具有多项式权和分段多项式角点轨迹的热带簇
Tropical varieties with polynomial weights and corner loci of piecewise polynomials
摘要：
我们发现了几个多面体的混合体积和它们的并集的凸包之间的关系，从以下事实推导出它：一个多面体集合的混合体积只取决于它们的支撑函数的乘积（而不是单个支撑函数）。对于整数多面体，这种依赖性本质上是两个著名的复曲面上同调组合模型之间同构的某种特殊化，然而，这种构造迄今为止还没有扩展到任意多面体（部分原因是缺乏能够替代复曲面几何的组合工具，当顶点不是理性的）。我们提供了这样一个扩展，这导致了一个明确的公式的混合体积的产品的支撑函数，也可能是有趣的，因为组合工具（热带品种多项式权重和他们的角落轨迹），出现在我们的建设。作为这些新对象的另一个可能应用的例子，我们注意到热带流形M中的每个热带子簇都可以局部地表示为M与另一个热带簇（可能具有负权重）的交集，并推测这一事实对奇异M的某些推广。关于热带流形的子簇的上述事实可能是独立的兴趣，因为它意味着最近由Allerman，Francois，Rau和Shaw构造的关于热带流形的相交理论是从环境向量空间局部导出的。
英文摘要：
We find a relation between mixed volumes of several polytopes and the convex hull of their union, deducing it from the following fact: the mixed volume of a collection of polytopes only depends on the product of their support functions (rather than on the individual support functions). For integer polytopes, this dependence is essentially a certain specialization of the isomorphism between two well-known combinatorial models for the cohomology of toric varieties, however, this construction has not been extended to arbitrary polytopes so far (partially due to the lack of combinatorial tools capable of substituting for toric geometry, when vertices are not rational). We provide such an extension, which leads to an explicit formula for the mixed volume in terms of the product of support functions, and may also be interesting because of the combinatorial tools (tropical varieties with polynomial weights and their corner loci) that appear in our construction. As an example of another possible application of these new objects, we notice that every tropical subvariety in a tropical manifold M can be locally represented as the intersection of M with another tropical variety (possibly with negative weights), and conjecture certain generalizations of this fact to singular M. The above fact about subvarieties of a tropical manifold may be of independent interest, because it implies that the intersection theory on a tropical manifold, which was recently constructed by Allerman, Francois, Rau and Shaw, is locally induced from the ambient vector space.
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