Mathematics Colloquia and Seminars
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Bertini theorems over finite fieldsAlgebra & Discrete Mathematics
|Speaker: ||Prof. Bjorn Poonen, Univ. of California Berkeley|
|Location: ||693 Kerr|
|Start time: ||Thu, Jan 31 2002, 3:10PM|
One form of Bertini's theorem states that if X is a smooth projective
variety of dimension m in P^n over an infinite field k, then there
exists a hyperplane H defined over k such that the intersection of X
and H is smooth of dimension m-1. This can fail if k is finite. Katz
asked whether the statement would remain true if "hyperplane" were
changed to "hypersurface". We give an affirmative answer. In fact,
as d tends to infinity, the fraction of hypersurfaces of degree d that
are good tends to a special value of the zeta function of X.
Sketch of proof: sieve out the bad hypersurfaces and count carefully
to show that something remains...
A generalization of our result answers another question of Katz,
about "space filling curves": if X is a smooth projective variety of
dimension m>1 over a finite field k, does there exist a smooth
projective curve Y over k in X with Y(k)=X(k)?