Four lectures on secant varieties

Abstract

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.

arXiv commentLectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhauser

References in corpus (27)

• Shifted Power Method for Computing Tensor Eigenpairs
• The Number of Eigenvalues of a Tensor
• Computing symmetric rank for symmetric tensors
• E-Determinants of Tensors
• Eigenvectors of tensors and algorithms for Waring decomposition
• Ranks of tensors and a generalization of secant varieties
• On the rank of a binary form
• The Hyperdeterminant and Triangulations of the 4-Cube
• Algebraic Factor Analysis: Tetrads, Pentads and Beyond
• Phylogenetic Algebraic Geometry
• Waring decompositions of monomials
• Toward a salmon conjecture
• On the Waring problem for polynomial rings
• Ideals of varieties parameterized by certain symmetric tensors
• On the dimensions of secant varieties of Segre-Veronese varieties
• On the variety parametrizing completely decomposable polynomials
• Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited
• New lower bounds for the border rank of matrix multiplication
• Hyperdeterminants of Polynomials
• Set-theoretic defining equations of the tangential variety of the Segre variety
• Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^m$ embedded in bi-degree $(1,d)$
• Higher secant varieties of ${\mathbb{P}}^n\times {\mathbb{P}}^1$ embedded in bi-degree $(a,b)$
• 3x3 Minors of Catalecticants
• A syzygetic approach to the smoothability of zero-dimensional schemes
• Secants of Lagrangian Grassmannians
• Secant varieties of P^2 x P^n embedded by O(1,2)
• Explicit tensors of border rank at least 2n-1

Cited by in corpus (11)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• Equations for the fifth secant variety of Segre products of projective spaces
• Secant non-defectivity via collisions of fat points
• Border Ranks of Monomials
• Computations and Equations for Segre-Grassmann hypersurfaces
• Learning Algebraic Models of Quantum Entanglement
• Homotopy techniques for tensor decomposition and perfect identifiability
• Border Waring Rank via Asymptotic Rank
• The monic rank
• Waring decompositions of the product of two quadrics: the small rank cases
• The Waring rank of binary binomial forms