We show that the free hyperplane arrangements form a Zariski-closed set in various parameter spaces. Given a geometric lattice L let V(L) be the parameter space of arrangements with intersection lattice isomorphic to L. Coupling our result with a theorem of Yuzvinsky, we conclude that in V(L) the free arrangements are parameterized by a union of connected components of V(L).

We construct a variation map on homotopy groups associated to a pencil with isolated stratified singularities on a singular space. As application, we prove a far reaching Zariski-van Kampen-Lefschetz type theorem, for higher homotopy groups of nongeneric pencils on singular (open) complex spaces.

Let MH be a moduli space of stable parabolic Higgs bundles of rank two over a Riemann surface X . It is a smooth variety over C equipped with a holomorphic symplectic form. Fix a projective structure P on X . Using P , we construct a quantization of a certain Zariski open dense subset of the symplectic variety MH .

We determine the topological Euler number of certain moduli space of 1-dimensional closed subschemes in a smooth projective variety which admits a Zariski-locally trivial fibration with 1-dimensional fibers. The main approach is to use virtual Hodge polynomials and torus actions. The results might shed some light on the corresponding Donaldson-Thomas invariants.

Purpose of the present paper is to reveal part of the beauty and delicacy of resolution of singularities in the case of excellent two-dimensional schemes embedded in three-space and defined over an algebraically closed field of arbitrary characteristic. The proof of strong embedded resolution we describe here combines arguments and techniques of O. Zariski, H. Hironaka, S. Abhyankar and the aut...

The purpose of this paper is to study topological properties both the set all $k$-prime ideals and congruences for any commutative semiring with zero identity. We first prove that spectrum, i.e. equipped Zariski topology a spectral space, then homeomorphic spectrum respect their topologies.

Let X be an algebraic variety over the algebraically closed field K and Ξ ⊆ X(K) a set of K-rational points on X. We say that a subvariety Y ⊆ Xn of some Cartesian power of X is Ξ-special if Ξn ∩ Y (K) is Zariski dense in Y . We show under certain hypotheses on Ξ, for instance, that the class of Ξ-special varieties is closed under intersections, that when subvarieties of X vary in an algebraic ...

Let X be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of X is ample. Using the cylinder homomorphism associated with the family of complete intersections contained in X, we prove that the vanishing cycles in the middle homology group of X are represented by topological cycles whose support is contained in a pr...