On Bertini's theorem
Let \(X\) be a smooth projective variety with \(\dim X\geq 2\). If \[ |L|= \bigl\{\lambda_0 Q_0+ \lambda_1Q_1+ \cdots+ \lambda_kQ_k \mid (\lambda_0, \dots, \lambda_k) \in\mathbb{P}^k \bigr\} \] is a \(k\)-dimensional linear system of divisors without fixed component on \(X\), then Bertini's theorem [see \textit{E. Bertini}, ``Introduzione alla geometria proiettiva degli iperspazi con appendice sulle curve algebriche e lero singolarità'' (Pisa 1907; JFM 38.0582.02)] says that: (1) The generic element \(D\in|L|\) is nonsingular away from the base locus \(B\) of \(|L|\). (2) If \(P\in X\) is an arbitrary point on \(X\) and \(D\in |L|\) is a generic element, then \[ \inf_{i\in \{0,1, \dots, k\}} \text{mult}_PQ_i \geq\text{mult}_PD-1, \] in particular, if \(P\notin B\) the base locus of \(|L|\), that is, \(P\notin Q_i\) for some \(i\), then we must have \(\text{mult}_P D\leq 1\), that is, the generic element \(D\) is smooth at any point \(P\) outside of the base locus \(B\) of \(|L|\). Therefore, one can view statement (1) in Bertini's theorem as a special case of (2). In this paper, we study the question of what type of singularities precisely a generic element of a linear system can afford inside the base locus, and give a generalization of statement (2) in Bertini's theorem.
scientific article; zbMATH DE number 866133