  We establish precise counts of cubic $\mathbb{F}_q[t]$-algebras and quartic $\mathbb{F}_q[t]$-algebras with cubic resolvent of discriminant $q^b$. To get these counts, the main geometric input is our construction of smooth compactifications of simply branched degree $d$ Hurwitz stacks in mixed characteristic for $d \in \{3,4,5\}$ as stacks of orbifold quasimaps to $B\mathbf{S}_d$, which is presented as a GIT quotient stack $[V_d^{ss}/G_d]$ for a prehomogeneous vector space $(G_d,V_d)$. The main number-theoretic input is Igusa's computations of the Igusa zeta functions for $(G_3,V_3)$ and $(G_4,V_4)$. Our compactifications of Hurwitz stacks of a curve $C$ admit proper extensions of the branch map to $\mathrm{Sym}^n(C)$. Because these extensions are small, we can compute the intersection cohomology of certain local systems on $\mathrm{Conf}^n(C) \subset \mathrm{Sym}^n(C)$ as the cohomology of our compactifications, answering questions of Ellenberg-Tran-Westerland and Kapranov-Schechtman. Using a comparison theorem of Kapranov-Schechtman, we compute the part of the cohomology of a 576-dimensional Nichols algebra $\mathfrak{B}_4$ that is invariant under a natural $\mathbf{S}_4$-action. Because our counts are precise, we obtain two answers to Venkatesh's question about the topological origin of secondary terms in arithmetic counts. 
Hurwitz spaces, Nichols algebras, and Igusa zeta functions