Estimates of the number of rational mappings from a fixed variety to varieties of general type

Bandman, Tanya; Dethloff, Gerd

doi:10.5802/aif.1581

Bandman, Tanya ; Dethloff, Gerd

Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 801-824.

Résumé
Abstract

Nous démontrons d’abord que le nombre d’applications rationnelles dominantes $f : X \to Y$ , entre deux variétés projectives fixes avec fibré canonique ample, peut être majoré par ${A \cdot K_{X}^{n}}^{{B \cdot K_{X}^{n}}^{2}}$ . Ici $n = \dim X$ , $K_{X}$ est le fibré canonique de $X$ et $A, B$ sont quelques constantes, dépendant seulement de $n$ .

Ensuite nous démontrons que, pour toute variété $X$ , il y a des constantes $c (X)$ et $C (X)$ avec les propriétés suivantes :

Pour toute variété $Y$ de dimension 3 et de type général le nombre d’applications rationnelles dominantes $f : X \to Y$ est majoré par $c (X)$ .

Le nombre de variétés $Y$ de dimension 3 et de type général, modulo équivalence birationnelle, pour lesquelles il existe des applications rationnelles dominantes $f : X \to Y$ , est majoré par $C (X)$ .

Si, de plus, $X$ est aussi une variété de dimension 3 et de type général, nous démontrons que $c (X)$ et $C (X)$ dépendent seulement de l’index $r_{X_{c}}$ du modèle canonique $X_{c}$ de $X$ et de $K_{X_{c}}^{3}$ .

First we find effective bounds for the number of dominant rational maps $f : X \to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type ${A \cdot K_{X}^{n}}^{{B \cdot K_{X}^{n}}^{2}}$ , where $n = \dim X$ , $K_{X}$ is the canonical bundle of $X$ and $A, B$ are some constants, depending only on $n$ .

Then we show that for any variety $X$ there exist numbers $c (X)$ and $C (X)$ with the following properties:

For any threefold $Y$ of general type the number of dominant rational maps $f : X \to Y$ is bounded above by $c (X)$ .

The number of threefolds $Y$ , modulo birational equivalence, for which there exist dominant rational maps $f : X \to Y$ , is bounded above by $C (X)$ .

If, moreover, $X$ is a threefold of general type, we prove that $c (X)$ and $C (X)$ only depend on the index $r_{X_{c}}$ of the canonical model $X_{c}$ of $X$ and on $K_{X_{c}}^{3}$ .

MR Zbl

DOI : 10.5802/aif.1581

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     author = {Bandman, Tanya and Dethloff, Gerd},
     title = {Estimates of the number of rational mappings from a fixed variety to varieties of general type},
     journal = {Annales de l'Institut Fourier},
     pages = {801--824},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {3},
     year = {1997},
     doi = {10.5802/aif.1581},
     zbl = {0868.14008},
     mrnumber = {98h:14016},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1581/}
}

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Bandman, Tanya; Dethloff, Gerd. Estimates of the number of rational mappings from a fixed variety to varieties of general type. Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 801-824. doi : 10.5802/aif.1581. https://aif.centre-mersenne.org/articles/10.5802/aif.1581/

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