We study the existence and properties of birationally equivalent models for elliptically fibered varieties. In particular these have either structure Mori fiber spaces or, assuming some standard conjectures, minimal with a Zariski decomposition compatible elliptic fibration. prove relations between birational invariants variety, base fibration its Jacobian.

In this paper, we formulate and prove a weighted homogeneous version of Zariski’s hyperplane section theorem on the fundamental groups of the complements of hypersurfaces in a complex projective space, and apply it to the study of π1(P 2 \C), where C ⊂ P is a projective plane curve. The main application is to prove a comparison theorem as follows. Let φ : P → P be the composition of the Verones...

Let $R$ be an associative ring and let $M$ be a left $R$-module.Let $Spec_{R}(M)$ be the collection of all prime submodules of $M$ (equipped with classical Zariski topology). There is a conjecture which says that every irreducible closed subset of $Spec_{R}(M)$ has a generic point. In this article we give an affirmative answer to this conjecture and show that if $M$ has a Noetherian spectrum, t...

We study the number of rational points of bounded height on a certain threefold. The accumulating subvarieties are Zariski-dense in this example. The computations support an extension of a conjecture of Manin to this situation.

Abstract We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to maximal representations. When the target is not tube type, we show that there cannot be Zariski-dense representations, whenever existence boundary map can guaranteed, representation preserves finite-dimensional totally geodesic s...

We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.