Families of subvarieties of manifolds with a canonical form

2023/12/28 by Yeuk Hay Joshua Lam, Federico Moretti, Lam, Yeuk Hay Joshua +3
Computer Science · Engineering · Mathematics · #14D07 #14J70 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2312.16974

openalex publication_date 2023/12/28 · openalex created_date 2023/12/30 · openalex updated_date 2026/07/28

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⊂ \mathbb Pⁿ⁺¹ of degree ≥ 2n+1, any map C → 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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