Probabilistic enumerative geometry over $p$-adic numbers: linear spaces on complete intersections

Ait El Manssour, Rida; Lerario, Antonio

Annales Henri Lebesgue, Tome 5 (2022), pp. 1329-1360.

Nous calculons l’espérance du nombre d’espaces linéaires dans des intersections complètes aléatoires dans l’espace projectif $p$-adique. Ici “aléatoire” signifie que les coefficients des polynômes définissant les intersections complètes sont tirés uniformément parmi les entiers $p$-adiques. Nous montrons que lorsque le nombre premier $p$ tend vers l’infini, le nombre moyen d’espaces linéaires dans une intersection complète aléatoire tend vers $1$. Dans le cas du nombre de droites sur une cubique aléatoire en trois dimensions, et de l’intersection de deux quadriques aléatoires en quatre dimensions, nous donnons une formule explicite pour cette espérance.

We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here “random” means that the coefficients of the polynomials defining the complete intersections are sampled uniformly from the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a random complete intersection tends to $1$. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.

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