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

Maria Gillespie, Andrew Reimer-Berg

TL;DR

The paper addresses enumerating linear series on $n$-pointed curves by translating Grassmannian intersection formulas into a combinatorial framework of $L$-tableaux that generalize the RSK correspondence. It proves that the number of $L$-tableaux with parameters $(g,r,d)$ equals $(r+1)^g$ for $d\ge g+r$, via a bijection to $(r+1)$-ary sequences; it also provides a parallel $L'$-tableaux interpretation yielding $2^g$ for $d\ge g+k$. The approach unifies Schubert calculus with explicit tableau combinatorics, and recovers the Castelnuovo case through a red-to-purple bijection connected to standard Young tableaux. The results yield new purely combinatorial proofs of geometric counts and offer a toolkit for variants with ramification and degenerations on $n$-pointed curves.

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 $d$.

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

TL;DR

The paper addresses enumerating linear series on -pointed curves by translating Grassmannian intersection formulas into a combinatorial framework of -tableaux that generalize the RSK correspondence. It proves that the number of -tableaux with parameters equals for , via a bijection to -ary sequences; it also provides a parallel -tableaux interpretation yielding for . The approach unifies Schubert calculus with explicit tableau combinatorics, and recovers the Castelnuovo case through a red-to-purple bijection connected to standard Young tableaux. The results yield new purely combinatorial proofs of geometric counts and offer a toolkit for variants with ramification and degenerations on -pointed curves.

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- morphisms from a general genus , -marked curve to , sending the marked points on to specified general points in , is equal to for sufficiently large . 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 -ary sequences of length , 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 and several marked points map to the same point in , the number of morphisms is still for sufficiently large .
Paper Structure (18 sections, 17 theorems, 29 equations, 3 figures)

This paper contains 18 sections, 17 theorems, 29 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgments
  4. Background
  5. The RSK correspondence on words
  6. Linear series and calculations in the Grassmannian
  7. The iterated Pieri rule
  8. $L$-tableaux and enumeration by $(r+1)^g$
  9. The $L$-tableaux
  10. Enumeration by $(r+1)^g$
  11. The case $r=1$
  12. The Castelnuovo case
  13. $L'$-tableaux and enumeration by $2^g$
  14. The $L'$-tableaux
  15. Enumeration by $2^g$
  16. ...and 3 more sections

Key Result

Theorem 1.3

The number of $L$-tableaux with parameters $(g,r,d)$ is $(r+1)^g$ whenever $d\ge g+r$.

Figures (3)

  • Figure 1: The Young diagram (shaded) of the partition $(5,5,2,1)$, drawn inside a $5\times 6$ grid. This corresponds to the Schubert class $\sigma_{(5,5,2,1)}$ in $A^\bullet(\mathrm{Gr}(5,11))$.
  • Figure 2: At left, the horizontal strip $(6,4,4,3)/(4,4,3,1)$. At right, the vertical strip $(4,3,3,3)/(3,2,2,2)$. Each is drawn as a set of shaded boxes.
  • Figure 3: A tableau in $\mathrm{TrSSYT}(5,4,3)$ and its image under $\varphi_3$ in $\mathrm{TrSSYT}(5,4,2)$.

Theorems & Definitions (55)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 2.1: RSK for words
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5: Pieri rules
  • ...and 45 more