The second part of this conjecture is analogous to Conjecture 2.2 (i) in [PV], which considers hypersurfaces in P. If C is a curve of genus 2 as above and P = (a : y : b) is a rational point on C (i.e., we have F (a, b) = y with a, b coprime integers), then we denote by H(P ) the height H(a : b) = max{|a|, |b|} of its x-coordinate. Conjecture 2. Let ε > 0. Then there is a constant Bε and a Zari...

where we view Gm(S) as the subgroup of invertible scalars in GLn(S). Define PGLn = s ◦ P̃GLn to be its sheafification (taken in the sense of Problem 8 on PS 2). Our goal will be to show that PGLn is an affine group scheme that represents automorphisms of projective space in the sense that PGLn(S) ' AutS(S × Pn−1). Before we begin, we require some preliminaries about Zariski sheaves. Let F : Sch→...

For a rank one Lie group G and a Zariski dense and geometrically finite subgroup Γ of G, we establish the joint equidistribution of closed geodesics and their holonomy classes for the associated locally symmetric space. Our result is given in a quantitative form for geometrically finite real hyperbolic manifolds whose critical exponents are big enough. In the case when G = PSL2(C), our results ...

We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with (N,+,×) if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in Z. Notably, by constructing a nontrivial...

In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary l poles of prescribed orders, such that for every geometric point f in U(Q), the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to a...

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

We describe the structure QHO = QHON (dependent on the positive integer number N) on the universe L which is a finite cover, of order N, of the projective line P = P(F), F an algebraically closed field of characteristic 0. We prove that QHO is a complete irreducible Zariski geometry of dimension 1. We also prove that QHO is not classical in the sense that the structure is not interpretable in a...

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( √

We construct a model space C(sp(R 2n)) for the variety of Abelian simply transitive groups of affine transformations of type Sp(R 2n). The model is stratified and its principal stratum is a Zariski-open subbundle of a natural vector bundle over the Grassmannian of Lagrangian subspaces in R 2n. Next we show that every flat special Kähler manifold may be constructed locally from a holomorphic fun...

In joint work with A. Küronya and T. Szemberg we study certain asymptotic invariants of linear series: the stable base locus and the volume. In particular we are interested in the question how these invariants behave under small perturbations in the Néron-Severi space. We show that both invariants lead to a partition of the big cone into suitable subcones, and that – somewhat surprisingly – the...