On secant varieties of Compact Hermitian Symmetric Spaces

TL;DR

This paper investigates the algebraic and geometric properties of secant varieties of rank three compact Hermitian symmetric spaces, establishing normality, singularity types, and degrees of ideal generators.

Contribution

It proves that these secant varieties are normal with rational singularities and identifies the degrees in which their ideals are generated, with a specific exception.

Findings

Secant varieties are normal with rational singularities.

Ideals are generated in degree three, except for one case.

The ideal of the spinor variety secant is generated in degree four.

Abstract

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three - with one exception, the secant variety of the $21$-dimensional spinor variety in $\pp{63}$ where we show the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.