Our goal for this lecture is to prove that morphisms of projective varieties are closed maps. In fact we will prove something stronger, that projective varieties are complete, a property that plays a role comparable to compactness in topology. For varieties, compactness as a topological space does not mean much because the Zariski topology is so coarse. Indeed, every subset of An (and hence of ...

We prove the Zariski dense orbit conjecture in positive characteristic for endomorphisms of G a N defined over F p ‾ .

The action of a noncompact semisimple Lie group G on a finite dimensional real vector space V is said to be stable if there exists a nonempty Zariski open subset O of V such that the orbit G(v) is closed in V for all v ∈ O. We study a Hilbert-Mumford numerical function M : V → R defined by A. Marian that extends the corresponding function in the complex setting defined by D. Mumford and studied...

Let X be an irreducible algebraic variety over C, endowed with an algebraic foliation F . In this paper, we introduce the notion of minimal invariant variety V (F , Y ) with respect to (F , Y ), where Y is a subvariety of X. If Y = {x} is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through x. First we prove that ...

Let V be a valuation domain of rank one with quotient field K. We study the set extensions to rational functions K(X) induced by pseudo-convergent sequences K from topological point view, endowing this either Zariski or constructible topology. In particular, we consider two subspaces prescribed breadth pseudo-limit. give some necessary conditions for space metrizable (under topology) in terms v...

We ask whether the notion of a homotopy class of a path on a complex algebraic variety admits a purely algebraic characterisation, and reformulate this question as a question of categoricity of the universal covering space of a complex algebraic variety in a natural countable in nitary language. We provide partial positive results towards the question. Assuming a conjecture of Shafarevich and s...

There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is natur...

Let $R$ be a $G$-graded ring and M $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to set all submodules $Q$ such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we topology on $\mathcal{PS}_G(M)$ having Zariski prime $Spec_G(M)$ as subspace topology, investigate several topological properties space.