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arXiv · 1003.5222

The probability that a complete intersection is smooth

Alina Bucur·Kiran S. Kedlaya
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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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Alina Bucur, Kiran S. Kedlaya. 2010-03-26. The probability that a complete intersection is smooth. https://arxiv.org/abs/1003.5222

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