$3 \times 3$ minors of catalecticants

Abstract

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 × 3 minors of any of the "middle" catalecticants. Part of this conjecture is the statement that the ideals of 3 × 3 minors are equal for most catalecticants, and this was known to hold set-theoretically. We prove the equality of 3 × 3 minors and derive Geramita's conjecture as a consequence of previous work by Kanev.

On the Gorenstein locus of the punctual Hilbert scheme of degree 11

“…More precisely in [17] the author conjectures that the ideal of the 2nd secant variety (the variety of secant lines) of the dth Veronese embedding of P n k is generated by the 3 × 3 minors of the ith catalecticant matrix for 2 i d − 2. Such a conjecture was confirmed in [27]. As pointed out in [3], Section 8.1, the above Main Theorem allows to extend the above result as follows: if r 11, 2r d and i = r, .…”

Secant varieties of Segre–Veronese varieties

“…(As the rest of the proof will imply, we have F = F 1 + F 2 + F 3 ; see [Raicu 2010;Raicu 2011, Chapter 6] for more precise results in this direction in the case n = 1 of the Veronese variety.) Recall that I = I d r denotes the space of generic multiprolongations of degree r (Definition 3.13), that is, I is the kernel of the map…”

Equations for secant varieties of Veronese and other varieties

“…Remark 3.1.3. C. Raicu [43] recently proved that in Corollary 3.1.2 it is possible to replace the (2, d − 2) flattening with any (i, d − i)-flattening such that 2 ≤ i ≤ d − 2.…”