Asymptotic geometric Tevelev degrees of hypersurfaces

TL;DR

This paper provides a simpler geometric method to compute asymptotic counts of maps from a fixed curve to hypersurfaces, extending previous Gromov-Witten theory results for large degrees and small hypersurface degrees.

Contribution

It introduces a straightforward projective geometric approach to compute Tevelev degrees of hypersurfaces, simplifying prior complex virtual count methods.

Findings

Derived explicit asymptotic formulas for Tevelev degrees

Validated the enumerative counts through geometric analysis

Extended results to a broader class of hypersurfaces

Abstract

Let $(C,p_1,\ldots,p_n)$ be a fixed general pointed curve and let $(X,x_1,\ldots,x_n)$ be a smooth hypersurface of degree $e$ and dimension $r$ with $n$ general points. We consider the problem of enumerating maps $f:C\to X$ of degree $d$ (as measured in the ambient projective space) such that $f(p_i)=x_i$. When $e$ is small compared to $r$ and $d$ is large compared to $g$, $e$, and $r$, these numbers have been computed first by passing to a virtual count in Gromov-Witten theory obtained by Buch-Pandharipande, and then by showing (in work of the author with Pandharipande) that the virtual counts are enumerative via an analysis of boundary contributions in the moduli space of stable maps. In this note, we give a simpler computation via projective geometry.