On linear series with negative Brill-Noether number

Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is ${\mathcal{W}}_{d,g}^r$, the moduli space of smooth genus $g$ curves with a choice of degree $d$ line bundle having at least $(r+1)$ independent global sections. The Brill-Noether theorem asserts that the map ${\mathcal{W}}_{d,g}^r\to {\mathcal{M}}_g$ is surjective with general fiber dimension given by the number $\rho = g - (r+1)(g-d+r)$,underthehypothesisthat$0 \leq \rho \leq g$. One may naturally conjecture that for $\rho <0$, this map is generically finite onto a subvariety of codimension $-\rho$ in ${\mathcal{M}}_g$. This conjecture fails in general, but seemingly only when $-\rho$ is large compared to $g$. This paper proves that this conjecture does hold for at least one irreducible component of ${\mathcal{W}}_{d,g}^r$, under the hypothesis that $0 < -\rho \leq \frac{r}{r+2} g - 3r+3$\mathrm{.}Weconjecturethatthisresultshouldholdforall$0 < -\rho \leq g

• C$forsomeconstant$C$, and we give a purely combinatorial conjecture that would imply this stronger result.