Bounds on the denominators in the canonical bundle formula
[Bornes sur les dénominateurs dans la formule du fibré canonique]
Floris, Enrica1
1 IRMA, Université de Strasbourg et CNRS 7 rue René-Descartes 67084 Strasbourg Cedex France
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1951-1969.

Dans cet article on considère la partie modulaire dans la formule du fibré canonique pour une fibration lc-triviale dont la fibre générique est une courbe rationnelle. Soit r l’indice de Cartier de la fibre. Il avait été conjecturé que 12r est une borne sur les dénominateurs de la partie modulaire. Nous démontrons qu’une telle borne ne peut même pas être polynômiale en r, nous calculons une borne N(r) et nous fournissons un exemple où la borne optimale sur les dénominateurs est N(r)/r. De plus nous montrons que même localement les dénominateurs dépendent quadratiquement de r.

In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If r is the Cartier index of the fibre, it was expected that 12r would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in r, we provide a bound N(r) and an example where the smallest integer that clears the denominators of the moduli part is N(r)/r. Moreover we prove that even locally the denominators depend quadratically on r.

DOI : 10.5802/aif.2819
Classification : 14J10 14J26
Keywords: lc-trivial fibration, moduli part, denominators
Mot clés : fibration lc-triviale, partie modulaire, dénominateurs
Floris, Enrica 1

1 IRMA, Université de Strasbourg et CNRS 7 rue René-Descartes 67084 Strasbourg Cedex France 
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Floris, Enrica. Bounds on the denominators in the canonical bundle formula. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1951-1969. doi : 10.5802/aif.2819. http://www.numdam.org/articles/10.5802/aif.2819/

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