Brill--Noether loci in genus $\le 12$

A refined Brill–Noether theory seeks to determine which linear series are admitted by a ``general’’ curve in a particular Brill–Noether locus. However, as Brill–Noether loci are not irreducible in general, a coarse answer is given by the relative positions of Brill–Noether loci. Via an analysis of unstable Lazarsfeld–Mukai bundles on K3 surfaces, we distinguish Brill–Noether loci and provide expectations for the relative positions of Brill–Noether loci in general. Together with classical results, the refined Brill–Noether theory for curves of fixed gonality and on Hirzebruch surfaces, and explicit constructions, we identify the relative positions of all Brill–Noether loci in genus $g\le 12$.