Matthew Hase-Liu : Bounding the singular locus of the moduli of curves on a hypersurface

Posted by Eamon Quinlan-Gallego , part of the Algebraic Geometry Seminar.

At

Feb. 9, 2026, 3 p.m.

In

636 SEO

Abstract

The space of rational curves on a Fano variety X serves as a powerful tool for probing the geometry of X. Even for hypersurfaces, characterizing these spaces is difficult; however, work by Riedl–Yang established they are irreducible and have the expected dimension. In this talk, I will discuss another aspect, namely the singular locus. Specifically, I will show the singular locus of the moduli space of smooth degree e curves on a general low-degree hypersurface is small, i.e. has codimension growing linearly with e. This turns out to use a weird combination of 1. Lehmann–Riedl–Tanimoto's recent work on geometric Manin’s conjecture and 2. Sawin's work on Waring's problem from analytic number theory.