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We describe the Hodge theory of brilliant families of K3 surfaces. Theircharacteristic feature is a close link between the Hodge structures of any twofibres over points in the Noether-Lefschetz locus. Twistor deformations, theanalytic Tate-Safarevic group, and one-dimensional Shimura special cycles arecovered by the theory. In this setting, the Brauer group is viewed as theNoether-Lefschetz locus of the Brauer family or as the specialization of theNoether-Lefschetz loci in a family of approaching twistor spaces. Passing fromone algebraic twistor fibre to another, which by construction is atranscendental operation, is here viewed as first deforming along the morealgebraic Brauer family and then along a family of algebraic K3 surfaces.

Brilliant families of K3 surfaces: Twistor spaces, Brauer groups, and Noether-Lefschetz loci