Bertini theorems over finite fields

Abstract

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.