Dimension counts for limit linear series on curves not of compact type

TL;DR

This paper extends the Brill-Noether theorem to certain non-compact type curves, analyzing limit linear series and their dimensions, with implications for tropical Brill-Noether theory.

Contribution

It proves a generalized Brill-Noether theorem for multivanishing sequences and establishes expected dimensions for limit linear series on specific non-compact type curves.

Findings

Spaces of limit linear series have expected dimension under maximal codimension gluing conditions

Identifies cases where the expected dimension holds in various curve families

Provides new insights into tropical Brill-Noether theory

Abstract

We first prove a generalized Brill-Noether theorem for linear series with prescribed multivanishing sequences on smooth curves. We then apply this theorem to prove that spaces of limit linear series have the expected dimension for a certain class of curves not of compact type, whenever the gluing conditions in the definition of limit linear series impose the maximal codimension. Finally, we investigate these gluing conditions in specific families of curves, showing expected dimension in several cases, each with different behavior. One of these families sheds new light on the work of Cools, Draisma, Payne and Robeva in tropical Brill-Noether theory.