Limit linear series and the Amini-Baker construction

TL;DR

This paper compares different theories of limit linear series across algebraic and metric graphs, demonstrating their shared properties and limitations, and explores implications for Brill-Noether theory and smoothability.

Contribution

It introduces a unified perspective on limit linear series theories, establishing their common satisfaction of Riemann and Clifford inequalities and analyzing Brill-Noether generality.

Findings

All theories satisfy Riemann and Clifford inequalities.

Negative results on Brill-Noether generality for certain metric graph families.

Further work shows smoothability of limit linear series and divisors.

Abstract

We draw comparisons between the author's recent construction of limit linear series for curves not of compact type and the Amini-Baker theory of limit linear series on metrized complexes, as well as the related theories of divisors on discrete graphs and on metric graphs. From these we conclude that the author's theory (like the others) satisfies the Riemann and Clifford inequalities. Motivated by our comparisons, we also develop negative results on Brill-Noether generality for certain families of metric graphs. Companion work of He develops our comparisons further and uses them to prove new results on smoothability of Amini-Baker limit linear series and of divisors on metric graphs.