Algebraic K3 Surfaces

This work surveys the theory of K3 surfaces from two complementary perspectives: geometric and arithmetic. On the geometric side, we review the foundational results of Saint-Donat on linear systems on K3 surfaces, including the characterization of ample line bundles and the absence of zero-dimensional base components. We then study applications to the classification of Fano 3-folds, presenting Shokurov’s proof of the existence of smooth anticanonical divisors and results of Iskovskikh on the classification of Fano 3-folds, highlighting the central role played by K3 surfaces in both. On the arithmetic side, we give a brief exposition of elliptic and Jacobian fibrations, torsors under elliptic curves and K3 surfaces, and the Weil-Chatelet, Tate-Shafarevich and Brauer groups, concluding this with a theorem of Grothendieck establishing that, for an elliptic K3 surface with a section, the Brauer, cohomological Brauer and Tate-Shafarevich groups are isomorphic. Finally, we look at the lattice and Hodge structures of the cohomology of K3 surfaces and discuss their relationships to automorphisms and elliptic fibrations, with a view to a result of Bogomolov and Tschinkel regarding the density of rational points on elliptic K3 surfaces or K3 surfaces with an infinite automorphism group.