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Tropical geometry over the tropical hyperfield. (English) Zbl 1499.14105

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This paper expands and analyzes properties of the tropical hyperfield and ordered blueprints. The tropical hyperfield, introduced by O. Viro [Proc. Steklov Inst. Math. 273, 252–282 (2011; Zbl 1237.14074)], is a delicate variant of the classical tropical semifield. The underlying set of the tropical hyperfield \(\mathbf{T}\) is \(\mathbb{R}_{\geq 0}\). Multiplication is the usual multiplication of real numbers and the hyperaddition is \[ a \boxplus b = \begin{cases} \{\max\{a, b\}\} \quad & \text{if} \, a \neq b \\ [0, a] \quad & \text{if} \, a=b \end{cases} \]
The tropical hyperfield \(\mathbf{T}\) has multiple advantages over the classical tropical semifield. In particular, the hyperaddition of \(\mathbf{T}\) allows the author to interpret a nonarchimedean absolute value \(v: k \to \mathbb{R}_{\geq 0}\) as a morphisms into the tropical hyperfield. Therefore, he describes scheme theoretic tropicalization of a classical variety, simply as a base change of a classical variety over \(k\) to \(\mathbf{T}\).
One of the results of this paper is that Berkovich analytification \(X^{\text{an}}\) and set theoretic tropicalization \(X^{\text{trop}}\) of a classical variety \(X\) can be described as a set of \(\mathbf{T}\)-rational points of scheme theoretic tropicalizations. In fact, in the following diagram, the natural bijections \(X^{\text{an}} \to \text{Trop}_v(\mathbf{X})(\mathbf{T})\) and \(X^{\text{trop}} \to \text{Trop}_v(Y)(\mathbf{T})\) are homeomorphisms with respect to Euclidean topologies that they inherit from \(\mathbf{T} = \mathbb{R}_{\geq 0}\)
\[ \begin{tikzcd}[column sep = 3cm, row sep= 0.75cm] X^{\text{an}} \ar{r}{\text{trop}} \ar{d} & X^{\text{trop}} \ar{d} \\ \text{Trop}_v(\mathbf{X})(\mathbf{T}) \ar{r}{f^*} & \text{Trop}_v(Y)(\mathbf{T}) \end{tikzcd} \]
Another interesting result of the paper is that the Giansiracusa bend relations in [J. Giansiracusa and N. Giansiracusa, Duke Math. J. 165, No. 18, 3379–3433 (2016; Zbl 1409.14100)] appear naturally from the scheme theoretic tropicalization by enforcing the relation \(1 + 1 = 1\).

MSC:

14T10 Foundations of tropical geometry and relations with algebra
06F05 Ordered semigroups and monoids
16Y60 Semirings
08A30 Subalgebras, congruence relations
12J20 General valuation theory for fields
14G22 Rigid analytic geometry
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References:

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