Applications of global Bertini theorems (Q2736611)

Let \(k\) be an infinite field of arbitrary characteristic. Let \((A, M, K)\) be a local \(k\)-algebra (\(M\) is the maximal ideal and \(K\) the residue field). Let \(\mathcal{P}\) be a local property of \(A\). Put \(\mathcal{P}(A) := \{ P \in \text{Spec} A \mid A_P\) is \(\mathcal{P}\}\). The author uses the following notation: NEWLINENEWLINENEWLINE\(LB_k (\mathcal{P})\) holds iff for the generic \(\alpha = (\alpha_1, \ldots, \alpha_n) \in k^n\) the inclusion \(\mathcal{P}(A/x_\alpha A)\supseteq \mathcal{P}(A) \cap V (x_\alpha) \cap U_{\mathcal{P}}\) holds (where \(x_\alpha = \sum \alpha_i x_i\), \(\langle x_1, \ldots, x_n\rangle = M\), \(U_{\mathcal {P}}\) is a nonempty Zariski open subset of Spec \(A\) and \(V(x_\alpha)\) is the standard closed Zariski subset of Spec \(A\) associated to \(x_\alpha\)). NEWLINENEWLINENEWLINEIf \(\mathcal {P}\) is a local property and \(A\) is a local ring containing a field \(k\), one says that \(A\) is geometrically \(\mathcal {P}\) iff \(A \otimes_k \bar k\) is \(\mathcal {P}\) (\(\bar k\) being the algebraic closure of \(k\)). If \(\mathcal {P}\) is a local property then the corresponding geometric property is denoted by \(\mathcal {G}\mathcal {P}\). The author proves: NEWLINENEWLINENEWLINE\(LB_k(\mathcal {P})\) holds \(\Rightarrow L B_k (\mathcal {GP})\) holds for any noetherian local \(k\)-algebra essentially of finite type with \(K \supset k\) a separable field extension and for any local property \(\mathcal {P}\). NEWLINENEWLINENEWLINEThe previous implication is then used in order to prove that \(LB_k(\mathcal {GP})\) holds for \(\mathcal {P}=\) regular, normal, reduced, \(R_s\), \(S_t\) (Serre's properties) for any \((A,M,K)\) as before. NEWLINENEWLINENEWLINEAs an application, a Bertini type theorem for hypersurface sections of a variety \(X \subset \mathbb{P}^n_k\) concerning the geometric properties is obtained.NEWLINENEWLINENEWLINE[Editor's comment: This paper is a direct copy of the paper by \textit{M. L.NEWLINESpreafico}, Local Bertini theorems for geometric properties over a nonperfectNEWLINEfield, Rend. Mat. Appl. (7) 13, No.3, 561-572 (1993; Zbl 0820.14003).]