no. 2
A boundedness theorem for morphisms between threefolds
Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 405-415.

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→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→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.

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     title = {A boundedness theorem for morphisms between threefolds},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Amerik, Ekatarina; Rovinsky, Marat; Van De Ven, Antonius. A boundedness theorem for morphisms between threefolds. Annales de l'Institut Fourier, Tome 49 (1999) no. 2, pp. 405-415. doi : 10.5802/aif.1679. https://aif.centre-mersenne.org/articles/10.5802/aif.1679/

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