Completeness in partial differential algebraic geometry
This paper studies various aspects of complete differential algebraic varieties. The setting is the following, introduced by Kolchin. If \(F\) is a field of characteristic \(0\) equipped with \(m\) commuting derivations, then one has the notion of \textit{differential algebraic varieties} over \(F\). For simplicity, we fix a \textit{universal domain} \(\mathcal U \supset F\) and we will identify a differential variety with its \(\mathcal U\)-rational points. In this setting, a universal domain \(\mathcal U\) has to be a model of \(\text{DCF}_{0,m}\), i.e., it should be \textit{differentially closed} -- the appropriate replacement for algebraically closed in the presence of the derivations. The Kolchin topology on \(\mathbb A^n = \mathcal U^n\) is the differential analogue of the Zariski topology: closed sets are zero sets of differential polynomials with coefficients in \(F\); in a similar way, one also defines the Kolchin topology on \(\mathbb P^n\). The present paper considers the notion of completeness of quasi-projective differential varieties in this setting, which is defined analogously to the non-differential setting. It generalizes results from \textit{W. Y. Pong} [J. Algebra 224, No. 2, 454--466 (2000; Zbl 1037.12009)] in the \(m=1\) case. The main results are the following: \(\bullet\) Every complete projective differential variety is isomorphic to a closed differential subvariety of \(\mathbb A^1\) (Corollary~4.9). (This is in strong contrast to the non-differential setting; a key ingredient is that \(\mathbb P^1\) is \textit{not} differentially complete.) \(\bullet\) A valuative criterion for completeness is given (Theorem~5.9), similar to the one in the non-differential setting (which says that a variety \(V\) is complete if for every valuation ring \(R\) with quotient field \(K\), the map \(V(R) \to V(K)\) is a bijection). The paper also contains some results about embedability of arbitrary (i.e., not necessarily projective) differential varieties into \(\mathbb A^n\) or \(\mathbb P^n\) (Proposition~4.4, Corollaries 4.15, 4.16). Finally, it contains many examples, which are very helpful to get acquainted with the differential algebraic world.
scientific article; zbMATH DE number 6350924
scientific article