Resolving the Singularities of Splitting Loci

We construct modular resolutions of singularities for splitting loci, and use them to show that tame splitting loci have rational singularities. As a corollary of our results and Hurwitz-Brill-Noether theory, we prove that if $C$ is a general $k$-gonal curve, the components of $W_d^r(C)$ have rational singularities. We also recover the classical Gieseker-Petri theorem. Along the way, we prove a cohomology vanishing statement for certain tautological vector bundles on ${\mathrm{Quot}}_{{\mathbb{P}}^1}^{r,d}({\mathcal{O}}^{\oplus N})$, which may be of independent interest.