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

Tanya Bandman; Gerd Dethloff

Annales de l'institut Fourier (1997)

  • Volume: 47, Issue: 3, page 801-824
  • ISSN: 0373-0956

Abstract

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

How to cite

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Bandman, Tanya, and Dethloff, Gerd. "Estimates of the number of rational mappings from a fixed variety to varieties of general type." Annales de l'institut Fourier 47.3 (1997): 801-824. <http://eudml.org/doc/75245>.

@article{Bandman1997,
abstract = {First we find effective bounds for the number of dominant rational maps $f:X \rightarrow Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\lbrace A \cdot K_X^n\rbrace ^\{\lbrace B \cdot K_X^n\rbrace ^2\}$, where $n=\{\rm dim\}\;X$, $K_X$ 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 $f:X \rightarrow Y$ is bounded above by $c(X)$.The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X \rightarrow Y$, 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 $K_\{X_c\}^3$.},
author = {Bandman, Tanya, Dethloff, Gerd},
journal = {Annales de l'institut Fourier},
keywords = {number of dominant rational maps; threefold of general type},
language = {eng},
number = {3},
pages = {801-824},
publisher = {Association des Annales de l'Institut Fourier},
title = {Estimates of the number of rational mappings from a fixed variety to varieties of general type},
url = {http://eudml.org/doc/75245},
volume = {47},
year = {1997},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 47
IS - 3
SP - 801
EP - 824
AB - First we find effective bounds for the number of dominant rational maps $f:X \rightarrow Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\lbrace A \cdot K_X^n\rbrace ^{\lbrace B \cdot K_X^n\rbrace ^2}$, where $n={\rm dim}\;X$, $K_X$ 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 $f:X \rightarrow Y$ is bounded above by $c(X)$.The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X \rightarrow Y$, 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 $K_{X_c}^3$.
LA - eng
KW - number of dominant rational maps; threefold of general type
UR - http://eudml.org/doc/75245
ER -

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