Bertini theorems over finite fields

• Autores: Bjorn Poonen
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 160, Nº 3, 2004, págs. 1099-1127
• Idioma: inglés
• DOI: 10.4007/annals.2004.160.1099
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• Resumen

  • 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.