Theorem of Bertini
In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0. Statement for hyperplane sections of smooth varieties. Let "X" be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space formula_1. Let formula_2 denote the complete system  of hyperplane divisors in formula_1. Recall that it is the dual space formula_4 of formula_1 and is isomorphic to formula_1. The theorem of Bertini states that the set of hyperplanes not containing "X" and with smooth intersection with "X" contains an open dense subset of the total system of divisors formula_2. The set itself is open if "X" is projective. If formula_8, then these intersections (called hyperplane sections of "X") are connected, hence irreducible. The theorem hence asserts that a "general" hyperplane section not equal to "X" is smooth, that is: the property of smoothness is generic. Over an arbitrary  field "k", there is a dense open subset of the dual space formula_4 whose rational points define hyperplanes smooth hyperplane sections of "X". When "k" is infinite, this open subset then has infinitely many rational points and there are infinitely many smooth hyperplane sections in "X". Over a finite field, the above open subset may not contain rational points and in general there is no hyperplanes with smooth intersection with "X". However, if we take hypersurfaces of sufficiently big degrees, then the theorem of Bertini holds. Outline of a proof. We consider the subfibration of the product variety formula_10 with  fiber above formula_11 the linear system of hyperplanes that intersect "X" non-transversally at "x". The rank of the fibration in the product is one less than the codimension of formula_12, so that the total space has lesser dimension than formula_13 and so its projection is contained in a divisor of the complete system formula_2. General statement. Over any infinite field formula_15 of characteristic 0, if "X" is a smooth quasi-projective formula_16-variety, a general member of a linear system of divisors on "X" is smooth away from the base locus of the system. For clarification, this means that  given a linear system formula_17, the preimage formula_18 of a hyperplane "H" is smooth -- outside the base locus of "f" -- for all hyperplanes "H" in some dense open subset of the dual projective space formula_19. This theorem also holds in characteristic p>0 when the linear system "f" is unramified. Generalizations. The theorem of Bertini has been generalized in various ways. For example, a result due to Steven Kleiman asserts the following (cf. Kleiman's theorem): for a connected algebraic group "G", and any homogeneous "G"-variety "X", and two varieties "Y" and "Z" mapping to "X", let "Y" be the variety  obtained by letting σ ∈ "G" act on "Y". Then, there is an open dense subscheme "H" of "G" such that for σ ∈ "H", formula_20 is either empty or purely of the (expected) dimension dim "Y" + dim "Z" − dim "X". If, in addition, "Y" and "Z" are smooth and the base field has characteristic zero, then "H" may be taken such that formula_20 is smooth for all formula_22, as well. The above theorem of Bertini is the special case where formula_23 is expressed as the quotient of SL by the parabolic subgroup of upper triangular matrices, "Z  is a subvariety and Y" is a hyperplane. Theorem of Bertini has also been generalized to discrete valuation domains or finite fields, or for étale coverings of "X". The theorem is often used for induction steps.