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Let X be a smooth quasiprojective subscheme of Pn of dimension m ≥ 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 ζX (m + 1)−1 , where ζX (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. 1. Introduction The classical Bertini theorems say that if a subscheme. | Annals of Mathematics Bertini theorems over finite fields By Bjorn Poonen Annals of Mathematics 160 2004 1099-1127 Bertini theorems over finite fields By Bjorn PooNEN Abstract Let X be a smooth quasiprojective subscheme of P of dimension m 0 over Fg. Then there exist homogeneous polynomials f over Fg 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 Zx m 1 - 1 where Zx 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. 1. Introduction The classical Bertini theorems say that if a subscheme X c P has a certain property then for a sufficiently general hyperplane H c Pn H n X has the property too. For instance if X is a quasiprojective subscheme of P that is smooth of dimension m 0 over a field k and U is the set of points u in the dual projective space J3n corresponding to hyperplanes H c P M such that H n X is smooth of dimension m 1 over the residue field k u of u then U contains a dense open subset of Pn. If k is infinite then U n Pn k is nonempty and hence one can find H over k. But if k is finite then it can happen that the finitely many hyperplanes H over k all fail to give a smooth intersection H n X see Theorem 3.1. N. M. Katz Kat99 asked whether the Bertini theorem over finite fields can be salvaged by allowing hypersurfaces of unbounded degree in place of hyperplanes. In fact he asked for a little more see Section 3 for details. We answer the question affirmatively below. O. Gabber Gab01 Corollary 1.6 has independently proved the existence of good hypersurfaces of any sufficiently large degree divisible by the characteristic of k . This research was supported by NSF grant DMS-9801104 and DMS-0301280 and a Packard Fellowship. Part of the research was done while the author was enjoying the hospitality of the Universite de Paris-Sud. 1100 BJORN POONEN Let Fg be a finite field of q pa .
