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 $d$.

Revised:

Accepted:

Published online:

Keywords: Young tableaux, RSK algorithm, Schubert calculus, Brill-Noether theory

Gillespie, Maria ^{1};
Reimer-Berg, Andrew ^{1}

@article{ALCO_2023__6_1_1_0, author = {Gillespie, Maria and Reimer-Berg, Andrew}, title = {A {Generalized} {RSK} for {Enumerating} {Linear} {Series} on $n$-pointed {Curves}}, journal = {Algebraic Combinatorics}, pages = {1--16}, publisher = {The Combinatorics Consortium}, volume = {6}, number = {1}, year = {2023}, doi = {10.5802/alco.250}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.250/} }

TY - JOUR AU - Gillespie, Maria AU - Reimer-Berg, Andrew TI - A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves JO - Algebraic Combinatorics PY - 2023 SP - 1 EP - 16 VL - 6 IS - 1 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.250/ DO - 10.5802/alco.250 LA - en ID - ALCO_2023__6_1_1_0 ER -

%0 Journal Article %A Gillespie, Maria %A Reimer-Berg, Andrew %T A Generalized RSK for Enumerating Linear Series on $n$-pointed Curves %J Algebraic Combinatorics %D 2023 %P 1-16 %V 6 %N 1 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.250/ %R 10.5802/alco.250 %G en %F ALCO_2023__6_1_1_0

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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