The probability that a complete intersection is smooth

2010/03/26 by Bucur, Alina, Kedlaya, Kiran S. · 1 citation
#11M38 #14G15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1003.5222

Abstract

Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.

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• Equidistribution and arithmetic Λ-distributions

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• Explicit Estimates for the Number of Rational Points of Singular Complete Intersections over a Finite Field
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• Irrational Complete Intersections