Étant donné un sous-schéma lisse d’un espace projectif sur un corps fini, nous calculons la probabilité que son intersection avec un nombre fixe d’hypersurfaces de grand degré soit lisse de la dimension attendue. Cela généralise le cas d’une seule hypersurface, considéré par Poonen. Nous utilisons ce résultat pour donner un modèle probabiliste pour le nombre de points rationnels d’une telle intersection complète. Un corollaire un peu surprenant est que le nombre de points rationnels sur une intersection lisse de deux surfaces de l’espace projectif de dimension est strictement inférieur au nombre de points sur la droite projective.
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.
@article{JTNB_2012__24_3_541_0,
author = {Bucur, Alina and Kedlaya, Kiran S.},
title = {The probability that a complete intersection is smooth},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {541--556},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {24},
number = {3},
year = {2012},
doi = {10.5802/jtnb.810},
zbl = {1268.14021},
mrnumber = {3010628},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.810/}
}
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 PY - 2012 SP - 541 EP - 556 VL - 24 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.810/ DO - 10.5802/jtnb.810 LA - en ID - JTNB_2012__24_3_541_0 ER -
%0 Journal Article %A Bucur, Alina %A Kedlaya, Kiran S. %T The probability that a complete intersection is smooth %J Journal de théorie des nombres de Bordeaux %D 2012 %P 541-556 %V 24 %N 3 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.810/ %R 10.5802/jtnb.810 %G en %F JTNB_2012__24_3_541_0
Bucur, Alina; Kedlaya, Kiran S. The probability that a complete intersection is smooth. Journal de théorie des nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 541-556. doi : 10.5802/jtnb.810. http://www.numdam.org/articles/10.5802/jtnb.810/
[1] P. Billingsley, Probability and Measure, second edition. Wiley, New York, 1986. | MR | Zbl
[2] 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. | MR | Zbl
[3] 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. | MR | Zbl
[4] 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. | MR | Zbl
[5] O. Gabber, On space filling curves and Albanese varieties. Geom. Funct. Anal. 11 (2001), 1192–1200. | MR | Zbl
[6] A. Granville, ABC allows us to count squarefrees. Internat. Math. Res. Notices No. 19 (1998), 991–1009. | MR | Zbl
[7] 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. | MR | Zbl
[8] 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. | MR
[9] S. Lang and A. Weil, Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819–827. | MR | Zbl
[10] B. Poonen, Squarefree values of multivariable polynomials. Duke Math. J. 118 (2003), 353–373. | MR | Zbl
[11] B. Poonen, Bertini theorems over finite fields. Ann. Math. 160 (2004), 1099–1127. | MR | Zbl
[12] M.M. Wood, The distribution of the number of points on trigonal curves over . Int. Math. Res. Not. IMRN 2012. To appear, doi: 10.1093/imrn/rnr256.
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