Low-rank tensor approximations are plagued by a well-known problem — a tensor may fail to have a best rank-r approximation. Over R, it is known that such failures can occur with positive probability, sometimes with certainty: in R2×2×2, every tensor of rank 3 fails to have a best rank-2 approximation. We will show that while such failures still occur over C, they happen with zero probability. I...

We prove that the stabiliser group GX of the Segre variety product in PG(V ) of three projective lines over a field F has four orbits on singular points of PG(V ), and that GX has five orbits on points of PG(V ) if F is finite.

1. Where we are: Segre classes of vector bundles, and Segre classes of cones 1 1.1. Segre classes of cones 1 2. What the “functoriality of Segre classes of subschemes” buys us 2 2.1. The multiplicity of a variety along a subvariety 2 3. Deformation to the normal cone 3 3.1. The construction 3 4. Specialization to the normal cone 5 4.1. Gysin pullback for local complete intersections 6 4.2. Inte...

We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a ‘Segre-Bertini’ theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general...

The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre–Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been ...

We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the Rees-Böger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of variou...

Let XΣ be a complete smooth toric variety of dimension n defined by a fan Σ where all Cartier divisors in Pic(XΣ) are nef and let V be a subscheme of XΣ. We show a new expression for the Segre class s(V,XΣ) in terms of the projective degrees of a rational map associated to V . In the case where the number of primitive collections of rays in the fan Σ is equal to the number of generating rays in...

This paper is an elementary introduction to the methods of Landsberg and Manivel, [3] for finding the ideals of secant varieties to Segre varieties. We cover only the most basic topics from [3], but hope that since this is a topic which is rarely made explicit, these notes will be of some use. We assume the reader is familiar with the basic operations of multilinear algebra: tensor, symmetric, ...