Seminari del Dipartimento

Geometria Algebrica

An important open problem in arithmetic geometry is to identify geometric properties of a variety X defined over Q that guarantee that the set X(k) is dense for some finite extension k of Q (or that admit dense entire curves). We will explain how the usual geometric classification in dimension at least 2 does not provide an adequate answer and present a new type of classification proposed by Campana that tries to answer these type of questions. In particular we will introduce the class of “special varieties” in terms of Bogomolov sheaves, and in terms of Campana Orbifolds, and explain the canonical (and functorial) decomposition of varieties in their “special” and “non-special” parts. Finally we will show how this questions are related to questions of Harris and Abramovich-Colliot-Thélène and discuss interesting constructions in dimension 3. (Based on joint work with Erwan Rousseau and Jiulie TY Wang)