# The probability that a complete intersection is smooth

Alina Bucur^{[1]}; Kiran S. Kedlaya^{[1]}

- [1] University of California at San Diego 9500 Gilman Dr #0112 San Diego, CA 92093

Journal de Théorie des Nombres de Bordeaux (2012)

- Volume: 24, Issue: 3, page 541-556
- ISSN: 1246-7405

## Access Full Article

top## Abstract

top## How to cite

topBucur, Alina, and Kedlaya, Kiran S.. "The probability that a complete intersection is smooth." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 541-556. <http://eudml.org/doc/251045>.

@article{Bucur2012,

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.},

affiliation = {University of California at San Diego 9500 Gilman Dr #0112 San Diego, CA 92093; University of California at San Diego 9500 Gilman Dr #0112 San Diego, CA 92093},

author = {Bucur, Alina, Kedlaya, Kiran S.},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Bertini theorem; smooth subscheme of a projective space; finite field; hypersurface of large degree; complete intersection; random smooth intersection; probabilistic model},

language = {eng},

month = {11},

number = {3},

pages = {541-556},

publisher = {Société Arithmétique de Bordeaux},

title = {The probability that a complete intersection is smooth},

url = {http://eudml.org/doc/251045},

volume = {24},

year = {2012},

}

TY - JOUR

AU - Bucur, Alina

AU - Kedlaya, Kiran S.

TI - The probability that a complete intersection is smooth

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2012/11//

PB - Société Arithmétique de Bordeaux

VL - 24

IS - 3

SP - 541

EP - 556

AB - 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.

LA - eng

KW - Bertini theorem; smooth subscheme of a projective space; finite field; hypersurface of large degree; complete intersection; random smooth intersection; probabilistic model

UR - http://eudml.org/doc/251045

ER -

## References

top- P. Billingsley, Probability and Measure, second edition. Wiley, New York, 1986. Zbl0411.60001MR830424
- B.W. Brock and A. Granville, More points than expected on curves over finite field extensions. Finite Fields Appl. 7 (2001), no. 1, 70–91. Zbl1023.11029MR1803936
- A. Bucur, C. David, B. Feigon, and M. Lalín, Statistics for traces of cyclic trigonal curves over finite fields. Int. Math. Res. Not. No. 5 (2010), 932–967. Zbl1201.11063MR2595014
- A. Bucur, C. David, B. Feigon, and M. Lalín, The fluctuations in the number of points of smooth plane curves over finite fields. J. Number Theory 130 (2010), 2528–2541. Zbl1210.11070MR2678860
- O. Gabber, On space filling curves and Albanese varieties. Geom. Funct. Anal. 11 (2001), 1192–1200. Zbl1072.14513MR1878318
- A. Granville, ABC allows us to count squarefrees. Internat. Math. Res. Notices No. 19 (1998), 991–1009. Zbl0924.11018MR1654759
- P. Kurlberg and Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field. J. Number Theory 129 (2009), 580–587. Zbl1221.11141MR2488590
- P. Kurlberg and I. Wigman, Gaussian point count statistics for families of curves over a fixed finite field. Int. Math. Res. Not. 2011 (2011), 2217–2229. Zbl1297.11063MR2806563
- S. Lang and A. Weil, Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819–827. Zbl0058.27202MR65218
- B. Poonen, Squarefree values of multivariable polynomials. Duke Math. J. 118 (2003), 353–373. Zbl1047.11021MR1980998
- B. Poonen, Bertini theorems over finite fields. Ann. Math. 160 (2004), 1099–1127. Zbl1084.14026MR2144974
- M.M. Wood, The distribution of the number of points on trigonal curves over ${\mathbb{F}}_{q}$. Int. Math. Res. Not. IMRN 2012. To appear, doi: 10.1093/imrn/rnr256.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.