2023/02/27 by Ethan Cotterill, Cotterill, Ethan, Renato Vidal Martins +1 · 1 citation
Engineering · Mathematics · #14H20 #14H45 #14H51 #20Mxx #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2302.13993
openalex publication_date 2023/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C0, one expects certain aspects of the global geometry of the smooth fibers to ``specialize" to the local geometry of the singularities of C0. Making this expectation quantitatively precise involves analyzing the arithmetic and combinatorics of semigroups \rm 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 ℕ.