# Bertini irreducibility theorems via statistics

### Bjorn Poonen (MIT)

Fri May 8, 19:00-20:00 (3 weeks ago)

Abstract: Let $X \subset \mathbb{P}^n$ be a geometrically irreducible subvariety with $\dim X \ge 2$, over any field. Let $\check{\mathbb{P}}^n$ be the moduli space parametrizing hyperplanes $H \subset \mathbb{P}^n$. Let $L \subset \check{\mathbb{P}}^n$ be the locus parametrizing $H$ for which $H \cap X$ is geometrically irreducible. The classical Bertini irreducibility theorem states that $L$ contains a dense open subset of $\check{\mathbb{P}}^n$, so the bad locus $L' := \mathbb{P}^n - L$ satisfies $\dim L' \le n-1$. Benoist improved this to $\dim L' \le \operatorname{codim} X + 1$.

We describe a new way to prove and generalize such theorems, by reducing to the case of a finite field and studying the mean and variance of the number of points of a random hyperplane section. This is joint work with Kaloyan Slavov.

algebraic geometry

Audience: researchers in the topic

( paper | slides )

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Series comments: Seminar meets 11-12:30 pm pacific time when there is just one talk, and 10:45-11:45 and 12-1 pm pacific time when there is a double header.

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 Organizers: Isabel Vogt*, Ravi Vakil* *contact for this listing

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