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

Ciliberto, Ciro; van der Geer, Gerard

Am. J. Math. 114, No. 3, 551-570 (1992).

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.

Reviewer: Oliver V. Debarre (Orsay)

MSC:

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

Keywords:

Jacobian with nontrivial endomorphism ring; moduli space of smooth projective irreducible complex curves

Citations:

Zbl 0806.14020