Enumerativity of virtual Tevelev degrees

TL;DR

This paper investigates when virtual Tevelev degrees, which count certain genus g maps in Gromov-Witten theory, are actual counts in the large curve class limit, proving this for specific varieties and applications.

Contribution

It proves the enumerativity of virtual Tevelev degrees for homogeneous varieties and low-degree hypersurfaces, and applies results to low-degree free curves in positive characteristic.

Findings

Virtual Tevelev degrees are enumerative for homogeneous varieties.

Virtual Tevelev degrees are enumerative for low-degree hypersurfaces.

Existence of very free low-degree curves on hypersurfaces in positive characteristic.

Abstract

Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus $g$ maps of fixed complex structure in a given curve class $\beta$ through $n$ general points of a target variety $X$. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class $\beta$. A simple speculation is that for all Fano $X$, the virtual Tevelev degrees are enumerative for sufficiently large $\beta$. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension). As an application, we prove a new result on the existence of very free curves of low degree on hypersurfaces in positive characteristic.