Bunches of cones in the divisor class group -- A new combinatorial language for toric varieties

As an alternative to the description of a toric variety by a fan in the lattice of one parameter subgroups, we present a new language in terms of what we call bunches -- these are certain collections of cones in the vector space of rational divisor classes. The correspondence between these bunches and fans is based on classical Gale duality. The new combinatorial language allows a much more natural description of geometric phenomena around divisors of toric varieties than the usual method by fans does. For example, the numerically effective cone and the ample cone of a toric variety can be read off immediately from its bunch. Moreover, the language of bunches appears to be useful for classification problems.Comment: Minor changes, to appear in Int. Math. Res. No

Geometric Invariant Theory based on Weil Divisors

Given an action of a reductive group on a normal variety, we construct all invariant open subsets admitting a good quotient with a quasiprojective or a divisorial quotient space. Our approach extends known constructions like Mumford's Geometric Invariant Theory. We obtain several new Hilbert-Mumford type theorems, and we extend a projectivity criterion of Bialynicki-Birula and Swiecicka for varieties with semisimple group action from the smooth to the singular case.Comment: Final version, to appear in Compositio Mat

Log del Pezzo C∗\mathbb{C}^*-surfaces, K\"ahler-Einstein metrics, K\"ahler-Ricci solitons and Sasaki-Einstein metrics

We consider two classes of non-toric log del Pezzo $\mathbb{C}^*$-surfaces: on the one side the 1/3-log canonical ones and on the other side those of Picard number one and Gorenstein index at most 65. In each of the two classes we figure out the surfaces admitting a K\"ahler-Einstein metric, a K\"ahler-Ricci soliton and those allowing a Sasaki-Einstein metric on the link of their anticanonical cone. We encounter examples that admit a K\"{a}hler-Ricci soliton but no Sasaki-Einstein cone link metric.Comment: 27 page

On Cox rings of K3-surfaces

We study Cox rings of K3-surfaces. A first result is that a K3-surface has a finitely generated Cox ring if and only if its effective cone is polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3-surfaces of Picard number two, and explicitly compute the Cox rings of generic K3-surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.Comment: minor corrections, to appear in Compositio Mathematica, 32 page

Genomic Landscape of Spitzoid Neoplasms Impacting Patient Management

Spitzoid neoplasms are a distinct group of melanocytic proliferations characterized by epithelioid and/ or spindle shaped melanocytes. Intermediate forms that share features of both benign Spitz nevi (SN) and Spitz melanoma, i.e., malignant Spitz tumor (MST) represent a diagnostically and clinically challenging group of melanocytic lesions. A multitude of descriptive diagnostic terms exist for these ambiguous lesions with atypical Spitz tumor (AST) or Spitz tumor of uncertain malignant potential (STUMP) just naming two of them. This diagnostic gray zone creates confusion and high insecurity in clinicians and in patients. Biological behavior and clinical course of this intermediate group still remains largely unknown, often leading to difficulties with uncertainties in clinical management and prognosis. Consequently, a better stratification of Spitzoid neoplasms in benign and malignant forms is required thereby keeping the diagnostic group of AST/STUMP as small as possible. Ancillary diagnostic techniques such as immunohistochemistry, comparative genomic hybridization, fluorescence in situ hybridization, next generation sequencing, micro RNA and mRNA analysis as well as mass spectrometry imaging offer new opportunities for the distinct diagnosis, thereby allowing the best clinical management of Spitzoid neoplasms. This review gives an overview on these additional diagnostic techniques and the recent developments in the field of molecular genetic alterations in Spitzoid neoplasms. We also discuss how the recent findings might facilitate the diagnosis and stratification of atypical Spitzoid neoplasms and how these findings will impact the diagnostic work up as well as patient management. We suggest a stepwise implementation of ancillary diagnostic techniques thereby integrating immunohistochemistry and molecular pathology findings in the diagnosis of challenging ambiguous Spitzoid neoplasms. Finally, we will give an outlook on pending future research objectives in the field of Spitzoid melanocytic lesions

On Quasiprojective Open Subsets of G-Varieties

Let X be a normal algebraic variety endowed with a regular action of a connected linear algebraic group G. We provide a simple proof for the fact that the union GU of all translates of a given quasiprojective open subset U X is again quasiprojective

A General Hilbert-Mumford Criterion : Revised Version

We provide a Hilbert-Mumford Criterion for actions of reductive groups G on Q-factorial complex varieties. The result allows to construct open subsets admitting a good quotient by G from certain maximal open subsets admitting a good quotient by a maximal torus of G. As an application, we indicate how to obtain all invariant open subsets with good quotient for a given G-action on a complete Q-factorial toric variety