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:XY, 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(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:XY 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:XY, 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:XY 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(X) and C(X) with the following properties:For any threefold Y of general type the number of dominant rational maps f:XY is bounded above by c(X).The number of threefolds Y, modulo birational equivalence, for which there exist dominant rational maps f:XY, 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 .
@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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