Completeness of Finite-Rank Differential Varieties

Abstract

Differential algebraic geometry offers tantalizing similarities to the algebraic version as well as puzzling anomalies. This thesis builds on results of Kolchin, Blum, Morrison, van den Dries, and Pong to study the problem of completeness for projective differential varieties. The classical fundamental theorem of elimination theory asserts that if V is a projective algebraic variety defined over an algebraically closed field K and W is any algebraic variety defined over K, then the projection VxW -> W takes Zariski-closed sets to Zariski-closed sets. Differential varieties defined by differential polynomial equations over a differentially closed field are more complicated. We give the first example of an incomplete finite-rank differential variety, as well as new instances of complete differential varieties. We also explain how model theory yields multiple versions of Pong's valuative criterion for completeness and reduces the differential completeness problem to one involving algebraic varieties over the complex numbers.