Secant varieties of Segre–Veronese varieties

@article{Raicu2010SecantVO,
title={Secant varieties of Segre–Veronese varieties},
author={Claudiu Raicu},
journal={Algebra \& Number Theory},
year={2010},
volume={6},
pages={1817-1868}
}
• Claudiu Raicu
• Published 26 November 2010
• Mathematics
• Algebra & Number Theory
Secant varieties of Segre and Veronese varieties (and more generally Segre-Veronese varieties, which are embeddings of a product of projective spaces via the complete linear system of an ample line bundle) are very classical objects that go back to the Italian school of mathematics in the 19-th century. Despite their apparent simplicity, little is known about their equations, and even less about the resolutions of their coordinate rings. The main goal of this thesis is to introduce a new method…
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References

SHOWING 1-10 OF 52 REFERENCES
Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes
• Mathematics
• 2010
We study the secant varieties of the Veronese varieties and of Veronese reembeddings of a smooth projective variety. We give some conditions, under which these secant varieties are set-theoretically
3x3 Minors of Catalecticants
Secant varieties to Veronese embeddings of projective space are classical varieties whose equations are not completely understood. Minors of catalecticant matrices furnish some of their equations,
3× 3 MINORS OF CATALECTICANTS
Secant varieties of Veronese embeddings of projective space are classical varieties whose equations are far from being understood. Minors of catalecticant matrices furnish some of their equations,
On the ideals and singularities of secant varieties of Segre varieties
• Mathematics
• 2006
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
On the Ideals of Secant Varieties of Segre Varieties
• Mathematics, Computer Science
Found. Comput. Math.
• 2004
Basic techniques for determining the ideals of secant varieties of Segre varieties are established and a conjecture on the generators of the ideal of the first secant variety in the case of three factors is solved.
Higher secant varieties of Segre-Veronese varieties
• Mathematics
• 2003
We study the dimension of the higher secant varieties $X^s$ of ${\Bbb X} = {\Bbb P}^{n_1}\times ...\times {\Bbb P}^{n_t}$ embedded the morphism given by ${\cal O}_{\Bbb X}({a_1,...,a_t})$. We call it
Secant varieties of ℙ2 × ℙn embedded by 𝒪(1, 2)
• Mathematics
J. Lond. Math. Soc.
• 2012
A more general construction for producing explicit matrix equations that vanish on secant varieties of products of projective spaces is proposed, based on previous work of Strassen and Ottaviani.