Bertini irreducibility theorems via statistics

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.