Let be an algebraic variety over an algebraically closed field of characteristic 0, let be a linear system without fixed components on and let be the image of the variety under the mapping given by . The following two theorems are known as the first and the second Bertini theorem, respectively.
1) If , then almost all the divisors of the linear system (i.e. all except a closed subset in the parameter space not equal to ) are irreducible reduced algebraic varieties.
2) Almost all divisors of have no singular points outside the basis points of the linear system and the singular points of the variety .
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 , Bertini's theorem is replaced by the following theorem: Almost all fibres of the mapping are irreducible and reduced if the function field is algebraically closed in the field under the imbedding . If the characteristic of is finite, the corresponding theorem is true if the extension 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=23762