An axiomatic approach to the second theorem of Bertini
As indicated by the title, this paper studies a set of axioms for a local property \({\mathcal P}\) sufficient for deducing a Bertini theorem for \({\mathcal P}\). These are: \((A1)\quad {\mathcal P}\) is stable under pullback by a flat morphism with regular fibers; \((A3)\quad {\mathcal P}\) is open on schemes of finite type over a field;\ and \((A2): If\) \(Y\to S\) is a morphism of finite type, where Y is excellent and S is integral with generic point \(\xi\) ; if \(Y_{\xi}\) is geometrically \({\mathcal P}\), then there exists an open neighbourhood U of \(\xi\) in S such that \(Y_ s\) is geometrically \({\mathcal P}\) for all \(s\in U.\) The following theorem is then proved: Let X be a scheme of finite type over an algebraically closed field k, let \(\phi: X\to {\mathbb{P}}^ n_ k\) be a morphism with separably generated (not necessarily algebraic) residue field extensions. Suppose X has a local property \({\mathcal P}\) verifying (A1) and (A2). Then there exists a nonempty open subset U of \(({\mathbb{P}}^ n_ k)^*\) such that \(\phi^{-1}(H)\) has the property \({\mathcal P}\) for each hyperplane \(H\in U.\) The axioms (A1-3) are then discussed in particular for the example \({\mathcal P}=WN\), the property of being weakly normal. The important part is theorem 2: that WN satisfies (A2) [see also \textit{C. Cumino}, \textit{S. Greco}, and \textit{M. Manaresi}, Compos. Math. 48, 351-362 (1983; Zbl 0511.14025)].
scientific article; zbMATH DE number 3991654