The number of genus 2 covers of an elliptic curve
نویسنده
چکیده
The main aim of this paper is to determine the number cN,D of genus 2 covers of an elliptic curve E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈ Div(E). In the case that D is reduced, this formula is due to Dijkgraaf. The basic technique here for determining cN,D is to exploit the geometry of a certain compactification C = CE,N of the universal genus 2 curve over the Hurwitz space HE,N which classifies (normalized) genus 2 covers of degree N of E. Thus, a secondary aim of this paper is to study the geometry of C. For example, the structure of its degenerate fibres is determined, and this yields formulae for the numerical invariants of C which are also of independent interest.
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تاریخ انتشار 2006