Recent developments in Brill–Noether theory

Abstract

We briefly survey recent results related to linear series on curves that are general in various moduli spaces, highlighting the interplay between algebraic geometry on a general curve and the combinatorics of its degenerations. Breakthroughs include the proof of the maximal rank theorem, which determines the Hilbert function of the general linear series of given degree and rank on the general curve in ${\mathcal{M}}_g$, and complete analogues of the standard Brill–Noether theorems for curves that are general in Hurwitz spaces. Other advances include partial results in a similar direction for linear series in the Prym locus of a general unramified double cover of a general $k$-gonal curve and instances of the strong maximal rank conjecture.