Deformation of canonical morphisms and the moduli of surfaces of general type

In this article we study the deformation of finite maps and show how to use this deformation theory to construct varieties with given invariants in a projective space. Among other things, we prove a criterion that determines when a finite map can be deformed to a one--to--one map. We use this criterion to construct new simple canonical surfaces with different $c_1^2$ and $\chi$. Our general results enable us to describe some new components of the moduli of surfaces of general type. We also find infinitely many moduli spaces $\mathcal M_{(x',0,y)}$ having one component whose general point corresponds to a canonically embedded surface and another component whose general point corresponds to a surface whose canonical map is a degree 2 morphism.Comment: 32 pages. Final version with some simplifications and clarifications in the exposition. To appear in Invent. Math. (the final publication is available at springerlink.com

Homemade algebraic geometry: celebrating Enrique Arrondo’s 60th birthday

In this survey we recognize Enrique Arrondo’s contributions over the whole of his career, recalling his professional history and collecting the results of his mathematical productioFunding for open access charge: Universidad de Málaga / CBU

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth century geometers. Finally, we characterize the associated normal congruences

New examples of rational Gushel-Mukai fourfolds

We construct new examples of rational Gushel-Mukai fourfolds, giving more evidence for the analog of the Kuznetsov Conjecture for cubic fourfolds: a Gushel--Mukai fourfold is rational if and only if it admits an associated K3 surface.Comment: Minor changes. Exposition improved. To appear in Mathematische Zeitschrif

A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold

In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi also showed that over $\mathbb{C}$ each of these Calabi-Yau threefolds $Y$ is derived equivalent to a Reye congruence Calabi-Yau threefold $X$. We show that these derived equivalences may also be constructed over $\mathbb{Q}$ and give sufficient conditions for $X$ to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi-Yau variety over $\mathbb{C}$