Prym-Brill-Noether Loci of special curves

TL;DR

This paper employs Young tableaux to analyze Prym-Brill-Noether loci of special curves, providing new bounds, topological properties, and explicit computations for these loci in tropical and algebraic settings.

Contribution

It introduces a tropical approach using Young tableaux to determine dimensions and properties of Prym-Brill-Noether loci for special curves, advancing understanding in algebraic geometry.

Findings

Derived a new upper bound on Prym-Brill-Noether locus dimensions.

Proved the loci are pure-dimensional and connected in codimension 1 when positive dimension.

Computed the first Betti number and cardinality for specific cases.

Abstract

We use Young tableaux to compute the dimension of $V^r$, the Prym-Brill-Noether locus of a folded chain of loops of any gonality. This tropical result yields a new upper bound on the dimensions of algebraic Prym-Brill-Noether loci. Moreover, we prove that $V^r$ is pure-dimensional and connected in codimension $1$ when $\dim V^r \geq 1$. We then compute the first Betti number of this locus for even gonality when the dimension is exactly $1$, and compute the cardinality when the locus is finite and the edge lengths are generic.