Resumen de Bertini theorems over finite fields

Bjorn Poonen

• Let X be a smooth quasiprojective subscheme of Pn of dimension m ¡Ý 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 ¿ÆX(m+1).1, where ¿ÆX(s) = ZX(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.