Bertini theorems over finite fields
Annals of mathematics, Tome 160 (2004) no. 3, pp. 1099-1127
Let $X$ be a smooth quasiprojective subscheme of $\mathbf{P}^n$ of dimension $m \ge 0$ over $\mathbf{F}_q$. Then there exist homogeneous polynomials $f$ over $\mathbf{F}_q$ 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 $\mathbf{Z}$ is proved, assuming the $abc$ conjecture and another conjecture.
@article{10_4007_annals_2004_160_1099,
author = {Bjorn Poonen},
title = {Bertini theorems over finite fields},
journal = {Annals of mathematics},
pages = {1099--1127},
year = {2004},
volume = {160},
number = {3},
doi = {10.4007/annals.2004.160.1099},
mrnumber = {2144974},
zbl = {1084.14026},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.1099/}
}
Bjorn Poonen. Bertini theorems over finite fields. Annals of mathematics, Tome 160 (2004) no. 3, pp. 1099-1127. doi: 10.4007/annals.2004.160.1099
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