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# Bertini Theorems over Finite Fields

Bjorn Poonen
Annals of Mathematics
Second Series, Vol. 160, No. 3 (Nov., 2004), pp. 1099-1127
Stable URL: http://www.jstor.org/stable/3597333
Page Count: 29
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## Abstract

Let X be a smooth quasiprojective subscheme of Pn of dimension m ≥ 0 over Fq. Then there exist homogeneous polynomials f over Fq 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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