Degenerations of complete collineations and geometric Tevelev degrees of $\mathbb{P}^r$

TL;DR

This paper computes the number of degree d maps from a genus g curve to projective space satisfying point conditions, extending known results to higher dimensions and arbitrary incidences using degenerations of complete collineations.

Contribution

It provides a complete enumeration of such maps for projective spaces of any dimension with arbitrary incidence conditions, expanding previous results limited to specific cases.

Findings

Explicit formulas for Tevelev degrees in all dimensions.

Extension of known counts to arbitrary incidence conditions.

Application of degeneration techniques to complete collineations.

Abstract

We consider the problem of enumerating maps $f$ of degree $d$ from a fixed general curve $C$ of genus $g$ to $\mathbb{P}^r$ satisfying incidence conditions of the form $f(p_i)\in X_i$, where $p_i\in C$ are general points and $X_i\subset\mathbb{P}^r$ are general linear spaces. We give a complete answer in the case where the $X_i$ are points, where the counts, the ``Tevelev degrees'' of $\mathbb{P}^r$, were previously known only when $r=1$, when $d$ is large compared to $r,g$, or virtually in Gromov-Witten theory. We also give a complete answer in the case $r=2$ with arbitrary incidence conditions. Our main approach studies the behavior of complete collineations under various degenerations.