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Oberseminar WiSe 12/13 Zusammenfassung des Vortrags von Nicola Pagani

Moduli of abelian covers of elliptic curves

For G any fixed finite abelian group, we study moduli of ramified G-covers of elliptic curves. We show an explicit description of it as a complement of divisors in the iterated universal curve over a modular curve. From this description, we obtain explicit lower bounds for the Kodaira dimension of each component of the moduli space. In the totally ramified case, we prove that the moduli space is birational to a moduli space of elliptic curves with marked points, and that its rational Picard group vanishes. In the case of moduli of bielliptic curves, we can also prove that the boundary divisors are a basis for the rational Picard group of the admissible coverings compactification. Our motivation and methods come from the Gromov-Witten theory of orbifolds, a topic of research that has been very fertile in recent years, particularly for the genus 1 case.