Class groups and differential function fields (Q795878)

Let \(V\) be a \(K\)-variety and \(d\) a derivation on the function field of \(V\) such that \(d(K)\subset K\). The paper is concerned with the set \(V_D\) of all points \(p\in V\) for which the local ring of \(V\) at \(p\) is a differential ring with differential maximal ideal. It is proved for instance that for \(K\) algebraically closed and \(V\) a smooth complete curve, the divisor class group of \(V\) is generated modulo torsion by \(V\backslash V_D\). This result is then used to prove that for arbitrary \(V\) and for any subfield \(K_0\) of \(K\) over which \(K\) has infinite transcendence degree, there exists a generic point of \(V\) over \(K_0\) lying outside \(V_D\).