通过一对 Gorenstein PL 锥构造多联晶格
A construction of polyptych lattices via a pair of Gorenstein PL cones
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中文总结 AI 辅助
本文提出一种通过一对Gorenstein PL锥构造严格对偶多联晶格的方法，引入Gorenstein PL锥概念，并证明若原多联晶格可去热带化，则新构造的也可去热带化，为严格对偶多联晶格提供丰富例子。
中文摘要 AI 辅助
多联晶格理论旨在将许多推广环面几何的理论中出现的分段线性双射（突变）整合到一个统一的组合框架中，例如Newton-Okounkov体、环面退化、簇簇等。与代数几何的联系来自多联晶格去热带化的伴随概念及其相关的紧化。本文的主要结果给出了多联晶格严格对偶对$(\mathcal{E}, \mathcal{F})$的一个具体构造，我们称之为Gorenstein PL锥扩张。为了定义我们的构造，我们引入了Gorenstein PL锥的概念，它是经典情形中Gorenstein锥的多联晶格类比。如果原始多联晶格是可去热带化的，那么新的多联晶格$\mathcal{E},\mathcal{F}$也是可去热带化的，通过一个简单的显式公式。我们的Gorenstein PL锥扩张为多联晶格的严格对偶对提供了丰富的例子来源。特别地，任意一对维数分别为$n,n'$的Gorenstein-Fano多胞体$\Delta,\Delta'$，都会产生一个秩为$n+n'+1$的可去热带化多联晶格的严格对偶对$(\mathcal{E}(\Delta,\Delta'), \mathcal{F}(\Delta,\Delta'))$。其他例子可以从簇数据构建。
英文摘要
The theory of polyptych lattices seeks to incorporate, into a single combinatorial framework, the piecewise-linear bijections (mutations) that appear in many theories that generalize toric geometry, such as Newton-Okounkov bodies, toric degenerations, cluster varieties, and other areas. The link to algebraic geometry comes from the accompanying notion of detropicalizations of polyptych lattices, and their associated compactifications. The main result of this manuscript gives a concrete construction of a strict dual pair $(\mathcal{E}, \mathcal{F})$ of polyptych lattices, which we call a Gorenstein PL cone extension. To define our construction, we introduce the notion of Gorenstein PL cones, which are a polyptych analogue of a Gorenstein cone in the classical setting. If the original polyptych lattices are detropicalizable, then the new polyptych lattices $\mathcal{E},\mathcal{F}$ are also detropicalizable, via a simple explicit formula. Our Gorenstein PL cone extensions give a rich source of examples of strict dual pairs of polyptych lattices. In particular, any pair of Gorenstein-Fano polytopes $Δ,Δ'$ of dimension $n,n'$ respectively, gives rise to a strict dual pair $(\mathcal{E}(Δ,Δ'), \mathcal{F}(Δ,Δ'))$ of detropicalizable polyptych lattices of rank $n+n'+1$. Other examples can be built from cluster data.
发表机构
- University of California Santa Cruz（加州大学圣克鲁兹分校）
- McMaster University（麦克马斯特大学）
- University of Kentucky（肯塔基大学）
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