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

Bandman, Tanya; Dethloff, Gerd

Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 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 $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$. 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 $c\left(X\right)$ 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 $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$, 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 $c\left(X\right)$ 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 $K_{X_c}^3$.

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