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Bjorn Poonen
Sieve methods for varieties over finite fields and arithmetic schemes
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 221-229, doi: 10.5802/jtnb.583
Article PDF | Reviews MR 2332063 | Zbl 1149.11031
Keywords: Bertini theorem, finite field, Lefschetz pencil, squarefree integer, sieve

Résumé - Abstract

Classical sieve methods of analytic number theory have recently been adapted to a geometric setting. In the new setting, the primes are replaced by the closed points of a variety over a finite field or more generally of a scheme of finite type over $\mathbb{Z}$. We will present the method and some of the surprising results that have been proved using it. For instance, the probability that a plane curve over ${\mathbb{F}}_2$ is smooth is asymptotically $21/64$ as its degree tends to infinity. Much of this paper is an exposition of results in [Poo04] and [Ngu05].

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