Geometric monodromy of double covers of $\mathbb{P}^n$ branched along hyperplane arrangements

Abstract

Let $n\ge1$, let $m\ge n+3$ be even, and let $\ell$ be an odd prime. Over an algebraically closed field in which $2\ell$ is invertible, we determine the geometric mod-$\ell$ and integral $\ell$-adic monodromy of double covers of $\mathbb{P}^n$ branched along ordered arrangements of $m$ hyperplanes in general position, on the negative eigenspace of their middle cohomology. The mod-$\ell$ image is the full symplectic group for odd $n$ and an index-two orthogonal subgroup determined by the spinor norm and determinant for even $n$. The integral image is the full inverse image of the finite image. Over finite fields, we prove generic irreducibility of the numerator of the zeta function in odd dimension and an equidistribution theorem for Frobenius fixed spaces. For $n=1$, this recovers the $\ell$-torsion part of the geometric Cohen-Lenstra distribution for quadratic function fields.