Prym–Brill–Noether Loci of Special Curves

Abstract

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 1st 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.