Geodesic
Bertini theorems over finite fields
Bjorn Poonen1
1 University of California, Berkeley, Berkeley, CA
Annals of mathematics, Tome 160 (2004) no. 3, pp. 1099-1127.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2004.160.1099

Bjorn Poonen 1

1 University of California, Berkeley, Berkeley, CA 
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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. http://geodesic.mathdoc.fr/articles/10.4007/annals.2004.160.1099/

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