Higher secants of spinor varieties

TL;DR

This paper investigates the dimensions of secant varieties of spinor varieties, providing a probabilistic algorithm for computation and establishing defectiveness in specific cases for dimensions 7 and 8.

Contribution

It introduces a probabilistic algorithm to compute secant variety dimensions and proves defectiveness for certain spinor varieties in dimensions 7 and 8.

Findings

The 3-secant variety of $S_h$ has the expected dimension for most $h$.

$S_7$ has a defective 3-secant variety.

$S_8$ has defective 3-secant and 4-secant varieties.

Abstract

Let $S_h$ be the even pure spinors variety of a complex vector space $V$ of even dimension $2h$ endowed with a non degenerate quadratic form $Q$ and let $\sigma_k(S_h) $ be the $k$-secant variety of $S_h$. We decribe a probabilistic algorithm which computes the complex dimension of $\sigma_k(S_h) $. Then, by using an inductive argument, we get our main result: $\sigma_3(S_h) $ has the expected dimension except when $h\in \{7,8\} $. Also we provide theoretical arguments which prove that $S_7$ has a defective 3-secant variety and $S_8$ has defective 3-secant and 4-secant varieties.