Equations for secant varieties of Veronese and other varieties

By J. M. Landsberg, Giorgio Ottaviani. 2011

149 citations recorded by the source corpus.

New classes of modules of equations for secant varieties of Veronese varieties are defined using representation theory and geometry. Some old modules of equations (catalecticant minors) are revisited to determine when they are sufficient to give scheme-theoretic defining equations. An algorithm to decompose a general ternary quintic as the sum of seven fifth powers is given as an illustration of our methods. Our new equations and results about them are put into a larger context by introducing vector bundle techniques for finding equations of secant varieties in general. We include a few homogeneous examples of this method.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9340808853f5ded4a2758703b60539c3051ba29e

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In Computational Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Irreducibility of the Gorenstein loci of Hilbert schemes via ray families (2014, background): “For simplicity and to refer to some results from [Landsberg and Ottaviani 2013], we assume that k = C, but the claim holds for all fields of characteristic either zero or large enough.”
Topological Noetherianity of polynomial functors (2017, methodology): “the theory of modules over the category FI of finite sets with injective maps [7]; Gröbner techniques [29] for modules over more general combinatorial categories that, among other things, led to a resolution of the Artinian conjecture [24] and to a resolution of a conjecture by Rauh-Sullivant [26] o”
A comparison of different notions of ranks of symmetric tensors (2012, background): “In [29, Theorems 1.2.3 and 4.2.7] , it is shown that Γ 2 , 3 5 ( P 2 ) has codimension 5 in P ( S 5 ) while the secant variety σ 55 ( P 2 ) has codimension 6.”
On generic identifiability of symmetric tensors of subgeneric rank (2015, methodology): “2 in [26] remains true if the third case (k, d, n) = (9, 5, 3) is added to the list of exceptions.”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “We borrow the following list of known results from [85].”