Bertini's theorem

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.

Let X be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space ${\displaystyle \mathbf {P} ^{n}}$. Let ${\displaystyle |H|}$ denote the complete system of hyperplane divisors in ${\displaystyle \mathbf {P} ^{n}}$. Recall that it is the dual space ${\displaystyle (\mathbf {P} ^{n})^{\star }}$ of ${\displaystyle \mathbf {P} ^{n}}$ and is isomorphic to ${\displaystyle \mathbf {P} ^{n}}$.

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