Linear series on general curves with prescribed incidence conditions

TL;DR

This paper computes the number of linear series with prescribed incidence conditions on general curves, using degeneration and Schubert calculus, providing complete formulas for large degrees and special cases.

Contribution

It offers new formulas and proofs for counting linear series with incidence conditions on general curves, extending previous results.

Findings

Complete formulas for linear series counts when degree is large

Explicit counts for cases where r=1 or n=r+2

Generalization of recent results by Tevelev and Cela-Pandharipande-Schmitt

Abstract

Using degeneration and Schubert calculus, we consider the problem of computing the number of linear series of given degree $d$ and dimension $r$ on a general curve of genus $g$ satisfying prescribed incidence conditions at $n$ points. We determine these numbers completely for linear series of arbitrary dimension when $d$ is sufficiently large, and for all $d$ when either $r=1$ or $n=r+2$. Our formulas generalize and give new proofs of recent results of Tevelev and of Cela-Pandharipande-Schmitt.