Algorithms for rational cones and toric geometry

该项目的第一个目标是进一步开发软件包Normaliz，该软件包结合了有理锥和仿射幺半群的几种算法，特别是三角剖分，希尔伯特基和h向量的计算。Hilbert基的计算在交换代数、代数几何、数论和组合优化等数学领域都有着重要的意义。同时Normaliz发现应用在理论物理和代数拓扑。申请的进一步开发旨在提高Normaliz的能力，并为Normaliz打开更多的应用程序。第二个目标是算法搜索反例，以开放的问题，在射影复曲面几何，特别是射影规范的光滑复曲面品种和他们的定义在度2。

The first goal of the project is the further development of the software package Normaliz that combines several algorithms for rational cones and affine monoids, in particular the computation of triangulations, Hilbert bases and h-vectors. The computation of Hilbert bases is of considerable interest in many areas of mathematics, for example, commutative algebra, algebraic geometry, number theory and combinatorial optimization. Meanwhile Normaliz has found applications in theoretical physics and algebraic topology. The further development applied for is supposed to improve the capacity of Normaliz and to open further applications for Normaliz. The second goal is the algorithmic search for counterexamples to open questions in projective toric geometry, notably the projective normality of smooth toric varieties and their definition in degree 2.