Mathematical Research Letters

Volume 20 (2013)

Number 4

$3 \times 3$ minors of catalecticants

Pages: 745 – 756

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n4.a10

Author

Claudiu Raicu (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.; and Institute of Mathematics “Simion Stoilow” of the Romanian Academy)

Abstract

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, and in some situations even generate their ideals. Geramita conjectured that this is the case for the secant line variety of the Veronese variety, namely that its ideal is generated by the $3 \times 3$ minors of any of the “middle” catalecticants. Part of this conjecture is the statement that the ideals of $3 \times 3$ minors are equal for most catalecticants, and this was known to hold set-theoretically. We prove the equality of $3 \times 3$ minors and derive Geramita’s conjecture as a consequence of previous work by Kanev.

Keywords

catalecticant matrices, Veronese varieties, secant varieties

2010 Mathematics Subject Classification

14M12

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