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

Abstract

The moduli space ${\mathcal{G}}_{g,d}^r\to {\mathcal{M}}_g$ 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 $\rho \ge 0$; we consider the non-surjective case $\rho <0$. We prove the existence of components of this moduli space with the expected relative dimension when $0>\rho \ge -g+3$, or $0>\rho \ge -C_{r}g+\mathcal{O}(g^{5/6})$, 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.

arXiv comment27 pages. v2: added Appendix A, on connections to Hurwitz-Brill-Noether theory, and made minor corrections