The probability that a complete intersection is smooth
Number Theory
2015-03-13 v3 Algebraic Geometry
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.
Cite
@article{arxiv.1003.5222,
title = {The probability that a complete intersection is smooth},
author = {Alina Bucur and Kiran S. Kedlaya},
journal= {arXiv preprint arXiv:1003.5222},
year = {2015}
}
Comments
14 pages; v3: final journal version