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arXiv · 1012.0628

Smooth hypersurface sections containing a given subscheme over a finite field

Bjorn Poonen
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Abstract

We use the "closed point sieve" to prove a variant of a Bertini theorem over finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect X is smooth of dimension l, we compute the fraction of homogeneous polynomials vanishing on Z that cut out a smooth subvariety of X. The fraction is positive if m>2l.

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Bjorn Poonen. 2010-12-03. Smooth hypersurface sections containing a given subscheme over a finite field. https://doi.org/10.4310/mrl.2008.v15.n2.a5

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