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( √ 5)/Q).

We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first examples, to our knowledge, of a simply connected smooth variety whose sets of integral points are never Zariski-dense (and no entire curve has Zariski-dense image). Some of our results are connected with divisibility problems, i.e. the problem of describing the integral p...

Definition. A couple of complex reduced projective plane curves C1 and C2 of a same degree is said to make a Zariski pair, if there exist tubular neighborhoods T (Ci) ⊂ P of Ci for i = 1, 2 such that (T (C1), C1) and (T (C2), C2) are diffeomorphic, while the pairs (P, C1) and (P , C2) are not homeomorphic; that is, the singularities of C1 and C2 are topologically equivalent, but the embeddings ...

Each Bers slice is a holomorphically embedded copy of Teichmüller space within XC(S). While it follows that BY can be locally described as the common zero locus of finitely many analytic functions on XC(S), it is known that the Bers slice is not a locally algebraic set [DK]—this is used to show that W. Thurston’s skinning map is not a constant function [DK]. We prove a stronger result about the...

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

A well known result of Forstnerić [18] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [19] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular,...

In this paper we prove that there exists a Zariski dense open subset U defined over Q in the parameter space of one-variable rational functions with prescribed l poles with fixed orders, such that for every geometric point f in U(Q), the L-function of exponential sum of f at p has Newton polygon approaches the Hodge polygon as p approaches infinity. This result generalizes some result in [13] a...

The multiview variety of an arrangement cameras is the Zariski closure images world points in cameras. prime vanishing ideal this complex projective called ideal. We show that bifocal and trifocal polynomials from generate when foci are distinct. In computer vision literature, many sets (determinantal) have been proposed to describe variety. establish precise algebraic relationships between the...

Let Φ d,n be the fundamental group of the space of smooth projective hypersurfaces of degree d and dimension n and let ρ be its natural monodromy representation. Then the kernel of ρ is large for d ≥ 3 with the exception of the cases (d, n) = (3, 0), (3, 1). For these and for d < 3 the kernel is finite. A large group is one that admits a homomorphism to a semisimple Lie group of noncompact type...

Let G = PSL2(F) where F = R,C, and consider the space Z = (Γ1 × Γ2)\(G × G) where Γ1 < G is a co-compact lattice and Γ2 < G is a finitely generated discrete Zariski dense subgroup. The work of Benoist-Quint [2] gives a classification of all ergodic invariant Radon measures on Z for the diagonal G-action. In this paper, for a horospherical subgroup N of G, we classify all ergodic, conservative, ...