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

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 \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 $r\to \mathrm{\infty }$. These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.