorphic. Then one of the two possibilities holds:1) H = H′ (possibly after reordering the hyperplanes).2) There exists a Veronese curve C ⊂P n such that all hyperplanes from H and H′ oscu-late this curve.

In this case E(H) and E(H′) are isomorphic to the Schwarzenbergerbundle E(C, m).7.3. To prove Theorem 7.2 we have to recover (as far as possible) the configuration Hfrom the bundle E(H).

The key idea is that the lines lying in each Hi are special jumpinglines.More precisely, we shall call a line l ⊂P n a super-jumping line for H if the restrictionE(H)|l contains as a direct summand a sheaf Ol(a) with a ≤0. As before, let us writem = nd + 1 + r with 0 ≤r < n. Clearly the line l is super-jumping if and only if therestriction Enorm(H)|l of the normalized bundle contains Ol(b) with b ≤−d.7.4.

Proposition. Any line l lying in one of the hyperplanes Hi of the arrangement H isa super-jumping line for H.Proof: We can assume that l ⊂H1.

Let F be a vector bundle on P 1. The propertythat F contains O(a) with a ≤0 as a direct summand, is equivalent to the fact thatH1(P 1, F(−2)) ̸= 0.32

We shall therefore prove that for F = E(H)|l the above cohomology does not vanish.Since the dimension of the cohomology groups varies semi-continuously with l, it is enoughfor our purpose to assume that l ⊂H1 is not contained in any other Hi, i ̸= 1. The residueexact sequence (2.1) gives a surjectionF →Ol ⊕mMi=2Cl∩Hi →0.Hence F(−2) maps surjectively onto Ol(−2).

Since for coherent sheaves on P 1 the functorH1 is right exact, we get a surjection H1(F(−2)) →H1(Ol(−2)) = C. Proposition isproven.We want now to reformulate the condition of being a super-jumpimg line in terms ofthe dual projective space ˇP n.7.5. Proposition.

Let H = (H1, ..., Hm) be as before and let l ⊂P n be a line not lyingin any Hi. Let pi be the point of the dual projective space ˇP n corresponding to Hi.

Letalso Z ⊂ˇP n be the codimension 2 flat corresponding to the line l. Then the following twoconditions are equivalent:(i) l is a super-jumping line for H;(ii) There exists a quadric Q ⊂P n (of rank ≤4) containing all the points p1, ..., pm andthe flat Z.Proof: Consider the dual bundle E∗to E = E(H). In other words, E∗is the bundleTP n(log H).

A line l is super-jumping for H if and only if H0(l, E∗) ̸= 0. By reasoninganalogous to those in the proof of Corollary 4.8 and Lemma 5.11, the existence of a sectionof E∗|l is equivalent to the existence of a regular map ψ : l →Hm of degree 1 such thatψ(l ∩Hi) ⊂Hm ∩Hi for i = 1, ..., m −1.

A map ψ is just an identification of l with someline l′ in Hm. Let Λ, Λ′ be codimension 2 flats in ˇP n corresponding to l, l′.

The map ψgives an identification Ψ of the projective lines (pencils) ]Λ[ , ]Λ′[ formed by hyperplanesthrough Λ and Λ′ respectively. Such an identification defines, by Steiner’s construction[GH], a quadric Q of rank ≤4.

Explicitly, Q is the union of codimension 2 subspaces ofthe form Π ∩Ψ(Π) where Π ∈]Λ[ is a hyperplane through Λ. This proves Proposition 7.5.7.6.

We would like now to characterize the hyperplanes Hi of H as those of which everyline is super-jumping for H. To do this, it is again convenient to use the dual projectivespace ˇP n and the points pi ∈ˇP n corresponding to Hi. Let us call a point q ∈ˇP n adjointto p1, ..., pm if, q does not coincide with any of pi and for any codimension 2 flat Z ⊂ˇP ncontaining q there is a quadric containing Z, p1, ..., pm.

For a fixed point q ∈ˇP n let H ⊂P nbe the corresponding hyperplane. Proposition 7.5 shows that q is adjoint to p1, ..., pm ifand only if any line in H is super-jumping for H = (H1, ..., Hm).

Thus Theorem 7.2 isequivalent to the following fact (in which we write P n instead of ˇP n).33

7.7. Theorem.

Let p1, ..., pm be points in P n in linearly general position and m ≥2n+3.Then:a) Unless all pi lie on one Veronese curve, there are no points adjoint to p1, ..., pm.b) If all pi do lie on one Veronese curve C then points of C and only they, are adjoint top1, ..., pm.Part b) is a consequence of Lemma 4.21. We shall concentrate on the proof of partb).

The proof will be based on the classical Castelnuovo lemma [GH].7.8. Lemma.

Let m ≥2n + 3 and p1, ..., pm be points in P n in linearly general positionwhich impose ≤2n + 1 conditions on quadrics. Then pi lie on a Veronese curve.We shall prove the following fact which, together with Castelnuovo lemma will implyTheorem 7.7.7.9.

Proposition. If m ≥2n and q is adjoint to p1, ..., pm then q, p1, ..., pm impose exactly2n + 1 conditions on quadrics.Proof: Let L be the linear system of all quadrics through first 2n points p1, ..., p2n.

Itis well known that L has codimension 2n in the linear system |O(2)| of all quadrics andquadrics from L cut out precisely p1, ..., p2n. For any codimension 2 flat ⊂P n let M(Z)be the linear system of all quadrics through Z.

It has dimension 2n (see n.4.2). L Toestablish Proposition 7.9 it suffices to prove the following fact.7.10.

Proposition. Let q be any point different from p1, ..., p2n.

Let L1 ⊂L be the linearsystem of all quadrics through p1, ..., p2n, q. Then:a) For a generic codimension 2 flat Z containing q the intersection L ∩M(Z) consists ofone point.b) The points L ∩M(Z) for generic Z through q as above, span L1 as a projectivesubspace.Proof of 7.9 from 7.10: Suppose we know Proposition 7.10.

Note that L1 has codimension2n + 1 in |O(2)|. This follows from the fact that quadrics from L (in fact, just rank 2quadrics from L) cut out p1, ..., p2n and nothing else.Let L2 ⊂L1 be the linear system of quadrics through all pi, i = 1, ..., m and q. Weshall show that L2 = L1.

Indeed, suppose that L2 is a proper subspace in L1. Since qis adjoint to p1, ..., pm, the intersection L2 ∩M(Z) is non-empty for any codimension 2flat Z through q.

But for generic such Z the intersection L ∩Z = L1 ∩Z consists of justone point and these points span L1. Hence L2 should miss some of these points, giving acontradiction.Proof of Proposition 7.10 a): Suppose that the statement is wrong.

Then for every Zthrough q the linear system L∩M(Z) contains a pencil. This means that for any additional34

point r ∈P n and any Z through q there will be a quadric containing p1, ..., p2n, r and Z.Now we shall move r and Z in a special way to get a contradiction.Let us number the points p1, ..., p2n so as to ensure that the hyperplane < p1, ..., pn >does not contain q. Let H =< q, p1, ..., pn−1 >.

Take 1-parameter families Z = Z(t), r =r(t), t ∈C with the following properties:1) Z(t), r(t) lie in H for any t.2) Z(0) =< q, p1, ..., pn−2 >, the point r(0) is a generic point inside Z(0).3) For t ̸= 0 the flat Z(t) intersects Z(0) transversely inside H and the curve r(t)intersects Z(0) transversely at t = 0.Let Q(t) be a quadric containing p1, ..., p2n, r(t), Z(t) (which exists by our assumption).We can choose Q(t) for t ̸= 0 to depend algebraically of t. Let Q = limt→0 Q(t). ThenQ contains the flat < q, p1, ..., pn−2 > and, in addition, the embedded tangent space toQ at points p1, ..., pn−2 and r(0) coincides the hyperplane H. This means that the lines< pi, pn−1 > and < r(0), pn−1 > will lie on Q.

This implies that the whole hyperplaneH w ill be part of Q. Hence the remaining n + 1 points pn, ..., p2n should lie on anotherhyperplane, which is impossible.

Part a) of Proposition 7.10 is proven.7.11. Now we shall concentrate on the proof of part b) of Proposition 7.10.

Note that wecan reformulate it as follows.7.12. Proposition.

The linear system L1 of quadrics through p1, ..., p2n and q is spannedby quadrics of rank ≤4 contained in this system.Indeed, the union of L1 ∩M(Z) for all codimension 2 flats Z through q coincide withthe part of L1 consisting of quadrics of rank ≤4.So we shall prove Proposition 7.12. Note that is a consequence of the following fact.7.13.

Lemma. The linear system L of quadrics through p1, ..., p2n is spanned by (1/2)2nnquadrics of rank 2 contained in L.Indeed, suppose we know Lemma 7.13.

Let Q be any quadric in L1. By lemma, Qis a linear combination of rank 2 quadrics Q1, ..., QN which lie in L but not necessarily inL1.

Since L1 is a hyperplane in L, any pencil < Qi, Qj > intersects L1 and Q lies in theprojective span of their intersection points (for all i, j). But any quadric from any pencil< Qi, Qj > has rank ≤4 since the Qi have rank 2.

This reduces our statement to Lemma7.13.To prove Lemma 7.13, let D ⊂|O(2)| be the locus of quadrics in P n of rank ≤4.Lemma 7.14. The dimension of D equals 2n, the degree of D equals 122nnand D spansthe projective space |O(2)|.35

Proof of Lemma 7.14: The space D is the image of the map f : ˇP n × ˇP n →|O(2)| whichtakes (H, H′) 7→H ∪H′. The map f is generically two - to - one.

The inverse image underf of the standard bundle O(1) on the projective space |O(2)| is O(1, 1) and its degree(self-intersection index) is2nn. This proves the statements about the dimension and thedegree.

The fact that D spans |O(2)| follows since D contains the Veronese variety ofdouble planes which by itself spans |O(2)|.End of the proof of Lemma 7.13: Note that L intersects D in finitely many points whosenumber is equal to the degree of D. These points are, moreover, smooth points of D (beingquadrics of rank exactly 2). Hence the i ntersection is transversal at any of the points.Suppose that the intersection points do not span L and are contained in some hyper-plane M ⊂L.Take any quadric X ∈D which does not belong to L and consider the codimension2n subspace W in L spanned by M and X.

Then W intersects D in more points thanthe degree of D. This means that the intersection will contain a curve (denote it C(X))passing through one of the (1/2)2nnrank 2 quadrics which belong to M. Clearly therewill be one of these quadrics, say, Q, which will be contained in any C(X).The embedded tangent space TQC(X) is contained in the tangent space to D at Q.Since L intersects D transversely, TQC(X) does not intersect M in any point other thanQ. On the other hand, since D spans |O(2)|, for a generic X ∈D the intersection of theprojective span < M, X > with TQD will consist of Q alone.

Since TQC(X) ⊂TQD∩, we get a contradiction.This finishes the chain of reductions proving Proposition 7.10 b). The proof of Theo-rem 7.7 is finished.36

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Math., 28 (1964), 21-52 (inRussian).Authors’ addresses:I.D. : Department of Mathematics, University of Michigan, Ann Arbor MI 48109, email:IGOR.DOLGACHEV@um.cc.umich.eduM.K.

: Department of Mathematics, Northwestern University, Evanston IL 60208, email:kapranov@chow.math.nwu.edu38

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