Duke Mathematical Journal
- Duke Math. J.
- Volume 164, Number 1 (2015), 1-38.
Semiample Bertini theorems over finite fields
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.
Duke Math. J., Volume 164, Number 1 (2015), 1-38.
First available in Project Euclid: 9 January 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14G15: Finite ground fields
Secondary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]
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