Periods and Moduli of Elliptic Surfaces I

Explore a comprehensive lecture on the periods and moduli of elliptic surfaces delivered by Nicholas Shepherd-Barron from Kings College London. Delve into the intricate relationship between the derivative of the period map associated with the weight 2 Hodge structure of a Kaehler elliptic surface and the derivative of the j-invariant. Discover how this derivative can be used to recover equations for the base curve of the elliptic fibration, with certain constraints on the geometric genus and irregularity of the surface. Examine the proof of a generic Torelli theorem for simple elliptic surfaces without multiple fibres. Gain insights into the plumbing construction for curves and their morphisms, tracing back to Fay's work, and understand how the resulting formulae lead to an interpretation of period map derivatives in terms of rank-one tensors. This 1-hour and 12-minute talk, presented at the University of Miami on March 31, 2021, offers a deep dive into the mathematical foundations of elliptic surfaces and their properties.