Algebraic Geometry Seminar
Jarek Bucynzki
Institute of Mathematics, Polish Academy of Sciences
Defining equations of secant varieties to high degree Veronese reembeddings
Abstract: We fix a projective variety $X\subset \mathbb{P}^n$ and an integer $r$. We are interested in the defining equations of the $r$-th secant variety to the $d$-uple Veronese reembedding of $X$, and we assume $d$ is sufficiently large. One of the interesting cases is when $X= \mathbb{P}^n$. With these assumptions we prove that the $(r+1)$-minors of the catalecticant matrix with linear entries are sufficient to define the secant variety set-theoretically if and only if the Hilbert scheme parametrising $0$-dimensional Gorenstein subschemes of $X$ of length $r$ is irreducible. In particular, if $X$ is smooth and either $\dim X$ is at most $3$ or $r$ is at most $13$, then the minors are sufficient. If $\dim X$ is at least $4$ and $r$ is sufficiently large, then the locus defined by the minors has some additional components. These results motivate introducing cactus varieties, which generalise the secant varieties, and received a lot of attention since then.
The talk will be based on joint works with:
1) Adam Ginensky and Joseph Landsberg (JLMS 2013);
2) Weronika Buczynska (JAG 2014);
3) Joachim Jelisiejew (in preparation).
Wednesday October 8, 2014 at 4:00 PM in SEO 427