## Duke Mathematical Journal

### Semiample Bertini theorems over finite fields

#### Abstract

We prove a semiample generalization of Poonen’s Bertini theorem over a finite field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by the relevant divisor. We give several applications including a negative answer to a question of Baker and Poonen by constructing a variety (in fact one of each dimension) which provides a counterexample to Bertini over finite fields in arbitrarily large projective spaces. As another application, we determine the probability of smoothness for curves in Hirzebruch surfaces and the distribution of points on those smooth curves.

#### Article information

Source
Duke Math. J., Volume 164, Number 1 (2015), 1-38.

Dates
First available in Project Euclid: 9 January 2015

https://projecteuclid.org/euclid.dmj/1420813013

Digital Object Identifier
doi:10.1215/00127094-2838327

Mathematical Reviews number (MathSciNet)
MR3299101

Zentralblatt MATH identifier
1349.14092

Subjects
Primary: 14G15: Finite ground fields
Secondary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]

#### Citation

Erman, Daniel; Wood, Melanie Matchett. Semiample Bertini theorems over finite fields. Duke Math. J. 164 (2015), no. 1, 1--38. doi:10.1215/00127094-2838327. https://projecteuclid.org/euclid.dmj/1420813013

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