Around Kummer surfaces

A Kummer surface was first found by A. Fresnel (1822) for special case and by E. Kummer (1864) for a general one. Later F. Klein (1870) discovered a relation between quadratic line complexes and Kummer surfaces, studied their automorphisms and raised a question on the automorphism group. In this lecture we discuss two topics: Part 1: The Leech lattice and the automorphism group of a generic Kummer surface. By applying the theory of Leech lattice, we present a generator of the automorphism group of a generic Kummer surface associated with a curve of genus 2. This gives an answer of Klein’s question. Main reference is S. Kondo, K3 surfaces, EMS 2020, the last chapter.