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Gabino González-Diez

Publications and source records attributed to Gabino González-Diez.

6 recordsLinked to original sources
arXiv2023

The minima of the geodesic length functions of uniform filling curves

There is a natural link between (multi-)curves that fill up a closed oriented surface and dessins d'enfants. We use this approach to exhibit explicitly the minima of the geodesic length function of a kind of curves (uniform filling curves) which include those that admit a homotopy equivalent representative such that all self-intersection points as well as all faces of their complement have the same multiplicity. We show that these minima are attained at the Grothendieck-Belyi surfaces determined by a natural dessin d'enfant associated to these filling curves. In particular they are all Riemann surfaces defined over number fields.

math.GT↗
arXiv2020

Galois action on Fuchsian surface groups and their solenoids

Let $C$ be a complex algebraic curve uniformised by a Fuchsian group $Γ$. In the first part of this paper we identify the automorphism group of the solenoid associated with $Γ$ with the Belyaev completion of its commensurator $\mathrm{Comm}(Γ)$ and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of $\mathrm{Gal}(\mathbb C/\mathbb Q)$ on algebraic curves. In turn this fact yields a proof of the Galois invariance of the arithmeticity of $Γ$ independent of Kazhhdan's. In the second part we focus on the case in which $Γ$ is arithmetic. The list of further Galois invariants we find includes: i) the periods of $\mathrm{Comm}(Γ)$, ii) the solvability of the equations $X^2+\sin^2 \frac{2π}{2k+1}$ in the invariant quaternion algebra of $Γ$ and iii) the property of $Γ$ being a congruence subgroup.

math.AG↗
arXiv2020

Complex surfaces with mutually non-biholomorphic universal covers

It is known that the universal cover of compact Riemann surface is either the projective line, the complex plane or the unit disk. In this article we construct a very explicit family of complex surfaces that gives rise to uncountably many mutually non-biholomorphic universal covers. The slope of these surfaces, which are going to be total spaces of Kodaira fibrations, is also determined

math.AG↗
arXiv2016

Families of Riemann Surfaces, Uniformization and Arithmeticity

A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible 2-dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk. In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the 2-dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read off from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk.

math.AG↗
arXiv2012

Beauville surfaces with abelian Beauville group

A Beauville surface is a rigid surface of general type arising as a quotient of a product of curves $C_{1}$, $C_{2}$ of genera $g_{1},g_{2}\ge 2$ by the free action of a finite group $G$. In this paper we study those Beauville surfaces for which $G$ is abelian (so that $G\cong \mathbb{Z}_{n}^{2}$ with $\gcd(n,6)=1$ by a result of Catanese). For each such $n$ we are able to describe all such surfaces, give a formula for the number of their isomorphism classes and identify their possible automorphism groups. This explicit description also allows us to observe that such surfaces are all defined over $\mathbb{Q}$.

math.AG↗