Let $V$ be an algebraic variety over an algebraically closed field $k$ of characteristic 0, let $L$ be a linear system without fixed components on $V$ and let $W$ be the image of the variety $V$ under the mapping given by $L$. The following two theorems are known as the first and the second Bertini theorem, respectively.
1) If $\dim W > 1$, then almost all the divisors of the linear system $L$ (i.e. all except a closed subset in the parameter space $P(L)$ not equal to $P(L)$) are irreducible reduced algebraic varieties.
2) Almost all divisors of $L$ have no singular points outside the basis points of the linear system $L$ and the singular points of the variety $V$>.
Both Bertini theorems are invalid if the characteristic of the field is non-zero.
Conditions under which Bertini's theorems are valid for the case of a finite characteristic of the field have been studied , . If $\dim W = 1$, Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping $\phi-L : V \to W$ are irreducible and reduced if the function field $k(W)$ is algebraically closed in the field $k(V)$ under the imbedding $\phi_L^* : k(W) \to k(V)$. If the characteristic of $k$ is finite, the corresponding theorem is true if the extension $k(V)/k(W)$ is separable , . The Bertini theorems apply to linear systems of hyperplane sections, without restrictions on the characteristic of the field .
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Bertini theorems. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bertini_theorems&oldid=41168