arXiv 2302.139932023-02-27math.AGmath.CO

尖点曲线的Brill-Noether理论

Towards Brill--Noether theory for cuspidal curves

Ethan Cotterill, Renato Vidal Martins

备注：: 10 pages, 1 figure; submitted to proceedings of the 2022 GS & MS workshop in Córdoba

摘要：

理解当一个抽象的复杂曲线的给定属来配备一个地图的固定程度的一个射影空间的固定尺寸是一个基本问题;和布里尔-诺特理论解决这个问题通过线性系列，代数编纂地图投影目标。经典的Brill-Noether理论主要研究平滑曲线，但对于奇异曲线，特别是非节点奇异曲线的研究却很少。在专用于奇异曲线$C_0$的单参数平滑曲线族中，人们期望光滑纤维的全局几何的某些方面”专用于“$C_0$的奇点的局部几何。要使这种期望在数量上精确，需要分析半群${\rm S}$的算术和组合学，这些半群附着在这些奇点（的局部环）上定义的离散赋值上。在这篇简要的笔记中，我们主要关注具有{ \it尖点}的曲线的Brill-Noether型结果，即，单支奇点;在这种情况下，相关的半群是在$\mathbb{N}$中具有有限补的{ \it数值}半群。

英文摘要：

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 ${\rm S}$ 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 $\mathbb{N}$.

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