2022/01/02 by Maria Gillespie, Gillespie, Maria, Andrew Reimer-Berg +1 · 2 citations
Mathematics · #05E14 (Primary) 05A05 #14N10 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2201.00416
openalex publication_date 2022/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℙr, sending the marked points on C to specified general points in ℙr, 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 ℙ1, the number of morphisms is still 2g for sufficiently large d.