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Calabi-Yau Manifolds Seminar

Speaker: Rikuto Ito (Nagoya University)

Title: Genus Theory for Principal Rank-21 Cubic Fourfolds

Abstract: Lattice theory plays a powerful role in algebraic geometry. For example, oriented positive-definite even lattices of rank 2 correspond bijectively, up to isomorphism, to singular K3 surfaces (i.e., K3 surfaces with maximal Picard number), as established by the Shioda–Inose correspondence. While isomorphism of lattices is a global notion, lattices that are locally isomorphic need not be globally so; the set of such locally isomorphic lattices constitutes a genus. Describing a genus explicitly is useful for studying the degrees of fields of definition of algebraic varieties. In fact, for singular K3 surfaces, the degree of the field of definition can be bounded below by the number of elements in a certain genus (see Shimada 2006; Schütt 2007).

In this talk, I will present an explicit computation of the genus for principal rank-21 cubic fourfolds and explain how this yields a lower bound for the degree of their fields of definition.