Singularities of the Secant Variety

TL;DR

This paper establishes conditions under which secant varieties of smooth projective varieties are normal, explores their relation to the Hodge conjecture, and analyzes their singular loci, advancing understanding of their geometric properties.

Contribution

It generalizes previous results on secant varieties by providing new positivity conditions for normality and links to the Hodge conjecture.

Findings

Positivity conditions ensure secant variety normality

Certain secant varieties satisfy the Hodge conjecture

Results on the singular locus of degenerate secant varieties

Abstract

We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties.