Bertini theorems over finite fields

Abstract

Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q 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 zeta_X(m+1)^{-1}, where zeta_X(s)=Z_X(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.

22 pages, Latex 2e

Cited by in corpus (39)

• Squarefree values of multivariable polynomials
• Discriminants in the Grothendieck Ring
• Hypersurfaces in projective schemes and a moving lemma
• Bertini irreducibility theorems over finite fields
• The fluctuations in the number of points of smooth plane curves over finite fields
• Generalized explicit descent and its application to curves of genus 3
• Curves of every genus with many points, II: Asymptotically good families
• Nice triples and Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes
• On descending cohomology geometrically
• Semiample Bertini theorems over finite fields
• p-adic Hodge-theoretic properties of étale cohomology with mod p coefficients, and the cohomology of Shimura varieties
• A probabilistic approach to systems of parameters and Noether normalization
• Transverse lines to surfaces over finite fields
• Sums of squares in function fields over Henselian local fields
• Pseudo-reflection groups and essential dimension
• The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3
• A Witt Nadel vanishing theorem for threefolds
• Curves in characteristic 2 with non-trivial 2-torsion
• Existence of pencils with nonblocking hypersurfaces
• The moduli space of hypersurfaces whose singular locus has high dimension
• Sharp Bertini theorem for plane curves over finite fields
• The Hodge ring of varieties in positive characteristic
• Squarefree values of polynomial discriminants II
• Parameter spaces for algebraic equivalence
• Distributions of traces of Frobenius for smooth plane curves over finite fields
• Simply connected varieties in characteristic $p>0$
• Distinguished models of intermediate Jacobians
• Analytic number theory for 0-cycles
• Most plane curves over finite fields are not blocking
• An h-principle for complements of discriminants
• Infinitely generated symbolic Rees algebras over finite fields
• Rank 2 local systems and abelian varieties II
• On the Tate-Shafarevich group of Abelian schemes over higher dimensional bases over finite fields
• Steenrod operations and algebraic classes
• Aspherical neighborhoods on arithmetic surfaces: the local case
• Godeaux-Serre Varieties with Prescribed Arithmetic Fundamental Group
• Differential Codes on Higher Dimensional Varieties Via Grothendieck's Residue Symbol
• Exceptional covers of surfaces
• A short proof that all linear codes are weakly algebraic-geometric using Bertini theorems of B. Poonen