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Session 2. Algebraic Geometry
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Hurwitz spaces of torus covers: irreducibility conjectures and degree calculations |
Martin Möller, Goethe Universität Frankfurt, Germany
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The talk is based on the joint work with André Kappes
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Hurwitz spaces for covers of the projective line or with many branch
points are connected and their degree in known by representation
theory. Here, on the contrary, we consider Hurwitz spaces for
branched covers of the torus branched over one point only. Interest
in this particular case stems from the theory of Teichmüller
curves.
Even for genus two covers, the components of these Hurwitz spaces
are only conjecturally known. We present these conjectures, compute
the degree of the Hurwitz spaces and their classes in the Picard
groups of split Hilbert modular surfaces. The method relies on
theta functions and intersection theory on the universal family of
abelian surfaces.
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