发表机构
Universidad de Santiago de Chile (USACH); Instituto de Matemática Pura e Aplicada (IMPA); Universität Bielefeld（圣地亚哥智利大学; 纯数学与应用数学研究所; 比勒费尔德大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明链式射影空间具有对角的多元希尔伯特多项式，通过给出法交叉方案的一般公式并利用Mustafin簇局部描述其结构。
AI 中文摘要
链式射影空间是某些箭图表示的一维子空间的箭图格拉斯曼流形。线性系的退化产生这些表示，其极限因子由相关的链式射影空间参数化。目前尚不清楚是否所有链式射影空间都以这种方式产生。如果它们确实如此产生，则它们是射影空间乘积中（小）对角的退化。无论如何，我们在此证明它们具有对角的（多元）希尔伯特多项式。为实现这一目标，我们首先给出射影空间乘积中具有无重数层的（简单）法交叉方案希尔伯特多项式的公式，该公式比Castillo等人发现的更一般且更简单。然后，我们通过用Mustafin簇局部描述链式射影空间，证明其具有法交叉性质。最后，我们利用链式射影空间的分量交集与关联链网的单形剖分中某些多胞体之间的关系，证明我们可以应用我们的希尔伯特多项式公式。
英文摘要
Linked projective spaces are quiver Grassmannians of subspaces of dimension 1 of certain quiver representations. Degenerations of linear series produce these representations, with the limit divisors parameterized by the associated linked projective spaces. It is not known whether all linked projective spaces arise this way. If they do, they are degenerations of the (small) diagonal in a product of projective spaces. In any case, we prove here that they have the (multivariate) Hilbert polynomial of the diagonal. To achieve this, we first extend a Hilbert-polynomial formula for multiplicity-free varieties to (simple) normal-crossings schemes with multiplicity-free strata in products of projective spaces. Then we prove that a linked projective space is normal-crossings, by describing it locally in terms of Mustafin varieties. Finally, we use a relation between intersections of components of the linked projective space and certain polytopes in the tiling of a simplex associated to the linked net to prove that our formula for the Hilbert polynomial applies.
Comments17 pages, in v_2 we added two references and a name in the Acknowledgments
