Aaron Landesman (Harvard/MIT)

PDL C-38

Title: The Kodaira-Parshin Trick

Abstract:

In the main seminar talk, given genera g and h, I will discuss whether there

exists a surface S mapping to a general curve of genus g with smooth and proper

fibers of genus h, not all of whose fibers are isomorphic.

A natural question is whether such an S ever exists for any values of g and h.

We will describe the classical Kodaira-Parshin trick, which allows one to

construct such surfaces when h is much larger than g.

Abstract:

In the main seminar talk, given genera g and h, I will discuss whether there

exists a surface S mapping to a general curve of genus g with smooth and proper

fibers of genus h, not all of whose fibers are isomorphic.

A natural question is whether such an S ever exists for any values of g and h.

We will describe the classical Kodaira-Parshin trick, which allows one to

construct such surfaces when h is much larger than g.

Title: Geometric local systems on very general curves

Abstract:

What is the smallest genus h of a non-isotrivial curve over the generic genus g curve?

In joint work with Daniel Litt, we show h is more than $\sqrt{g}$ by proving a

more general result about variations of Hodge structure on sufficiently general curves.

As a consequence, we show that local systems on a sufficiently general curve of geometric origin are not Zariski dense in the character variety

parameterizing such local systems. This gives counterexamples to conjectures of Esnault-Kerz and Budur-Wang.

Abstract:

What is the smallest genus h of a non-isotrivial curve over the generic genus g curve?

In joint work with Daniel Litt, we show h is more than $\sqrt{g}$ by proving a

more general result about variations of Hodge structure on sufficiently general curves.

As a consequence, we show that local systems on a sufficiently general curve of geometric origin are not Zariski dense in the character variety

parameterizing such local systems. This gives counterexamples to conjectures of Esnault-Kerz and Budur-Wang.