A short way of counting maps to hypersurfaces in Grassmannians

TL;DR

This paper develops a method using Quot scheme compactification to compute the virtual count of maps from curves to hypersurfaces in Grassmannians, exploring when these counts are genuinely enumerative for large degrees.

Contribution

It introduces a new approach to calculate and analyze virtual counts of maps to Grassmannian hypersurfaces, including conditions for their enumerativity.

Findings

Calculated virtual counts for maps to Grassmannian hypersurfaces.

Identified conditions under which virtual counts are actual enumerations.

Extended understanding of map counts in high-degree regimes.

Abstract

Using a Quot scheme compactification, we calculate the virtual count of maps of degree $d$ from a smooth projective curve of genus $g$ to a hypersurface in a Grassmannian, sending specified points of the curve to special Schubert subvarieties restricted to the hypersurface. We study the question of whether this virtual count is in fact enumerative under suitable conditions on the hypersurface, in the regime when the map degree $d$ is large.