Tropical schemes, tropical cycles, and valuated matroids. (English) Zbl 1509.14120

Maclagan, Diane; Rincón, Felipe

J. Eur. Math. Soc. (JEMS) 22, No. 3, 777-796 (2020).

To a subvariety \(Y\) of the \(n\)-dimensional algebraic torus \(T\), it is possible to associate a polyhedral complex \(\operatorname{trop}(Y)\). This polyhedral complex is called the “tropicalization” of \(Y\) and encodes some fundamental data about the variety \(Y\). In [Duke Math. J. 165, No. 18, 3379–3433 (2016; Zbl 1409.14100)] J. Giansiracusa and N. Giansiracusa proposed an upgrade of this idea for varieties to the more general framework of schemes.
In the paper under review, Maclagan and Rincón study the relation between these “tropical schemes”, the ideals in the semiring of tropical polynomials, and the framework of valuated matroids of A. W. M. Dress and W. Wenzel [Adv. Math. 93, No. 2, 214–250 (1992; Zbl 0754.05027)]. Fixing an ideal \(I\) and denoting by \(Y\) the tropical scheme defined by \(I\), their Theorem 1.1 proves that any of the following three data determines the other two:

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The structure of the tropical scheme. (More precisely, the congruence \(\operatorname{trop}(I)\)).

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The tropical ideal \(\operatorname{trop}(I)\).

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The collection of all valuated matroids associated to the graded components of the homogenization of \(I\).

For a variety \(Y\), the top-dimensional cells of \(\operatorname{trop}(Y)\) are endowed with certain weights that turn it into a tropical cycle. The second main result, Theorem 1.2, proves that the tropical cycle structure on a tropical variety is determined from any of the three listed objects above. This answers a question posed in [J. Giansiracusa and N. Giansiracusa, Duke Math. J. 165, No. 18, 3379–3433 (2016; Zbl 1409.14100)].

Reviewer: Luis Ferroni (Stockholm)

MSC:

14T20	Geometric aspects of tropical varieties
14T15	Combinatorial aspects of tropical varieties
14T10	Foundations of tropical geometry and relations with algebra
05B35	Combinatorial aspects of matroids and geometric lattices

Keywords:

tropical scheme; valuated matroid

Citations:

Zbl 1409.14100; Zbl 0754.05027

Software:

Binomials.m2; TropLi

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