Invariants of the singularities of secant varieties of curves

TL;DR

This paper investigates the singularity invariants of secant varieties of curves, computes their intersection cohomology, and shows that secant varieties of rational normal curves are rational homology manifolds, providing explicit cohomological calculations.

Contribution

It introduces new computations of intersection cohomology for secant varieties of arbitrary curves and proves that secant varieties of rational normal curves are rational homology manifolds.

Findings

Intersection cohomology expressed in terms of the curve's cohomology

Secant varieties of rational normal curves are rational homology manifolds

Computed nearby and vanishing cycles for the largest secant hypersurface

Abstract

Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety being singular along the last. We study invariants of the singularities for these varieties. In the case of an arbitrary curve, we compute the intersection cohomology in terms of the cohomology of the curve. We then turn our attention to rational normal curves. In this setting, we prove that all of the secant varieties are rational homology manifolds, meaning their singular cohomology satisfies Poincar\'e duality. We then compute the nearby and vanishing cycles for the largest nontrivial secant variety, which is a projective hypersurface.