Shigeru Mukai, Moduli of Abelian Surfaces, Polyhedral Groups and Fano 3-folds of Degree 22 (Part 1)

I will discuss the moduli space of abelian surfaces with bi-level structure of type (1, d) for d = 2, 3, 4, 5.

Part 1: Their Satake compactification is the projective 3-space P³ for d = 2, 3, 4.

Part 2: It is a small contraction of the blow-up of P³ at 60 points for d = 5. I give two proofs. One is analytic and uses good automorphic forms constructed by Gritsenko-Nilulin. The other is algebraic and uses U22, one of the Umemura 3-folds, and Reye congruences.