Some remarks on Brauer Classes of K3-type

TL;DR

This paper investigates Brauer group elements of prime order on K3 surfaces, exploring their relation to transcendental lattices and moduli space mappings, with explicit degree computations and Picard lattice implications.

Contribution

It characterizes prime order Brauer classes that induce Hodge-isometric transcendental lattices between K3 surfaces and computes the degree of the associated moduli space map.

Findings

Finite map between moduli spaces of polarized K3 surfaces.

Degree of the map computed explicitly.

Picard lattice of X determines that of S when Picard number of X is two.

Abstract

An element in the Brauer group of a general complex projective $K3$ surface $S$ defines a sublattice of the transcendental lattice of $S$. We consider those elements of prime order for which this sublattice is Hodge-isometric to the transcendental lattice of another K3 surface $X$. We recall that this defines a finite map between moduli spaces of polarized K3 surfaces and we compute its degree. We show how the Picard lattice of $X$ determines the Picard lattice of $S$ in the case that the Picard number of $X$ is two.