Secant non-defectivity via collisions of fat points

https://doi.org/10.1016/j.aim.2022.108657Get rights and content

Abstract

The computation of dimensions of secant varieties of projective varieties is classically approached via dimensions of linear systems with multiple base points in general position. Non-defectivity can be proved via degenerations. In this paper, we use a technique that allows some of the base points to collapse together in order to deduce a new general criterion for non-defectivity. We apply this criterion to prove a conjecture by Abo and Brambilla: for c3 and d3, the Segre-Veronese embedding of Pm×Pn in bidegree (c,d) is non-defective.

Introduction

A classical problem in algebraic geometry that goes back to late XIX century concerns the classification of defective varieties, i.e., algebraic varieties whose secant varieties have dimension strictly smaller than the one expected by a direct parameter count. In the last decades, this problem gained a lot of attention due to its relation with additive decompositions of tensors which are used in many areas of applied mathematics and engineering. Indeed, Segre varieties parametrize decomposable tensors; similarly, Veronese varieties and Segre-Veronese varieties are the symmetric and partially symmetric analogous. We refer to [15] and [8] for an overview on the geometric problem and to [23] for relations between secant varieties and questions on tensors.

The most celebrated result in this area of research is the Alexander-Hirschowitz Theorem, proven in [5], which classifies defective Veronese varieties by completing the work started more than 100 years earlier by the classical school of algebraic geometry. Denote by Vnd the Veronese variety given by the embedding of Pn via the linear system of degree d divisors. Several examples of defective Veronese varieties were known already at the time of Clebsch, Palatini and Terracini, but we had to wait until a series of enlightening papers by Alexander and Hirschowitz which culminated in 1995 with the proof that those are the only exceptional cases among Veronese varieties. We refer to [9, Section 7] for an historical overview on this theorem.

Theorem 1.1 Alexander-Hirschowitz

Let n, d and r be positive integers. The Veronese variety Vnd is r-defective if and only if either

  • (1)

    d=2 and 2rn, or

  • (2)

    (n,d,r){(2,4,5),(3,4,9),(4,3,7),(4,4,14)}.

After this result, the community tried to extend the classification of defective varieties to Segre and Segre-Veronese varieties by applying and refining the powerful methods introduced by Alexander and Hirschowitz. In the present paper, we focus on Segre-Veronese varieties with two factors, i.e., the image SVm×nc,d of the embedding of Pm×Pn via the linear system of divisors of bidegree (c,d). Among them, a long list of defective cases have been found (see [2, Table 1]) by Catalisano, Geramita and Gimigliano [12], [14], Abrescia [4], Bocci [10], Dionisi and Fontanari [17], Abo and Brambilla [1], Carlini and Chipalkatti [11] and Ottaviani [24]. In [2, Conjecture 5.5], Abo and Brambilla conjectured that these are the only defective cases. The fact that in all defective examples either c or d is strictly smaller than three suggested a weaker conjecture, stated in [2, Conjecture 5.6]. In this paper we prove this conjecture by proving the following result which is a step towards a complete classification of defective Segre-Veronese varieties with two factors.

Theorem 1.2

Let m and n be positive integers. If c3 and d3, then SVm×nc,d is not defective.

Abo and Brambilla themselves managed to greatly reduce the problem. Thanks to [2, Theorem 1.3], in order to prove Theorem 1.2 it is enough to prove the base cases (c,d){(3,3),(3,4),(4,4)}. This reminds what happened with Alexander-Hirschowitz Theorem, where the last to be overcome was the case of cubics which constituted the base case of the inductive proof. Low degrees are difficult to handle because they are rich of defective cases.

Dimensions of secant varieties of polarized varieties are classically computed by translating the problem to the computation of dimensions of certain linear systems of divisors with multiple base points in general position; see Section 2.2. In this context, defectivity means that the linear system has not the dimension expected by a direct parameter count. Degeneration techniques are very powerful tools to tackle these problems. Indeed, the expected dimension is always a lower bound for the actual one, while a specialization of the base points can only increase it. Hence, the whole approach boils down to finding a good specialization for which we can prove that the dimension is equal to the expected one. The seminal idea of Castelnuovo was to specialize some of the supports on a hypersurface. Then, via the so-called Castelnuovo exact sequence, one can proceed with a double induction on dimensions and degrees; see Section 2.3. However, this approach has arithmetic constraints: when the virtual dimension, i.e., the parameter count, is close to zero then there might be not enough freedom to find a good specialization to make the double induction work. At the same time, the cases with small virtual dimension are particular compelling because most defective cases appear among them. In the 1980s, Alexander and Hirschowitz improved drastically this method by introducing a new degeneration technique, called differential Horace method, which allowed them to complete the classification of defective Veronese varieties overcoming the arithmetic issues. The drawback of the differential Horace method is that, when the virtual dimension is close to zero, it might lead to consider linear systems whose base locus has a complicated non-reduced structure. In order to overcome the latter complication, in the literature, there are several examples of results on linear systems having virtual dimension sufficiently distant from zero; see e.g. [7, Theorem 2.3] or [3, Theorem 1.3].

In this paper, we employ a different degeneration technique, introduced by Evain in [18]: the base points are not only degenerated to a special position, but also allowed to collide together; see Section 2.4 for details. The degenerated linear system has a 0-dimensional base point with a very special, yet understood, non-reduced structure which is contained in a 4-fat point and contains a 3-fat point. Apparently a disadvantage, this new scheme turned out to be very useful to prove the following criterion for non-defectivity. Given a linear system L on a variety, we denote by L(m,2r) the linear subsystem of L obtained by imposing a base point of multiplicity m and r general base double points.

Theorem 1.3

Let (V,L) be a polarized smooth irreducible projective variety of dimension n. Suppose that L embeds V as a proper closed subvariety of PL and W is the image of such embedding. Assume that

  • (1)

    L(3,2rn1) is regular, i.e., dimL(3,2rn1)=dimL(n+22)(rn1)(n+1),

  • (2)

    L(4,2rn1) is zero,

  • (3)

    dimL(3)dimL(4)(n+12) and

  • (4)

    dimL(n+1)2

for every r{dimLn+1,dimLn+1}. Then W is not defective.

The key of success of the latter criterion should be made clear, in light of the historical remarks above, when we consider one of the problematic cases where L(2r) has virtual dimension close to zero. By letting collide n+1 of the general 2-fat base points, we get a base locus with r(n+1) 2-fat points and one component which is between a 3-fat and a 4-fat point. Then, regularity follows from proving the conditions (1) and (2), where the latter component of the base locus is replaced once with the 3-fat point and once with the 4-fat point. Even if in the latter two cases we no longer have only double points, the virtual dimension becomes sufficiently distant from zero to let us find further classic degenerations to complete the proof.

Our proof of Theorem 1.2 is indeed an application of Theorem 1.3. Our result adds up to previous successful applications of this degeneration technique: in [19] for linear systems of plane curves, in [21] for linear systems in P3 and in [22] in the context of Waring decompositions of polynomials. For this reason, we believe that our general result can be used to attack also other questions on non-defectivity of projective varieties for which previous methods presented technical obstacles.

Our Theorem 1.2, Theorem 1.3 have consequences also on identifiability. Given a non-degenerate projective variety WPN and a point pPN in the ambient space, the rank of p with respect to W is the smallest cardinality of a set of distinct points on W whose linear span contains p. The k-th secant variety of W is the Zariski-closure of the set of points of rank at most k. The point p is k-identifiable with respect to W if such spanning set of minimal cardinality is unique. The variety W is called k-identifiable if the general rank-k point of the ambient space is k-identifiable. In [16], the authors relate the study of defectivity to the study of identifiability. Combining their main result with our Theorem 1.2, Theorem 1.3, we obtain the following identifiability statement.

Corollary 1.4

Under the same hypothesis as in Theorem 1.3, the variety W is (k1)-identifiable for every 2dim(W)<kdimLdim(W)+1.

In the particular case of Segre-Veronese varieties of two factors, we conclude the following.

Corollary 1.5

Let m, n, c and d be positive integers such that c3 and d3. Ifk1m+n+1(m+cc)(n+dd), then SVm×nc,d is (k1)-identifiable.

As mentioned, when the algebraic variety W with respect to which we consider the notion of rank is a variety like Veronese, Segre, or Segre-Veronese varieties, then the geometric notion of rank corresponds to the one of symmetric rank, tensor rank and partially symmetric rank of tensors. If we consider partially symmetric tensors TSymcCm+1SymdCn+1, our results are rephrased as follows.

  • (Theorem 1.2) If T is general, then it has rank k=(m+cm)(n+dn)/(m+n+1), i.e., there exists {v1,,vk}Cm+1 and {w1,,wk}Cn+1 such thatT=i=1kvicwid.

  • (Corollary 1.5) If T is general of rank k<(m+cm)(n+dn)/(m+n+1) then the additive decomposition (1.2) is uniquely determined up to permutations of the summands and where the vi's (respectively, wi's) are determined up to c-th (respectively d-th) roots of unity.

Structure of the paper. In Section 2 we recall the basic definitions for secant varieties and linear systems with multiple base points. We also illustrate the tools we use in our computation, such as Castelnuovo exact sequence and collisions of fat points. In Section 3 we prove Theorem 1.3 via collisions of fat points. The rest of the paper is an application of Theorem 1.3 to Segre-Veronese varieties with two factors SVm×nc,d in order to prove Theorem 1.2. The three key cases (c,d){(3,3),(3,4),(4,4)} are solved in Sections 4, 5 and 6 respectively. This completes the proof of Theorem 1.2. In Appendix A we describe the software computations we performed to check the initial cases of our inductive proofs. In Appendix B we collect the long and tedious arithmetic computations needed in our proofs.

Acknowledgments. The project started while FG was a postdoc at MPI MiS Leipzig (Germany) and AO was a postdoc at OVGU Magdeburg (Germany). AO thanks MPI MiS Leipzig for providing a perfect environment during several visits at the beginning of the project. The authors also thank Maria Virginia Catalisano, Alessandro Gimigliano and Massimiliano Mella for useful discussions. The authors also wish to thank the anonymous referee for pointing out a mistake in the first draft of the article. FG is supported by the National Science Center, Poland, project “Complex contact manifolds and geometry of secants”, 2017/26/E/ST1/00231, and acknowledges partial support from the fund FRA 2018 of University of Trieste - project DMG, grant no. j961c1800138001. AO acknowledges partial financial support from A. v. Humboldt Foundation/Stiftung through a fellowship for postdoctoral researchers. AO is a member of GNSAGA of INdAM (Italy).

Section snippets

Basics and background

We recall useful definitions and constructions in the context of secant varieties and linear systems of divisors with multiple base points. We will work over the field of complex numbers C.

Non-defectivity via collisions of fat points

In this section we prove Theorem 1.2 and Theorem 1.3 by using the deformation methods described in the previous section. We compute the dimension of a linear system of divisors with 2-fat base points on a smooth variety by specializing the points, allowing some of them to collapse. Theorem 3.2 is the key of our proofs of Theorem 1.2 and Theorem 1.3. We first recall some auxiliary results that will be useful in the proof.

Lemma 3.1

[13, Lemma 1.9]

Let V be a projective variety and let HV be a positive dimensional

SVm×n3,3 is not defective

In this section we show that Theorem 1.2 holds for c=d=3. The specializations need to be chosen carefully and satisfy several arithmetic properties: in order to make our proofs easier to read, we moved some elementary but tedious computations to Appendix B.

Proposition 4.1

If m and n are positive integers, then Lm×n3,3(2r) is nonspecial for every rN.

Proof

We may assume mn by symmetry and 2m by [6, Theorem 3.1]. We apply Theorem 1.3 by checking its hypothesis: condition (1) in Proposition 4.2, condition (2) in

SVm×n3,4 is not defective

Now that we have established Proposition 4.1, it will be easier to show that SVm×n3,4 is also non-defective. As in Section 2.2, we can phrase this statement in terms of linear systems.

Proposition 5.1

If m and n are positive integers, then Lm×n3,4(2r) is nonspecial for every rN.

Proof

For n=1, the statement is known by [6, Theorem 3.1]. For n2 we apply Theorem 1.3: condition (1) is Proposition 5.2, condition (2) is Proposition 5.5, condition (3) is checked directly because a single j-fat point always impose

SVm×n4,4 is not defective

The last step to complete our proof of Theorem 1.2 is to prove that SVm×n4,4 is non-defective.

Proposition 6.1

If m and n are positive integers, then Lm×n4,4(2r) is nonspecial for every rN.

Proof

For m=1, the statement is known by [6, Theorem 3.1]. For m2 we apply Theorem 1.3: condition (1) is Proposition 6.2, condition (2) is Proposition 6.5, condition (3) is checked directly because a single j-fat point always impose independent conditions on Lm×nc,d for c,dj1 and condition (4) is trivial. 

Proposition 6.2

If m and n are

References (24)

  • E. Ballico et al.

    Higher secant varieties of Pn×P1 embedded in bi-degree (a,b)

    Commun. Algebra

    (2012)
  • A. Bernardi et al.

    The hitchhiker guide to: secant varieties and tensor decomposition

    Mathematics

    (2018)
  • View full text