Minimal types in separably closed field

Abstract

In [1], examples of types of U-rank 1 (i.e., minimal types) in the theories of separably closed fields were constructed, en route to displaying certain dimension phenomena. We construct here additional examples with U-rank 1 and of various transcendence degrees over arbitrary separably closed fields. Our examples include ones which are minimal but of infinite transcendence degree, i.e., not thin. Our interest in building new examples was piqued after seeing the role played by minimal types over separably closed fields in Hrushovski's analysis of abelian varieties. This article is the result of several working sessions between the authors at Wesleyan University and Paris 7, and was completed during the Model Theory of Fields program at MSRI in 1998. We are grateful for the hospitality and support of all three institutions. We thank Elisabeth Bouscaren and Françoise Delon for reading an earlier version of this paper, providing useful suggestions and corrections.