Singularities of Secant Varieties

Abstract

Abstract. We study the singularities of the secant variety Σ(X, L) associated to a smooth variety X embedded by a sufficiently positive adjoint line bundle L. We show that Σ(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 Σ(X, L) to have rational singularities.