We prove a multiple-points higher-jets nonvanishing theorem by the use of local Seshadri constants. Applications are given to effectivity problems such as constructing rational and birational maps into Grassmannians, and the global generation of vector bundles.

The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investigated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory,...

First we find effective bounds for the number of dominant rational maps $f:X\to Y$ 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 $c\left(X\right)$ and $C\left(X\right)$ with the following properties:For any threefold $Y$ of general type the number of dominant rational maps $f:X\to Y$ is bounded above by $c\left(X\right)$.The number of threefolds $Y$, modulo birational...