Brill-Noether Generality of Binary Curves

TL;DR

This paper proves that the space of linear series of arbitrary rank on a general binary curve has the expected dimension, extending previous results from low rank to all ranks and including ramification conditions.

Contribution

It generalizes Caporaso's theorem from rank ≤ 2 to arbitrary rank and analyzes the dimension of Osserman-limit linear series on binary curves.

Findings

The space of linear series has the expected dimension when nonempty.

The expected dimension persists under certain ramification conditions.

The results extend known theorems to higher rank cases.

Abstract

We show that the space of linear series of certain multi-degree (including the balanced ones) and rank $r$ on a general binary curve has the expected dimension if nonempty. This generalizes Theorem 24 of Caporaso's paper about binary curves from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$. In addition, we show that this space of linear series is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points lying on different components of the curve.