$3\times 3$ Minors of Catalecticants

Details $3\times 3$ Minors of Catalecticants $3\times 3$ Minors of Catalecticants

2010-11-06

arxiv.org/abs/1011.1564v2

Mathematics

Algebraic Geometry

v2: we simplify the presentation of the previous version. The main technical details are moved to arXiv:1011.5867

Scientific paper

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, 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.

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