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...

We present a constructive interpretation of some results in [6]. To any integral domain R we associate a distributive lattice V (R) which is a point-free presentation of the space of valuations over the ring R. Our definition, by generators and relations, is similar to Joyal’s definition of the Zariski lattice of a ring R, which is a point-free presentation of the Zariski spectrum of R. The spa...

This paper concentrates on understanding the first order theory of universal specializations of Zariski structures. Models of the theory are pairs, a Zariski structure and an elementary extension with a map (specialization) from the extension to the structure that preserves positive quantifier free formulas. The reader will find that this context generalizes both the study of algebraically clos...

We prove that the outer Lipschitz geometry of the germ of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in C3: Zariski equ...

Let X be a compact connected Riemann surface of genus g ≥ 1 equipped with nonzero holomorphic 1-form. MX(r) denote the moduli space semistable Higgs bundles on rank r and degree r(g − 1) + 1; it is complex symplectic manifold. Using translation structure open subset where 1-form does not vanish, we construct natural deformation quantization certain nonempty Zariski MX(r).

We study the Zariski tangent cone TX π −→ X to an affine variety X and the closure TX of π−1(Reg(X)) in TX . We focus on the comparison between TX and TX , giving sufficient conditions on X in order that TX = TX . One aspect of the results is to understand when this equality takes place in the presence of the reducedness of the Zariski tangent cone. Our other interest is to consider conditions ...

The polar curves of foliations F having a curve C of separatrices generalize the classical polar curves associated to hamiltonian foliations of C. As in the classical theory, the equisingularity type ℘(F) of a generic polar curve depends on the analytical type of F , and hence of C. In this paper we find the equisingularity types ǫ(C) of C, that we call kind singularities, such that ℘(F) is com...

Let X be a smooth complex projective variety and A an abelian group. Degree i unramified cohomology H i nr(X,A) of X with coefficients in A can be defined as the direct limit of the sets of data αk ∈ H i B(Uk, A), αk|Uk∩Ul = αl|Uk∩Ul , where the Uk’s are sufficiently small Zariski open sets covering X. Here the notation H i B stands for Betti cohomology of the underlying complex analytic space....