Flatness criteria
Let \(f : (X,x) \to (Y,y)\) be a morphism of local schemes and \((Z,x)\) the central fiber. The flatness of \(f\) is a subtle geometric notion and it is difficult to find some flatness criteria which do not require infinitesimal properties. A well-known result of \textit{H. Hironaka} [Ill. J. Math. 2, 355-366 (1958; Zbl 0099.15701)] gives such a criterion: Assume that (i) \(Z\) is generically reduced and \(\text{red} Z\) is normal; (ii) \(Y\) is regular and \(X\) has no embedded points, and (iii) \(f\) is of pure dimension; then \(f\) is flat and \(Z\) is normal. But this criterion is not always sufficient in the study of the deformations of canonical varieties or when we compare the Hilbert schemes and the Chow varieties [cf. \textit{J. Kollár}, ``Rational curves on algebraic varieties'', Springer, Berlin (1995)]. In the paper under review, the author gives some generalized criteria of flatness. For example, when \(Y\) is excellent, we can assume in the Hironaka criteria that \(Y\) is normal (instead of regular) and, if \(\text{char} k(y) > 0\), that \(f\) is essentially of finite type, with the same conclusion: \(f\) is flat and \(Z\) normal. Some versions with sheaves are also given.
scientific article; zbMATH DE number 783944