Smooth hypersurface sections containing a given subscheme over a finite field

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.

7 pages. This paper appeared a few years ago. (I'm posting it in response to a request for the TeX file.)

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