Singularities of secant varieties

TL;DR

This paper investigates the singularities of secant varieties of smooth projective varieties embedded by positive adjoint bundles, establishing conditions for Du Bois and rational singularities.

Contribution

It proves secant varieties are always Du Bois and characterizes when they have rational singularities, linking positivity of the embedding line bundle to singularity types.

Findings

Secant varieties are always Du Bois.

Conditions for secant varieties to have rational singularities.

Examples showing worse singularities with weaker positivity.

Abstract

We study the singularities of the secant variety $\Sigma(X,L)$ associated to a smooth variety $X$ embedded by a sufficiently positive adjoint bundle $L$. We show that $\Sigma(X,L)$ is always Du Bois singular. Examples of secant varieties with worse singularities when $L$ has weak positivity are provided. We also give a necessary and sufficient condition for $\Sigma(X, L)$ to have rational singularities.