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

Gillespie, Maria¹ ; Reimer-Berg, Andrew¹

¹ Colorado State University Department of Mathematics 1874 Campus Delivery. Fort Collins CO 80523 (USA)

Algebraic Combinatorics, Tome 6 (2023) no. 1, pp. 1-16

Résumé

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 $d$.

Détail

Reçu le : 2022-01-02
Révisé le : 2022-05-27
Accepté le : 2022-05-27
Publié le : 2023-02-24

DOI : 10.5802/alco.250

Classification : 05E14, 05A05, 14N10
Keywords: Young tableaux, RSK algorithm, Schubert calculus, Brill-Noether theory

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits