Fundamental groups of complements of wobbly divisors in intersections of two quadrics

Source abstract

A smooth complete intersection $X\subset\mathbb{P}^{n+2}_{\mathbb{C}}$ of two quadrics admits an interpretation as a moduli space of bundles. For $n=2$, it parametrizes stable rank $2$ parabolic bundles on $\mathbb{P}^1$ with five marked points, while for $n=3$ it parametrizes stable rank $2$ bundles of odd degree with fixed determinant on a genus $2$ curve. For general $n$, it can be interpreted as a moduli space of semistable twisted $\mathrm{Spin}$-bundles. We study $X\setminus W$, where $W$ is the wobbly locus, consisting of bundles that admit a nonzero nilpotent Higgs field. We prove that $W$ is an irreducible divisor for $n\geq 3$ and compute $H_1(X\setminus W,\mathbb{Z})$. Using a root-stack description, we identify $π_1(X\setminus W)$ with the kernel of a monodromy homomorphism from an orbifold mixed braid group. The Reidemeister-Schreier method then yields an explicit presentation in every dimension. For $n=2$, where $X$ is a degree $4$ del Pezzo surface and $W$ is the union of its sixteen $(-1)$-curves, we obtain a presentation with $10$ generators and $25$ relators and prove that both numbers are minimal among all finite presentations. For $n=3$, we show that $π_1(X\setminus W)$ admits a presentation with $4$ generators and $78$ relators.