On the Bertini regularity theorem for arithmetic varieties (Q2127188)

Bertini's famous theorem, in its classical form, proves that singular hyperplane sections of a smooth projective variety are ``usually'' smooth themselves. The paper under review proves a version of this for sections of ample Hermitian line bundles on a regular arithmetic variety. More specifically, the author's result is an abstract analogue of results of \textit{B. Poonen} [Ann. Math. (2) 160, No. 3, 1099--1127 (2005; Zbl 1084.14026)] and \textit{F. Charles} and \textit{B. Poonen} [J. Am. Math. Soc. 32, No. 2, 605--607 (2019; Zbl 1405.14114)], an analogue that does not involve the explicit choice of an embedding of the arithmetic variety in projective space. The precise statement is somewhat technical, but the upshot is that the author calculates precisely -- in terms of a particular zeta function -- the density of sections of the Hermitian line bundle whose divisor has no singular point of small residue characteristic, and shows that this density is positive.