arXiv subjects
Publications and source records attributed to G. Dethloff.
In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate.
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 $K_{X_c}^3$.
Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincaré group $π_1(\pr^2 \se C)$ contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Plücker curves. Such a curve $C$ can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski--Lefschetz type arguments we deduce the result from `the bigness' of the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.