In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are equivalent to certain sets of filtrations of vector spaces. We systematically construct the theory from the point of view of… (More)
King's conjecture states that on every smooth complete toric variety X there exists a strongly exceptional collection which generates the bounded derived category of X and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface F 2 iteratively blown up three times, and we show by explicit… (More)
The compilation of RELFUN programs consists of two main stages, horizontal transformations and vertical translations. The horizontal transformer performs both source-to-source steps into a subset of RELFUN and source-to-intermediate steps into a RELFUN-like language. The vertical translator is also divided into two phases, the classiier and the code… (More)
A knowledge-based process-planning system for generating workplans for idealized lathe CNC machines is discussed. It transforms CAD-like diagrams of rotational-symmetric workpieces into abstract NC programs. To simplify development and evolution , declarative representations are used for all processing stages and steps, including a qualitative simulation of… (More)
This work discusses combinatorial and arithmetic aspects of cohomology vanishing for divisorial sheaves on toric varieties. We obtain a refined variant of the Kawamata-Viehweg theorem which is slightly stronger. Moreover, we prove a new vanishing theorem related to divisors whose inverse is nef and has small Iitaka dimension. Finally , we give a new… (More)
We give a complete classification of equivariant vector bundles of rank two over smooth complete toric surfaces and construct moduli spaces of such bundles. This note is a direct continuation of an earlier note where we developed a general description of equivariant sheaves on toric varieties. Here we give a first application of that description.
In this work we construct global resolutions for general coherent equivariant sheaves over toric varieties. For this, we use the framework of sheaves over posets. We develop a notion of gluing of posets and of sheaves over posets, which we apply to construct global resolutions for equivariant sheaves. Our constructions give a natural correspondence between… (More)
In this article we survey recent results of joint work with Lutz Hille on exceptional sequences of invertible sheaves on rational surfaces. This survey is supplemented by explicit examples.
We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen-Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a… (More)
The effective use, preservation and creation of knowledge in design-type work processes is of vital interest for an enterprise. We present the CLOCKWORK approach to organizational learning which comprises a web-based toolset to structure, store and retrieve knowledge elements created during all phases of the product development. The examination and re-use… (More)