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

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.