tailieunhanh - Đề tài " Bertini theorems over finite fields "

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 . 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 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 .