Order 1 strongly minimal sets in differentially closed fields

Abstract

We give a classification of non-orthogonality classes of trivial order 1 strongly minimal sets in differentially closed fields. A central idea is the introduction of $\tau$-forms, functions on the prolongation of a variety which are analogous to 1-forms. Order 1 strongly minimal sets then correspond to smooth projective curves with $\tau$-forms. We also introduce $\tau$-differentials, algebraic versions of $\tau$-forms which are analogous to usual differentials, and develop their basic properties. This enables us to reformulate our classification scheme-theoretically in terms of curves with $\tau$-invertible sheaves. This work partially generalizes and extends results of Hrushovski and Itai.