Oberseminar SoSe 2010 Zusammenfassung des Vortrags von H. Ohashi (RIMS, JAP)
Hutchinson-Weber involutions degenerate exactly when the Jacobian is Comessatti
We will consider the K3 surface X naturally constructed from a curve C of genus two, known as the Jacobian Kummer surface. We define the notion of a Weber hexad W, which is a six-elemented set of 2-torsion points of the Jacobian J(C). For every such W, we can define an involutive automorphism on X called the Hutchinson-Weber (HW) involution.
When C and X are general enough, HW involutions are fixed-point-free and determine Enriques surfaces. But at the boundary of the moduli of Enriques surfaces they degenerate to have fixed points.
In the seminar, starting with the construction of HW involutions, we show that the degeneration leads us to find the Comessatti abelian surfaces, namely abelian surfaces with the automorphism (1+√ 5 ) /2 . Interpreting this condition as the existence of special curves on various models of X, we can find the classical theorem of Humbert.
If we have time, we also introduce an equivalence relation among Weber hexads and show that it is related to the outer-automorphism of the symmetric group of degree six. This fact is one of the reasons for the complexity of HW involutions and the subtle condition of Humbert's theorem.