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Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (G, X)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (G, X)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (G, X)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (G, X)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities. [ABSTRACT FROM AUTHOR]

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely A -determined map-germs from the plane to the plane can be represented by harmonic maps. [ABSTRACT FROM AUTHOR]

Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduced by Bowman and Valdez: linear approaches and rotational components. We show that there is a large flexibility in the spaces of rotational components and even more in the spaces of linear approaches. In particular, we prove that every finite topological space arises as space of rotational components. However, this space will still not contain enough information to describe an infinite translation surface. We showcase this through an uncountable family with the same space of rotational components but different spaces of linear approaches. Additionally, we study several known and new examples to illustrate the concept of linear approaches and rotational components. [ABSTRACT FROM AUTHOR]

We study the Z-module of first order local Vassiliev type invariants of stable maps of oriented 3-manifolds into R4. As a previous step, we determine a complete classification of the codimension two germs and multigerms as well as their corresponding bifurcation diagrams. This allows us to show the existence of 4 generators for this module. In particular, we see that the number of pairs of quadruple point, the number of transversal intersections of the crosscap suspension with immersive branches and the Euler number of the image are first order local Vassiliev type invariants for these maps. We also prove that the total number of connected components of the triple point curve is a non local Vassiliev type invariant. [ABSTRACT FROM AUTHOR]

Using a construction due to C. Casagrande and further developed by the author (Int. Math. Res. Notices, ), we prove that the Picard number of a non-smooth Fano 3-fold with isolated factorial canonical singularities is at most 6. [ABSTRACT FROM AUTHOR]

Let X be a Fano manifold of dimension n and index n − 3. Kawamata proved the non vanishing of the global sections of the fundamental divisor in the case n = 4. Moreover he proved that if Y is a general element of the fundamental system then Y has at most canonical singularities. We prove a generalization of this result in arbitrary dimension. [ABSTRACT FROM AUTHOR]

We study the caustic, evolute, Minkowski symmetry set and parallels of a smooth and regular curve in the Minkowski plane. [ABSTRACT FROM AUTHOR]

In dimension 2 there is a very useful criterion, due to Miles Reid, giving a necessary and sufficient condition for an adjoint linear system on a nonsingular complex projective surface X to be free at a point $${p\in X}$$ , in terms of the coherent cohomology of certain divisors on the blowup of X at p. In this note we extend Miles Reid's result to higher dimensions. [ABSTRACT FROM AUTHOR]

We enumerate cones over singular varieties satisfying appropriate incidence conditions to linear subspaces. [ABSTRACT FROM AUTHOR]

We define the concepts of binormal and asymptotic directions for submanifolds embedded with codimension 2 into Euclidean spaces and obtain necessary conditions, in terms of the existence of such directions, for the convexity and the sphericity of these submanifolds. [ABSTRACT FROM AUTHOR]

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams. [ABSTRACT FROM AUTHOR]