On the local cohomology of secant varieties

TL;DR

This paper investigates the local cohomological properties of secant varieties of smooth projective varieties, revealing conditions under which these varieties have quotient or Gorenstein singularities, and classifying cases with specific singularity types.

Contribution

It determines the local cohomological dimension and Hodge filtration generation level of secant varieties, and classifies when these varieties have quotient or Gorenstein singularities.

Findings

The local cohomological dimension equals the codimension if and only if X is P^1.

Secant varieties of P^1 and elliptic curves are the only cases with local complete intersection singularities.

A complete classification of (X,L) for Gorenstein singularities of secant varieties.

Abstract

Given a sufficiently positive embedding $X\subset\mathbb{P}^N$ of a smooth projective variety $X$, we consider its secant variety $\Sigma$ that comes equipped with the embedding $\Sigma\subset\mathbb{P}^N$ by its construction. In this article, we determine the local cohomological dimension $\textrm{lcd}(\mathbb{P}^N,\Sigma)$ of this embedding, as well as the generation level of the Hodge filtration on the topmost non-vanishing local cohomology module $\mathcal{H}^{q}_{\Sigma}(\mathcal{O}_{\mathbb{P}^N})$, i.e., when $q=\textrm{lcd}(\mathbb{P}^N,\Sigma)$. Additionally, we show that $\Sigma$ has quotient singularities (in which case the equality $\textrm{lcd}(\mathbb{P}^N,\Sigma)=\textrm{codim}_{\mathbb{P}^N}(\Sigma)$ is known to hold) if and only if $X\cong\mathbb{P}^1$. We also provide a complete classification of $(X,L)$ for which $\Sigma$ has ($\mathbb{Q}$-)Gorentein singularities. As a consequence, we deduce that if $\Sigma$ is a local complete intersection, then either $X$ is isomorphic to $\mathbb{P}^1$, or an elliptic curve.