Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes
Abstract
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 cut out by determinantal equations. More precisely, they are given by minors of a catalecticant matrix. These conditions include the case when the dimension of the projective variety is at most 3 and the degree of reembedding is sufficiently high. This gives a positive answer to a set-theoretic version of a question of Eisenbud in dimension at most 3. For dimension four and higher we produce plenty of examples when the catalecticant minors are not enough to set-theoretically define the secant varieties to high degree Veronese varieties. This is done by relating the problem to smoothability of certain zero-dimensional Gorenstein schemes.
Keywords
Cite
@article{arxiv.1012.3563,
title = {Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes},
author = {Weronika Buczyńska and Jarosław Buczyński},
journal= {arXiv preprint arXiv:1012.3563},
year = {2011}
}
Comments
28 pages, 10 figures; v2: improved presentation in accordance with referee's suggestions. The corrections were made while the first named author was at Mittag-Leffler Institute; v3: added Remark 1.5 (thanks to Anthony Iarrobino) on previously known results; v4: added new references; To appear in Journal of Algebraic Geometry