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( √

Let $f \colon X \to X$ be a surjective endomorphism of normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure $X$. Using this, extend second author's result to singular surfaces extent that either $X$ has $f$-invariant non-constant rational function, or $f$ infinitely many Zaris...

Leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras. Using this fact we obtain maximal algebras of right quotients from those (Leavitt) path algebras whose associated graph satisfies that every vertex connects to a line point (equivalently, the Leavitt path algebra has essential socle). We also introduce and characterize the algebraic counterpa...