Insertion algorithms for moduli spaces of curves

Abstract

Moduli spaces of curves are geometric spaces whose points parameterize some class of geometric object. Two such examples are the Grassmannian Gr(k,n) of k-planes through the origin and the space M_{0,n+3} of stable complex curves with n+3 marked points. Schubert calculus has been hugely successful in using combinatorics to understand the geometry and intersection theory of Gr(k,n). A more recent trend in enumerative geometry has been to extend the techniques of Schubert calculus to create a similar story for M_{0,n+3}, using new combinatorial techniques to understand the intersection theory of M_{0,n+3}. This thesis contains two main parts, that each relate respectively to this story for the two aforementioned spaces. For our first main result, 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. In our second main result, we resolve a question of Gillespie, Griffin, and Levinson that asks for a combinatorial bijection between two classes of trivalent trees, tournament trees and slide trees, that both naturally arise in the intersection theory of the moduli space M_{0,n+3} of stable genus zero curves with n+3 marked points. Each set of trees enumerates the same intersection product of certain pullbacks of psi classes under forgetting maps. We give an explicit combinatorial bijection between these two sets of trees using an insertion algorithm that mimics the RSK algorithm for Young tableaux, providing another step towards the broader goal of developing a combinatorial understanding of the theory of moduli spaces of curves that mirrors the success of Schubert calculus. We also classify the words that appear on the slide trees of caterpillar shape via pattern avoidance conditions.