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Presented By: Department of Mathematics

Boundedness of hyperbolic varieties

Jackson Morrow (UC Berkeley)

Abstract: Let C_1, C_2 be smooth projective curves over an algebraically closed field K of characteristic zero. What is the behavior of the set of non-constant maps from C_1 to C_2? Is it infinite, finite, or empty? It turns out that the answer to this question is determined by an invariant of curves called the genus. In particular, if C_2 has genus g(C_2) at least 2 (i.e., C_2 is hyperbolic), then there are only finitely many non-constant morphisms from C_1 to C_2, where C_1 is any curve, and moreover, the degree of any map from C_1 to C_2 is bounded linearly in g(C_1) by the Riemann--Hurwitz formula.

In this talk, I will explain the above story and discuss a higher dimensional generalization of this result. To this end, I will describe the conjectures of Demailly and Lang which predict a relationship between the geometry of varieties, topological properties of Hom-schemes, and the behavior of rational points on varieties. To conclude, I will sketch a proof of a variant of these conjectures, which roughly says that if X/K is a hyperbolic variety, then for every smooth projective curve C/K of genus g(C), the degree of any map from C to X is bounded uniformly in g(C).

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