Singular pleated surfaces and CP1–structures

Shearing Hyperbolic Surfaces, Bending Pleated Surfaces and Thurston's Symplectic Form

The article develops a system of local holomorphic coordinates for the space of hyperbolic 3–manifolds with the fundamental group of a surface. These coordinates depend on the choice of a geodesic lamination on the surface, and are a complexified version of Thurston's shear coordinates for Teichmüller space. The imaginary part of these coordinates measures the bending of a pleated surface reali...

DEPTH OF PLEATED SURFACES IN TOROIDALCUSPS OF HYPERBOLIC 3 - MANIFOLDSYing

On Singular Cubic Surfaces

We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kähler–Einstein metric on two singular cubic surfaces.

Circle Patterns on Singular Surfaces

We consider “hyperideal” circle patterns, i.e. patterns of disks which do not cover the whole surface, which are associated to hyperideal hyperbolic polyhedra. The main result is that, on a Euclidean or hyperbolic surface with conical singularities, those hyperideal circle patterns are uniquely determined by the intersection angles of the circles and the singular curvatures. This is related to ...

Enumerating singular curves on surfaces

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basi...