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Bertini theorems over finite fields

Bjorn Poonen
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Abstract

Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q for which the intersection of X and the hypersurface f=0 is smooth. In fact, the set of such f has a positive density, equal to zeta_X(m+1)^{-1}, where zeta_X(s)=Z_X(q^{-s}) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.

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Bjorn Poonen. 2002-03-29. Bertini theorems over finite fields. https://doi.org/10.4007/annals.2004.160.1099

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