Secant non-defectivity via collisions of fat points
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 the Veronese variety given by the embedding of 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 is r-defective if and only if either and , or .
Theorem 1.2
Let m and n be positive integers. If and , then 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 . 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 on a variety, we denote by the linear subsystem of obtained by imposing a base point of multiplicity m and r general base double points. Theorem 1.3 Let be a polarized smooth irreducible projective variety of dimension n. Suppose that embeds V as a proper closed subvariety of and W is the image of such embedding. Assume that is regular, i.e., , is zero, and
for every . Then W is not defective.
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 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 and a point 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 -identifiable for every .
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 and . If then is -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 , our results are rephrased as follows.
- •
(Theorem 1.2) If T is general, then it has rank , i.e., there exists and such that
- •
(Corollary 1.5) If T is general of rank then the additive decomposition (1.2) is uniquely determined up to permutations of the summands and where the 's (respectively, 's) are determined up to c-th (respectively d-th) roots of unity.
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 .
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 Let V be a projective variety and let be a positive dimensional[13, Lemma 1.9]
is not defective
In this section we show that Theorem 1.2 holds for . 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 is nonspecial for every . Proof We may assume by symmetry and by [6, Theorem 3.1]. We apply Theorem 1.3 by checking its hypothesis: condition (1) in Proposition 4.2, condition (2) in
is not defective
Now that we have established Proposition 4.1, it will be easier to show that 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 is nonspecial for every . Proof For , the statement is known by [6, Theorem 3.1]. For 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
is not defective
The last step to complete our proof of Theorem 1.2 is to prove that is non-defective. Proposition 6.1 If m and n are positive integers, then is nonspecial for every . Proof For , the statement is known by [6, Theorem 3.1]. For 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 for and condition (4) is trivial. □ Proposition 6.2 If m and n are
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