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Geometric local systems on very general curves

Aaron Landesman (Harvard/MIT)
Tuesday, May 21, 2024 - 1:30pm
PDL C-38
Title: The Kodaira-Parshin Trick

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

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.
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