Nous démontrons d’abord que le nombre d’applications rationnelles dominantes , entre deux variétés projectives fixes avec fibré canonique ample, peut être majoré par . Ici , est le fibré canonique de et sont quelques constantes, dépendant seulement de .
Ensuite nous démontrons que, pour toute variété , il y a des constantes et avec les propriétés suivantes :
Pour toute variété de dimension 3 et de type général le nombre d’applications rationnelles dominantes est majoré par .
Le nombre de variétés de dimension 3 et de type général, modulo équivalence birationnelle, pour lesquelles il existe des applications rationnelles dominantes , est majoré par .
Si, de plus, est aussi une variété de dimension 3 et de type général, nous démontrons que et dépendent seulement de l’index du modèle canonique de et de .
First we find effective bounds for the number of dominant rational maps between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type , where , is the canonical bundle of and are some constants, depending only on .
Then we show that for any variety there exist numbers and with the following properties:
For any threefold of general type the number of dominant rational maps is bounded above by .
The number of threefolds , modulo birational equivalence, for which there exist dominant rational maps , is bounded above by .
If, moreover, is a threefold of general type, we prove that and only depend on the index of the canonical model of and on .
@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}, volume = {47}, year = {1997}, pages = {801-824}, doi = {10.5802/aif.1581}, mrnumber = {98h:14016}, zbl = {0868.14008}, language = {en}, url = {http://dml.mathdoc.fr/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) pp. 801-824. doi : 10.5802/aif.1581. http://gdmltest.u-ga.fr/item/AIF_1997__47_3_801_0/
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