Moduli of lattice-polarized K3 surfaces and boundedness of Brauer groups

Abstract

Inspired by constructions over the complex numbers of Dolgachev and Alexeev-Engel, we define moduli stacks ${\mathcal{M}}_{(L,\mathcal{A})/\mathbb{Z}}$ of lattice-polarized K3 surfaces over arbitrary bases, paying particular attention to the open locus ${\mathcal{P}}_{(L,\mathcal{A})/\mathbb{Z}}$ of primitive lattice polarizations. We introduce the notion of very small ample cones $a$, after Alexeev and Engel's small cones, to construct smooth, separated stacks of lattice polarized K3 surfaces ${\mathcal{P}}_{(L,a)/\mathbb{Z}[1/N]}$ over suitable open subsets of $\text{Spec}(\mathbb{Z})$. We add level structures, coming from classes in ${\mathrm{H}}^2(X,{\mu }_n)$, to build moduli stacks ${\mathcal{P}}_{(L,\mathcal{A})/\mathbb{Z}}^{[n]}$ with a natural action by ${\mathcal{P}}_{(L,\mathcal{A})}\otimes \mathbb{Z}/n\mathbb{Z}$ whose associated quotient ${\mathcal{Q}}_{(L,\mathcal{A})}^{[n]}$ contains an open substack ${\mathcal{Q}}_{(L,\mathcal{A})}^{(n)}$ whose points parametrize pairs K3 surfaces $X$ such that $\text{Pic}(X)\simeq L$, together with a class $\alpha \in \text{Br}(X)$ of order $n$. When $L$ has rank 19, we show that the coarse moduli space ${\mathrm{Q}}_{(L,a),\mathbb{C}}^{(n)}$ is a union of quasi-projective curves, each isomorphic to an open subvariety of the quotient of the upper half plane by a discrete subgroup of ${\mathrm{SL}}_2(\mathbb{R})$. Fixing a prime $\ell$, we use this comparison to prove that the genus and the gonality of the components of ${\mathrm{Q}}_{(L,a),\mathbb{C}}^{({\ell }^m)}$ grows with $m$, and hence that they have finitely many points over number fields of bounded degree. As an application, we furnish a new proof of a result by Cadoret--Charles, showing uniform boundedness of the $\ell$-primary torsion of Brauer groups of K3 surfaces over number fields varying in a $1$-dimensional lattice-polarized family.

Preliminary version. 55 pages