Towards Brill--Noether theory for cuspidal curves

Understanding when an abstract complex curve of given genus comes equipped with a map of fixed degree to a projective space of fixed dimension is a foundational question; and Brill–Noether theory addresses this question via linear series, which algebraically codify maps to projective targets. Classical Brill–Noether theory, which focuses on smooth curves, has been intensively explored; but much less is known for singular curves, particularly for those with non-nodal singularities. In a one-parameter family of smooth curves specializing to a singular curve $C_0$, one expects certain aspects of the global geometry of the smooth fibers to ``specialize" to the local geometry of the singularities of $C_0$. Making this expectation quantitatively precise involves analyzing the arithmetic and combinatorics of semigroups $\mathrm{S}$ attached to discrete valuations defined on (the local rings of) these singularities. In this largely-expository note we focus primarily on Brill–Noether-type results for curves with {\it cusps}, i.e., unibranch singularities; in this setting, the associated semigroups are {\it numerical} semigroups with finite complement in $\mathbb{N}$.