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 $W^r_d(C)$ 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 $\operatorname{Quot}^{r,d}_{\mathbb{P}^1}(\mathcal{O}^{\oplus N})$, which may be of independent interest.