Canonical singularities on moduli spaces of rational curves via the circle method

TL;DR

This paper develops a specialized circle method to prove that the moduli space of degree e rational curves on a smooth hypersurface has only canonical singularities when the dimension is sufficiently large relative to e and d.

Contribution

It introduces a new version of the circle method tailored for analyzing singularities in moduli spaces of rational curves.

Findings

The moduli space has only canonical singularities under certain dimension conditions.

The method applies to smooth hypersurfaces of degree d.

Provides a link between number-theoretic techniques and algebraic geometry.

Abstract

By developing a suitable version of the circle method, we show that the space of degree $e$ rational curves on a smooth hypersurface of degree $d$ has only canonical singularities provided its dimension is sufficiently large with respect to $e$ and $d$.