23703 articles – 18149 references  [version française]
 Detailed view Article in peer-reviewed journal
 Ann. Inst. Fourier, 47 (1997) 801-824
 Estimates of the number of rational mappings from a fixed variety to varieties of general type
 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 $\{A \cdot K_X^n\}^{\{B \cdot K_X^n\}^2}$, where $n=dimX$, $K_X$ 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 $f:X \r Y$ is bounded above by $c(X)$. The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X \r Y$, 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 $K_{X_c}^3$.
 hal-00467720, version 1 http://hal.archives-ouvertes.fr/hal-00467720 oai:hal.archives-ouvertes.fr:hal-00467720 From: Gerd Dethloff <> Submitted on: Sunday, 28 March 2010 15:25:46 Updated on: Sunday, 28 March 2010 15:53:08