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Covers of elliptic curves and the moduli space of stable curves. (English) Zbl 1208.14024
Consider genus $$g$$ curves that admit degree $$d$$ covers of an elliptic curve. Varying a branch point, the author gets a one-parameter family $$W$$ of simply branched covers. Varying the target elliptic curve, he gets another one-parameter family $$Y$$ of covers that have a unique branch point. The author also investigates the geometry of $$W$$ and $$Y$$ by using admissible covers to study their slopes, genera and components. The results can be applied to study slopes of effective divisors on the moduli space of stable genus $$g$$ curves.
Contents: 1. Introduction and main results. 2. Geometry of $$W_{d,g}$$; 2.1. Slope; 2.2 Monodromy; 2.3. Application. 3. Geometry of $$Y_{d,g, \sigma}$$; 3.1. Slope; 3.2 Monodromy; 3.3. Application. 4. Density. 5. Related results and open problems.
Motivation is to study the moduli space of stable genus $$g$$ curves $$\overline{\mathcal{M}}_{g}$$ by using one-parameter families of covers of elliptic curves.

##### MSC:
 14H52 Elliptic curves 14H30 Coverings of curves, fundamental group 14H45 Special algebraic curves and curves of low genus 14D20 Algebraic moduli problems, moduli of vector bundles
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