Rational points of varieties with ample cotangent bundle over function fields
type: Item
instance of: scholarly article
title: Rational points of varieties with ample cotangent bundle over function fields
zbMATH Open document ID: 1419.14027
published in: Mathematische Annalen
publication date: 2018-08-23T00:00:00Z
review text: The paper addresses the following variation of Bombieri-Lang conjecture over function fields. Conjecture 1. Let $K_0$ be the function field of a variety defined over an algebraically closed field $k_0$. Let $Z$ be a variety of general type over $K_0$. Suppose that $\text{Zar}(Z(K_0))=Z$, where Zar denotes the Zariski closure. Then there exists a variety $Z_0$ over $k_0$ and a rational, dominant, generically finite $K_0^{\text{sep}}$-map $g: Z_{0,K_0^{\text{sep}}} \to Z_{K_0^{\text{sep}}}$. Similarly to the original Bombieri-Lang conjecture, Conjecture 1 has been proved (with several contributions) in the case where $Z$ is embeddable in an abelian variety. Another case where Conjecture 1 has been proved (again with several contributions) is when $\dim(Z)=1$. The paper under review proves Conjecture 1.1 in the situation where $K_0$ has transcendence degree 1 over its prime field and $Z$ is a subvariety of a larger variety $Z'$, where $Z'$ is smooth and has ample cotangent bundle over $K_0$. It is worth noticing that this result allows to give examples of varieties of general type in positive characteristic which satisfy Conjecture 1 and are not embeddable in abelian varieties.
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 6923798
zbMATH Keywords: function fields
zbMATH Keywords: Bombieri-lang conjecture
zbMATH Keywords: varieties of general type
Wikidata QID: Q59614007
author: Henri Gillet
author: D. Rössler
MaRDI profile type: Publication
full work available at URL: s00208-017-1569-4
OpenAlex ID: W2181528398
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DOI: 10.1007/S00208-017-1569-4
description: scientific article; zbMATH DE number 6923798