Triangulations of the sphere and degenerations of K3 surfaces

Secant Degree of Toric Surfaces and Delightful Planar Toric Degenerations

The $k$-secant degree is studied with a combinatorial approach. A planar toric degeneration of any projective toric surface $X$ corresponds to a regular unimodular triangulation $D$ of the polytope defining $X$. If the secant ideal of the initial ideal with respect to $D$ coincides with the initial ideal of the secant ideal, then $D$ is said to be delightful and the $k$-secant degree of $X$ can be easily computed. All delightful triangulations of toric surfaces having sectional genus $g\leq1$ are completely classified and, for $g\geq2$, a lower bound for the $2$- and $3$-secant degree, by mean…

Pillow Degenerations of K3 Surfaces

In this article we construct a specific projective degeneration of K3 surfaces of degree 2g-2 in P^g to a union of 2g-2 planes, which meet in such a way that the combinatorics of the configuration of planes is a triangulation of the 2-sphere. Abstractly, such degenerations are said to be Type III degenerations of K3 surfaces. Although the birational geometry of such degenerations is fairly well understood, the study of projective degenerations is not nearly as completely developed. In this article we construct degenerations for which the general member is embedded by a multiple of the primitiv…

Toric degenerations of integrable systems on Grassmannians and polygon spaces

We introduce a completely integrable system on the Grassmannian of 2-planes in an n-space associated with any triangulation of a polygon with n sides, and compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.

Semistable K3-surfaces with icosahedral symmetry

In a Type III degeneration of K3-surfaces the dual graph of the central fibre is a triangulation of the 2-sphere. We realise the tetrahedral, octahedral and especially the icosahedral triangulation in families of K3-surfaces, preferably with the associated symmetry groups acting.

Triangulations of spheres and discs

The main ob jective of this research is to find the different types of elliptic triangulations for planar discs and spheres. We begin in Chapter 1 with the mandatory introduction. In the second chapter we define and study the notion of a patch, that is, a triangulation of a planar disc. By introducing a suitable notion of degree for the vertex points, we focus on those patches with points of degrees less tah or equal to 6. Such patches are called el liptic. We show that the elliptic patches with precisely three points of degree four, denoted by (0, 3, 0), can be classified. The number of verte…

通过Belyi函数和拉普拉斯算子行列式研究奇异常曲率球面的三角剖分

Triangulations of singular constant curvature spheres via Belyi functions and determinants of Laplacians

本文通过Belyi函数和拉普拉斯行列式研究奇异常曲率球面的三角剖分，并计算了不同正多面体的行列式。