# ALGEBRAIC COMBINATORICS

A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 1-16.

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 ${ℙ}^{r}$, sending the marked points on $C$ to specified general points in ${ℙ}^{r}$, is equal to ${\left(r+1\right)}^{g}$ 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 $\left(r+1\right)$-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 ${ℙ}^{1}$, the number of morphisms is still ${2}^{g}$ for sufficiently large $d$.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.250
Classification: 05E14, 05A05, 14N10
Keywords: Young tableaux, RSK algorithm, Schubert calculus, Brill-Noether theory
Gillespie, Maria 1; Reimer-Berg, Andrew 1

1 Colorado State University Department of Mathematics 1874 Campus Delivery. Fort Collins CO 80523 (USA)
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Gillespie, Maria; Reimer-Berg, Andrew. A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 1-16. doi : 10.5802/alco.250. https://alco.centre-mersenne.org/articles/10.5802/alco.250/

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