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, p. 801-824
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 $\left\{A·{K}_{X}^{n}{\right\}}^{\left\{B·{K}_{X}^{n}{\right\}}^{2}}$. Ici $n=\mathrm{dim}\phantom{\rule{0.277778em}{0ex}}X$, ${K}_{X}$ 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 $c\left(X\right)$ et $C\left(X\right)$ 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 $c\left(X\right)$.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 $C\left(X\right)$.Si, de plus, $X$ est aussi une variété de dimension 3 et de type général, nous démontrons que $c\left(X\right)$ et $C\left(X\right)$ 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 $\left\{A·{K}_{X}^{n}{\right\}}^{\left\{B·{K}_{X}^{n}{\right\}}^{2}}$, where $n=\mathrm{dim}\phantom{\rule{0.277778em}{0ex}}X$, ${K}_{X}$ 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 $c\left(X\right)$ and $C\left(X\right)$ 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 $c\left(X\right)$.The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X\to Y$, is bounded above by $C\left(X\right)$.If, moreover, $X$ is a threefold of general type, we prove that $c\left(X\right)$ and $C\left(X\right)$ only depend on the index ${r}_{{X}_{c}}$ of the canonical model ${X}_{c}$ of $X$ and on ${K}_{{X}_{c}}^{3}$.
@article{AIF_1997__47_3_801_0,
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},
publisher = {Association des Annales de l'institut Fourier},
volume = {47},
number = {3},
year = {1997},
pages = {801-824},
doi = {10.5802/aif.1581},
zbl = {0868.14008},
mrnumber = {98h:14016},
language = {en},
url = {http://www.numdam.org/item/AIF_1997__47_3_801_0}
}

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. http://www.numdam.org/item/AIF_1997__47_3_801_0/

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