论文标题
Gorenstein Fano曲折变性
Gorenstein Fano toric degenerations
论文作者
论文摘要
我们提出了一个精致但自然的曲折变性概念，该概念尊重给定的嵌入，并表明在此框架内，戈伦斯坦·范诺（Gorenstein Fano）品种只能通过其抗态嵌入将其嵌入Gorenstein Fano Toric品种中。这也给出了出现反射性多面体的精确标准，这可能是镜像对称性中的应用所需的。为了证明此陈述的证明，我们将研究极性双重晶格多层的多面体。作为副产品，我们将有理凸多属的晶格点的数量与相关的圆环不变的理性威尔分离器的欧拉特特征之间概括，这使我们可以证明Ehrhart-Macdonald互惠性和串行二元性是多种品种类别的陈述。此外，我们猜想了有理凸多属性的Ehrhart准级 - 多项式的必要条件，即是多项式。最后，我们表明，在Gorenstein Fano品种上具有最坏理性概念的抗态线束由其Hilbert多项式的组合条件唯一决定。
We propose a refined but natural notion of toric degenerations that respect a given embedding and show that within this framework a Gorenstein Fano variety can only be degenerated to a Gorenstein Fano toric variety if it is embedded via its anticanonical embedding. This also gives a precise criterion for reflexive polytopes to appear, which might be required for applications in mirror symmetry. For the proof of this statement we will study polytopes whose polar dual is a lattice polytope. As a byproduct we generalize a connection between the number of lattice points in a rational convex polytope and the Euler characteristic of an associated torus invariant rational Weil divisor, allowing us to show that Ehrhart-Macdonald Reciprocity and Serre Duality are equivalent statements for a broad class of varieties. Additionally, we conjecture a necessary and sufficient condition for the Ehrhart quasi-polynomial of a rational convex polytope to be a polynomial. Finally, we show that the anticanonical line bundle on a Gorenstein Fano variety with at worst rational singularities is uniquely determined by a combinatorial condition of its Hilbert polynomial.
