A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves

TL;DR

This paper provides a combinatorial proof of a geometric result on the expected number of degree-$d$ morphisms from a general pointed curve to projective space, using a generalized RSK correspondence and Young tableaux.

Contribution

It introduces a bijection extending RSK to interpret intersection numbers on Grassmannians combinatorially, linking algebraic geometry with Young tableaux and sequences.

Findings

Expected morphism count is $(r+1)^g$ for large $d$

Established a bijection between tableaux and $(r+1)$-ary sequences

Provided a combinatorial proof for the case $r=1$ with multiple points

Abstract

We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-$d$ morphisms from a general genus $g$, $n$-marked curve $C$ to $\mathbb{P}^r$, sending the marked points on $C$ to specified general points in $\mathbb{P}^r$, is equal to $(r+1)^g$ for sufficiently large $d$. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the $(r+1)$-ary sequences of length $g$, and we explore our bijection's combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which $r=1$ and several marked points map to the same point in $\mathbb{P}^1$, the number of morphisms is still $2^g$ for sufficiently large $d$.