Genera of Brill-Noether curves and staircase paths in Young tableaux

TL;DR

This paper computes the genus of certain linear series varieties on general curves using degenerations, limit linear series, and staircase paths in Young tableaux, providing explicit descriptions in specific cases.

Contribution

It introduces a novel approach combining degenerations and combinatorial staircase paths to analyze the genus of Brill-Noether varieties.

Findings

Computed genus of 1-dimensional Brill-Noether varieties with ramification

Provided explicit scheme-theoretic descriptions for limit linear series

Linked combinatorial staircase paths to algebraic geometry of curves

Abstract

In this paper, we compute the genus of the variety of linear series of rank $r$ and degree $d$ on a general curve of genus $g$, with ramification at least $\alpha$ and $\beta$ at two given points, when that variety is 1-dimensional. Our proof uses degenerations and limit linear series along with an analysis of random staircase paths in Young tableaux, and produces an explicit scheme-theoretic description of the limit linear series of fixed rank and degree on a generic chain of elliptic curves when that scheme is itself a curve.