The probability that a complete intersection is smooth
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, p. 541-556

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.

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 3 est strictement inférieur au nombre de points sur la droite projective.

DOI : https://doi.org/10.5802/jtnb.810
Classification:  14G15,  11M38
@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},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     pages = {541-556},
     doi = {10.5802/jtnb.810},
     mrnumber = {3010628},
     zbl = {1268.14021},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 24 (2012) no. 3, pp. 541-556. doi : 10.5802/jtnb.810. http://www.numdam.org/item/JTNB_2012__24_3_541_0/

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