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 .
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 .
@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}, 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/} }
TY - JOUR AU - Bandman, Tanya AU - Dethloff, Gerd TI - Estimates of the number of rational mappings from a fixed variety to varieties of general type JO - Annales de l'Institut Fourier PY - 1997 SP - 801 EP - 824 VL - 47 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1581/ DO - 10.5802/aif.1581 LA - en ID - AIF_1997__47_3_801_0 ER -
%0 Journal Article %A Bandman, Tanya %A Dethloff, Gerd %T Estimates of the number of rational mappings from a fixed variety to varieties of general type %J Annales de l'Institut Fourier %D 1997 %P 801-824 %V 47 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1581/ %R 10.5802/aif.1581 %G en %F 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, Volume 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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