This paper studies a class of singular fibrations, called self-crossing boundary fibrations, which play an important role in semitoric and generalized complex geometry. These singular fibrations can be conveniently described using the language of Lie algebroids. We will show how these fibrations arise from (nonfree) torus actions, and how to use them to construct and better understand self-crossing stable generalized complex four-manifolds. We moreover show that these fibrations are compatible with taking connected sums, and use this to prove a singularity trade result between two types of singularities occurring in these types of fibrations (a so-called nodal trade).

We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a Möbius automorphism group of dimension at least two. Our theorem generalizes the classical classification of Dupin cyclides.

In this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hyper-surface germ and P is the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of f = 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Lê et al.