# ANNALES DE L'INSTITUT FOURIER

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 ${b}_{2}\left(X\right)={b}_{2}\left(Y\right)=1$. 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 ${c}_{1}\left(Y\right)=0$, 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 ${b}_{2}\left(X\right)={b}_{2}\left(Y\right)=1$. 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 ${c}_{1}\left(Y\right)=0$, 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 ${b}_{2}=1$.

@article{AIF_1999__49_2_405_0,
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},
publisher = {Association des Annales de l'institut Fourier},
volume = {49},
number = {2},
year = {1999},
pages = {405-415},
doi = {10.5802/aif.1679},
mrnumber = {2000f:14056},
zbl = {0923.14008},
language = {en},
url = {aif.centre-mersenne.org/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) no. 2, pp. 405-415. doi : 10.5802/aif.1679. https://aif.centre-mersenne.org/item/AIF_1999__49_2_405_0/

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