Bertini theorems over finite fields
By Bjorn Poonen. 2002
231 citations recorded by the source corpus.
Let X be a smooth quasiprojective subscheme of Pn of dimension m iÝ 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 ?AEX(m+1).1, where ?AEX(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.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b810fd7d7da748a6c9e2d8ed34073ec23ea9a423
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In Algebra and Number Theory
- Manjul Bhargava
- Bjorn Poonen
- Dino J. Lorenzini
- A. Shankar
- Qing Liu
- M. Raynaud
- Xiaoheng Wang
- Ofer Gabber
In Geometry and Topology
- Manjul Bhargava
- Bjorn Poonen
- Dino J. Lorenzini
- A. Shankar
- Qing Liu
- Xiaoheng Wang
- M. Raynaud
- Ofer Gabber
Citations in the corpus
- Integral p$p$-adic Hodge theory (2016, unknown): “Let X ⊂ D be a smooth and (sufficiently) ample hypersurface; this can be chosen over Z2: One has to arrange smoothness only over F2, so the result follows from the Bertini theorem over finite fields due to Gabber, [36], and more generally Poonen, [54].”
- Hypersurfaces in projective schemes and a moving lemma (2014, background): “Fakhruddin illustrating this point can be found in [59], 5.”
- Squarefree values of multivariable polynomials (2002, methodology): “I thank Johan de Jong and Ofer Gabber for independently suggesting that I try to use ideas from [Poo02] to compute the density of squarefree values in the Fq[t] case.”
- On the Brauer group of a surface (2005, methodology): “Replacing the embedding by a d-uple embedding if necessary, we can assume, using [ 13 ], 1.1, and Remark 3), that a geometrically integral hyperplane section of X is smooth.”
- Squarefree values of polynomial discriminants I (2016, background): “The non-monic case has a number of applications as well, including new results towards counting number fields and also to the resolution of a conjecture of Poonen [17] on an arithmetic Bertini theorem for the projective line.”