Mathematical Research Letters
Volume 20 (2013)
$3 \times 3$ minors of catalecticants
Pages: 745 – 756
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.
catalecticant matrices, Veronese varieties, secant varieties
2010 Mathematics Subject Classification