Work

Invariants of the singularities of secant varieties of curves

Year: 2025

Type: article

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 previous one. 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 of even degree. In this setting, we prove that all of the secant varieties are rational homology manifolds, meaning their singular cohomology satisfies Poincaré duality. We then compute the nearby and vanishing cycles for the largest nontrivial secant variety, which is a projective hypersurface.

Source: Journal de l’École polytechnique — Mathématiques

DOI: 10.5802/jep.321

Author: Daniel Brogan

Institution: University of Michigan

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: National Science Foundation

Award: Higher Multiplier Ideals and Other Applications of Hodge Theory in Algebraic Geometry