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 → 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 .