Induction for secant varieties of Segre varieties

By Hirotachi Abo, Giorgio Ottaviani, C. Peterson. 2006

172 citations recorded by the source corpus.

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.

Subfields: Computational Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/fdcfe9840689870400583d7255de4783585f4701

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In Computational Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On the dimensions of secant varieties of Segre-Veronese varieties (2009, unknown): “In [4], it was proved that there are no defective cases if a = (1(4)).”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “In [92], with a very clever idea, an explicit description of this degree seven invariant was found by relating its cube to the determinant of a 21× 21 symmetric matrix.”
Tensors : A brief introduction (2014, unknown): “In fact, there are exceptions [3], [1], [23], [58].”
Refined methods for the identifiability of tensors (2013, background): “Indeed, with the notations of [1], condition T (a1, .”
Generic and typical ranks of multi-way arrays (2009, background): “Recently, Abro and Ottaviani attempted to generalize the AH theorem to non symmetric complex tensors [26], and provided an almost exhaustive list of exceptions.”
Identifiability of parameters in latent structure models with many observed variables (2008, unknown): “MODELS WITH MANY OBSERVED VARIABLES1 BY ELIZABETH S. ALLMAN2, CATHERINE MATIAS AND JOHN A. RHODES2”
Non-defectivity of Grassmannians of planes (2009, result): “While the theorem does not reach this result, it does have a relatively small value for s2 s1. Such a result is reminiscent of one obtained in [CGG2] for X = P1 ::: P1 where they produced functions s1(t);s2(t) (with t denoting the number of factors of P1) such that s2 s1 1. This was extended in [ AO”
On generic identifiability of symmetric tensors of subgeneric rank (2015, background): “, that a general nonsymmetric tensor of subgeneric rank admits a unique tensor rank decomposition, unless it is one of the exceptional cases that have already been proved in [1, 5, 6, 10, 11].”
Tensor decompositions, alternating least squares and other tales (2009, background): “Results of [49] one ﬁnd several typical ranks, and indeed how many can there be?”
On the typical rank of real binary forms (2009, background): “Regarding general complex tensors, some partial results are contained in [1], while [9] provides a complete answer for tensors of format 2 2.”
Ranks of tensors and a generalization of secant varieties (2009, background): “nsors, just as one diagonalizes matrices in order to study them better. A ﬁrst question is how many decomposable tensors one needs. This question has been answered for generic p and most (a,b,c), see [1] for the state of the art. The next question is what is the maximum number of decomposable tensor”