A boundedness theorem for morphisms between threefolds

Amerik, Ekatarina; Rovinsky, Marat; Van De Ven, Antonius

Amerik, Ekatarina ; Rovinsky, Marat ; Van De Ven, Antonius

Annales de l'Institut Fourier, Tome 49 (1999), p. 405-415 / Harvested from Numdam

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Résumé
Abstract

Le résultat principal de cet article est le théorème suivant : soient $X, Y$ des variétés lisses projectives complexes de dimension trois telles que $b_{2} (X) = b_{2} (Y) = 1$ . Si $Y$ n’est pas l’espace projectif, alors le degré d’un morphisme $f : X \to Y$ est borné (par des invariants discrets de $X$ et de $Y$ ). En plus, supposons $X, Y$ lisses projectives de dimension quelconque et telles que leurs groupes de Néron-Severi soient cycliques. Si $c_{1} (Y) = 0$ , nous montrons que le degré de $f$ est borné si et seulement si $Y$ n’est pas une variété plate. Une partie de la preuve du théorème principal revient donc à montrer la non-existence d’une variété projective plate de dimension trois avec $b_{2} = 1$ .

The main result of this paper is as follows: let $X, Y$ be smooth projective threefolds (over a field of characteristic zero) such that $b_{2} (X) = b_{2} (Y) = 1$ . If $Y$ is not a projective space, then the degree of a morphism $f : X \to Y$ is bounded in terms of discrete invariants of $X$ and $Y$ . Moreover, suppose that $X$ and $Y$ are smooth projective $n$ -dimensional with cyclic Néron-Severi groups. If $c_{1} (Y) = 0$ , then the degree of $f$ is bounded iff $Y$ is not a flat variety. In particular, to prove our main theorem we show the non-existence of a flat 3-dimensional projective variety with $b_{2} = 1$ .

Publié le : 1999-01-01

MR 2000f:14056 | Zbl 0923.14008

DOI : https://doi.org/10.5802/aif.1679

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     author = {Amerik, Ekatarina and Rovinsky, Marat and Van De Ven, Antonius},
     title = {A boundedness theorem for morphisms between threefolds},
     journal = {Annales de l'Institut Fourier},
     volume = {49},
     year = {1999},
     pages = {405-415},
     doi = {10.5802/aif.1679},
     mrnumber = {2000f:14056},
     zbl = {0923.14008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1999__49_2_405_0}
}

Amerik, Ekatarina; Rovinsky, Marat; Van De Ven, Antonius. A boundedness theorem for morphisms between threefolds. Annales de l'Institut Fourier, Tome 49 (1999) pp. 405-415. doi : 10.5802/aif.1679. http://gdmltest.u-ga.fr/item/AIF_1999__49_2_405_0/

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