Generalized RSK for enumerating projective maps from n-pointed curves
Date
2022
Authors
Reimer-Berg, Andrew, author
Gillespie, Maria, advisor
Ghosh, Sudipto, committee member
Hulpke, Alexander, committee member
Shoemaker, Mark, committee member
Journal Title
Journal ISSN
Volume Title
Abstract
Schubert calculus has been studied since the 1800s, ever since the mathematician Hermann Schubert studied the intersections of lines and planes. Since then, it has grown to have a plethora of connections to enumerative geometry and algebraic combinatorics alike. These connections give us a way of using Schubert calculus to translate geometric problems into combinatorial ones, and vice versa. In this thesis, we define several combinatorial objects known as Young tableaux, as well as the well-known RSK correspondence between pairs of tableaux and sequences. We also define the Grassmannian space, as well as the Schubert cells that live inside it. Then, we describe how Schubert calculus and the Littlewood-Richardson rule allow us to turn problems of intersecting geometric spaces into ones of counting Young tableaux with particular characteristics. 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.