Giorgio Ottaviani

Induction for secant varieties of Segre varieties

This paper studies the dimension of secant varieties to Segre varieties. The problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry. An inductive procedure is built around the ideas of successive specializations of points and projections. This reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases. As applications of the technique: We give a complete classification of defective p-secant varieties to Segre varieties for p < 6. We generalize a theorem of Catalisano-Geramita-Gimigliano on non-defectivity of tensor powers of Pn. We determine the set of p for which unbalanced Segre varieties have defective p-secant varieties. In addition, we completely describe the dimensions of the secant varieties to the deficient Segre varieties P1 x P1 x Pn x Pn and P2 x P3 x P3. In the final section we propose a series of conjectures about defective Segre varieties.

The Euclidean Distance Degree of an Algebraic Variety

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart–Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.

SPINOR BUNDLES ON QUADRICS

We define some stable vector bundles on the complex quadric hypersurface Qn of dimension n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. When n = 2fc — 1 there is one spinor bundle of rank 2k~1. When n = 2k there are two spinor bundles of rank 2k~1. Their behavior is slightly different according as n = 0 (mod 4) or n = 2 (mod 4). As an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qe- Introduction. Let Qn be the smooth quadric hypersurface of the complex pro- jective space Pn+1. In this paper we define in a geometrical way some vector bundles on the quadric Qn as the natural generalization of the universal bundle and the dual of the quotient bundle on Q4 ~ Gr(l,3). We call them spinor bundles. On Q4 this definition is equivalent to the usual one. Spinor bundles are homogeneous and stable (according to the definition of Mum- ford-Takemoto). We study their first properties using the geometrical description given and some standard techniques available in (OSS). We also use a theorem of Ramanan (see (Um)) about the stability of homoge- neous bundles induced by irreducible representations. When n is odd there is only one spinor bundle, while when n is even there are two nonisomorphic spinor bun- dles. When n is even the behavior of spinor bundles is slightly different according as n = 0 (mod4) or n = 2 (mod4). In (Ot2) we have given a cohomological splitting criterion for vector bundles on quadrics involving spinor bundles. Qn ~ Spin(n + 2)/P(cty) (St) is a homogeneous manifold, and the semisimple part of the Lie algebra of -P(ai) is o(n). At the level of Lie algebras, spinor bundles are defined from the spin and half-spin representations of o(n). The paper is divided into three sections. In §1 we give some preliminary results and we define the spinor bundles. In §2 we study the first properties of spinor bundles. In §3, as an application, we describe some moduli spaces of rank 3 vector bundles on Q5 and Qq.

Eigenvectors of tensors and algorithms for Waring decomposition

A Waring decomposition of a (homogeneous) polynomial f is a minimal sum of powers of linear forms expressing f. Under certain conditions, such a decomposition is unique. We discuss some algorithms to compute the Waring decomposition, which are linked to the equations of certain secant varieties and to eigenvectors of tensors. In particular we explicitly decompose a cubic polynomial in three variables as the sum of five cubes (Sylvester Pentahedral Theorem).

Some extensions of horrocks criterion to vector bundles on Grassmannians and quadrics

SummaryIn this paper we prove that a vector bundle E on a grassmannian (resp. on a quadric) splits as a direct sum of line bundles if and only if certain cohomology groups involving E and the quotient bundle (resp. the spinor bundle) are zero. When rank E=2 a better criterion is obtained considering only finitely many suitably chosen cohomology groups.

On Generic Identifiability of 3-Tensors of Small Rank

We introduce an inductive method for the study of the uniqueness of decompositions of tensors, by means of tensors of rank

New lower bounds for the border rank of matrix multiplication

1

Syzygies of Veronese Embeddings

. The method is based on the geometric notion of weak defectivity. For three-dimensional tensors of type

On the typical rank of real binary forms

(a,b,c)

AN ALGORITHM FOR GENERIC AND LOW-RANK SPECIFIC IDENTIFIABILITY OF COMPLEX TENSORS ∗

,