Paper detail

Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)

We consider the moduli space $\mathcal{R}_n$ of pairs of monic, degree $n$ polynomials whose resultant equals $1$. We relate the topology of these algebraic varieties to their geometry and arithmetic. In particular, we compute their étale cohomology, the associated eigenvalues of Frobenius, and the cardinality of their set of $\mathbb{F}_q$-points. When $q$ and $n$ are coprime, we show that the étale cohomology of $\mathcal{R}_{n/\bar{\mathbb{F}}_q}$ is pure, and of Tate type if and only if $q\equiv 1$ mod $n$. We also deduce the values of these invariants for the finite field counterparts of the moduli spaces $\mathcal{M}_n$ of $SU(2)$ monopoles of charge $n$ in $\mathbb{R}^3$, and the associated moduli space $X_n$ of strongly centered monopoles. An appendix by Cazanave gives an alternative and elementary computation of the point counts.