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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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