let $r$ be a commutative ring with identity and $m$ be an$r$-module. let $fspec(m)$ denotes the collection of all prime fuzzysubmodules of $m$. in this regards some basic properties of zariskitopology on $fspec(m)$ are investigated. in particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$fspec(m)$ is a $t_0-$spa...

We prove that for any affine variety S defined over Q, there exists an Artin group G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset U’ of the character variety X(G,P0(3)) = Hom(G,P0(3))//P0(3). The subset U contains all real points of 5’. As an application, we construct new examples of finitely-presented groups which are not fundamental groups of smooth ...

Let X be a variety defined over an algebraically closed field k of characteristic 0, let N ∈ N, let g : X99KX be a dominant rational self-map, and let A : AN−→AN be a linear transformation defined over k(X), i.e., for a Zariski open dense subset U ⊂ X, we have that for x ∈ U(k), the specialization A(x) is an N -by-N matrix with entries in k. We let f : X × AN99KX × AN be the rational endomorphi...

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...

In this paper we show Whitney’s fibering conjecture in the real and complex, local analytic and global algebraic cases. For a given germ of complex or real analytic set, we show the existence of a stratification satisfying a strong (real arc-analytic with respect to all variables and analytic with respect to the parameter space) trivialization property along each stratum. We call such a trivial...