Equations of tropical varieties
Equations of tropical varieties
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DOI：
10.1215/00127094-3645544
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发表时间：
2013-07
影响因子：
2.5
通讯作者：
Jeffrey Giansiracusa;Noah Giansiracusa
中科院分区：
文献类型：
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作者：
Jeffrey Giansiracusa;Noah Giansiracusa
我们介绍了一个图式理论丰富的热带几何的主要对象。利用一类半环方案，我们将热带超曲面构造为幂等元半环上的方案，例如$\mathbb{T}=(\mathbb{R}\cup\-\inty，\mathm{max}，+)，将它们实现为由幂等元理论唯一确定的显式热带方程组的解集.然后，我们定义了一个热带化函数，它将环R上具有非阿基米德赋值的环簇的闭子格式发送到相应的热带环簇的闭子格式。当传递到$\mathbb{T}$点集时，这归结为Kajiwara-Payne的扩展热带化，并且在射影超曲面的情况下，我们证明了方案结构决定了附加到顶维单元的多重性。通过改变赋值，这些热带化形成了由我们构造的R上的赋值的模空间参数化的代数族$\mathbb{T}$-方案。对于投影子格式，希尔伯特多项式被热带化保持不变，而与赋值无关。最后，我们给出了一些例子，并讨论了方案理论背景下的热带基地。
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.