Limit linear series and ranks of multiplication maps

TL;DR

This paper introduces a new technique using limit linear series and degenerations to analyze multiplication map ranks, providing elementary criteria for the Maximal Rank Conjecture and new insights into the moduli space of curves.

Contribution

It develops an elementary criterion for the Maximal Rank Conjecture using limit linear series, enabling proofs for cases involving quadrics and cubics without approach restrictions.

Findings

Proved cases of the Maximal Rank Conjecture for quadrics and cubics.

Provided new conditions for the locus where maximal rank fails.

Developed a technique that avoids restrictions on approach directions.

Abstract

We develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus-1 curves. We use this approach to prove a purely elementary criterion for proving cases of the Maximal Rank Conjecture, and then apply the criterion to several ranges of cases, giving a new proof of the case of quadrics, and also treating several families in the case of cubics. Our proofs do not require restrictions on direction of approach, so we recover new information on the locus in the moduli space of curves on which the maximal rank condition fails.