Linear series with $\rho < 0$ via thrifty lego-building

TL;DR

This paper investigates the existence of components in the moduli space of algebraic curves with linear series when the expected dimension is negative, extending classical Brill-Noether theory to new parameter ranges.

Contribution

It proves the existence of such components for certain negative ca0c g, using a two-marked-point generalization and the regeneration theorem for limit linear series.

Findings

Existence of components with expected dimension for ca0c g when ca0c ho e; -g+3

Extension of Brill-Noether theory to ca0c ho < 0 cases

Development of inductive methods using two-marked-point generalization

Abstract

The moduli space $\mathcal{G}^r_{g,d} \to \mathcal{M}_g$ parameterizing algebraic curves with a linear series of degree $d$ and rank $r$ has expected relative dimension $\rho = g - (r+1)(g-d+r)$. Classical Brill-Noether theory concerns the case $\rho \geq 0$; we consider the non-surjective case $\rho < 0$. We prove the existence of components of this moduli space with the expected relative dimension when $0 > \rho \geq -g+3$, or $0 > \rho \geq -C_r g + \mathcal{O}(g^{5/6})$, where $C_r$ is a constant depending on the rank of the linear series such that $C_r \to 3$ as $r \to \infty$. These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.