## 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

Zbl 0806.14020
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