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arXiv · 2312.16974

On the locus of curves mapping to a fixed target

Yeuk Hay Joshua Lam·Federico Moretti·Giovanni Passeri
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Abstract

Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications are presented. On the one hand, we show that for a very general curve $C$ and a very general hypersurface $Y\subset \mathbb P^{n+1}$ of degree $\ge 2n+1$, any map $C \to Y$ is constant. On the other hand, we give a lower bound on the genus of a family of curves with an isotrivial factor in the associated family of Jacobians; we also characterize the families of curves attaining this bound as the families of degree $2$ branched covers of a fixed curve.

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Yeuk Hay Joshua Lam, Federico Moretti, Giovanni Passeri. 2023-12-28. On the locus of curves mapping to a fixed target. https://arxiv.org/abs/2312.16974

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