Bertini irreducibility theorems via statistics

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 )

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

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Stanford algebraic geometry seminar

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