Let F be a field, let D be a subring of F , and let X be the Zariski-Riemann space of valuation rings containing D and having quotient field F . We consider the Zariski, inverse and patch topologies on X when viewed as a projective limit of projective integral schemes having function field contained in F , and we characterize the locally ringed subspaces of X that are affine schemes.

Let Γ2 ⊆ Γ1 be finitely generated subgroups of GLn0 (Z[1/q0]). For i = 1 or 2, let Gi be the Zariski-closure of Γi in (GLn0 )Q, Gi be the Zariski-connected component of Gi, and let Gi be the closure of Γi in ∏ p-q0 GLn0 (Zp). In this article we prove that, if G1 is the smallest closed normal subgroup of G1 which contains G2 and Γ2 y G2 has spectral gap, then Γ1 y G1 has spectral gap.

We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type A10 + A9 that are defined over Q( √ 5) and are conjugate to each other by the action of Gal(Q( √

This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental ...

We continue to study Zariski pairs in sextics. In this paper, we study Zariski pairs of sextics which are not irreducible. The idea of the construction of Zariski partner sextic for reducible cases is quit different from the irreducible case. It is crucial to take the geometry of the components and their mutual intersection data into account. When there is a line component, flex geometry (i.e.,...

We compute the fundamental groups of all irreducible plane sextics constituting classical Zariski pairs

A new class of noncommutative k-algebras (for k an algebraically closed field) is defined and shown to contain some important examples of quantum groups. To each such algebra, a first order theory is assigned describing models of a suitable corresponding geometric space. Model-theoretic results for these geometric structures are established (uncountable categoricity, quantifier elimination to t...