# Bertini irreducibility theorems via statistics

*Bjorn Poonen (MIT)*

**08-May-2020, 19:00-20:00 (2 years 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

**Comments: **The discussion for Bjorn Poonenâ€™s talk is taking place not in the zoom-chat, but at tinyurl.com/stagMay08b (and will be deleted after 3-7 days).

**Stanford algebraic geometry seminar **

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Organizer: | Ravi Vakil* |

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