On the existence of projective, absolutely normal models of an algebraic variety (Q2530642)

This paper studies the characterization of the absolutely irreducible algebraic varieties, \(V\), of a projective space over an universal domain that are absolutely normal, that is, for which the method of normalization of \(V\) over a given domain of separability of \(V\) is independent of the definition field of \(V\). For \(\dim V = 1\), this question is solved by the invariability of the geometric genus [\textit{M. Rosenlicht}, Ann. Math. (2) 56, 169--191 (1952; Zbl 0047.14503); \textit{R. Mallol}, Math. Ann. 140, 344--350 (1960; Zbl 0096.36301)]. The question is reduced to the unidimensional case, obtaining the following result: (Theorem 9) If \(V\) is a \(\Delta\)-algebraic variety separable over \(\Delta\), and for some character of homogeneity, \(m\), of \(V\) [\textit{O. Zariski}, Am. J. Math. 61, 249--294 (1939; Zbl 0020.39101); Trans. Am. Math. Soc. 53, 490--542 (1943; Zbl 0061.33004); Bull. Am. Math. Soc. 48, 402--413 (1942; Zbl 0063.08388)] \[ g_{\Delta}^{(m)} = g_{\Delta'}^{(m)} \] \((g_{\Delta}^{(m)}\) is the \(\Delta\)-genus of intersection of order \(m\) of \(V\), and \(\Delta'\) the perfect closure of \(\Delta\) in the universal domain), then every locally normal model of \(V\) over \(\Delta\) is absolutely normal.