A Generalized RSK for Enumerating Linear Series 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 ℙ r , sending the marked points on C to specified general points in 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.