New examples (and counterexamples) of complete finite-rank differential varieties

TL;DR

This paper explores the concept of completeness in differential algebraic varieties, introduces new criteria for completeness, and provides novel examples and counterexamples, highlighting differences from classical algebraic varieties.

Contribution

It extends existing work by establishing differential valuative criteria and presenting the first examples of incomplete, finite-rank projective differential varieties.

Findings

New differential valuative criteria for completeness

Examples of complete differential varieties

First examples of incomplete, finite-rank projective differential varieties

Abstract

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a "differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties.