Jarosław Buczyński


Secant varieties, rank and cactus ranks


Given a projective variety $X\subset \mathbb{P}(W)$, and a point $p\in \mathbb{P}(W)$, the rank of $p$ is the minimal number $r$ such that $p$ is contained in the linear span of $r$ distinct points of $X$. The $r$-th secant variety is the closure of the set of points of rank at most $r$. I will discuss basic properties of ranks, secant varieties, their applications, and also define the cactus rank, which often is an obstruction to calculate rank. One of the main cases of interest is when $W=S^d V$ (that is elements of $W$ are homogeneous polynomials in $\dim V$ variables) and $X$ is the Veronese variety. I will present a simple idea for lower bounds for rank, which may distinguish between rank and cactus rank. Among the consequences, we construct an example of a ternary quintic or rank $10$ --- previously, it was not known, whether maximal possible rank for ternary quintics is $9$ or $10$.

The talk is based on a joint work in progress with Zach Teitler.