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Subvarieties of the moduli space of curves parametrizing Jacobians with non-trivial endomorphisms. (English) Zbl 0766.14016

Let \({\mathcal M}_ g\) be the moduli space of smooth projective irreducible complex curves of genus \(g\). The authors and M. Teixidor i Bigas [J. Algebr. Geom. 1, No. 2, 215–229 (1992; Zbl 0806.14020)] proved that any irreducible subvariety of \({\mathcal M}_ g\) whose generic point corresponds to a curve the Jacobian of which has a nontrivial endomorphism ring (i.e. \(\neq\mathbb Z)\), has dimension \(\leq 2g-2\). Moreover, for \(g\geq 3\), the subvarieties with dimension \(2g-2\) correspond to curves that are coverings of elliptic curves. The main result of this article is that for \(g\geq 5\), \(g\neq 6\), the subvarieties with dimension \(2g-3\) correspond to curves that are either coverings of elliptic curves or double covers of curves of genus 2. The proof proceeds as in the paper cited above: An infinitesimal calculation exhibits the tangent space to the set of Jacobians with nontrivial endomorphisms. If this space is large, it imposes strong geometric conditions on the curve.

MSC:

14H10 Families, moduli of curves (algebraic)
14H40 Jacobians, Prym varieties

Citations:

Zbl 0806.14020
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