On the ideals and singularities of secant varieties of Segre varieties

By J. M. Landsberg, Jerzy Weyman. 2006

82 citations recorded by the source corpus.

We find minimal generators for the ideals of secant varieties of Segre varieties in the cases of σk(ℙ1 × ℙn × ℙm) for all k, n, m, σ2(ℙn × ℙm × ℙp × ℙr) for all n, m, p, r (GSS conjecture for four factors), and σ3(ℙn × ℙm × ℙp) for all n, m, p and prove they are normal with rational singularities in the first case and arithmetically Cohen–Macaulay in the second two cases.

Subfields: Algebra and Number Theory, Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/37dae6dc312381c6db03ac31f907c0f1844093c4

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In Algebra and Number Theory

In Computational Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Generalizations of Strassen's Equations for Secant Varieties of Segre Varieties (2006, unknown): “Defining equations are known only for the following cases: all secant varieties of the two factor Segre (classical: these are just the ( r + 1) × ( r + 1) minors of the space of a × b matrices), the n -factor Segre itself ℙ A 1 ×···× ℙ A n ⊂ ℙ ( A 1 ⊗···⊗ A n ) (classical”
On the ideals of equivariant tree models (2007, background): “tions are equations for star models. These are very hard to come by: [9] posed several conjectures concerning these for the general Markov model, and special cases of these conjectures were proved in [1, 13, 14]. For certain important equivariant models equations were found in [4, 16]. Roughly speak”
On the third secant variety (2011, unknown): “[14].”
Secant varieties of Segre–Veronese varieties (2010, unknown): “ conjecture was obtained by Landsberg and Manivel [LM04], as well as the case of a 3–factor Segre product. The 2–factor case is classical, while the 4–factor case was resolved by Landsberg and Weyman [LW07]. The 5–factor case was proved by Allman and Rhodes [AR08]. We prove the GSS conjecture in Cor”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “In this case, the ideal is rather well understood; see, e.g., [215] and also the extensive bibliography given in the book of Weyman [216].”