A guide to tropicalizations.

*(English)*Zbl 1318.14061
Brugallé, Erwan (ed.) et al., Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9146-9/pbk; 978-1-4704-0940-1/ebook). Contemporary Mathematics 589, 125-189 (2013).

Let \(K\) be a non-Archimedean field and \(T\) be a split algebraic torus over \(K\). In tropical geometry one associates to a closed subscheme \(X\) of \(T\) its tropicalization \(\mathrm{Trop}(X)\), a weighted rational polyhedral complex in the real vector space \(N_\mathbb{R}\) generated by the cocharacter lattice \(N\) of \(T\). The basic idea of tropical geometry is to relate the combinatorial properties of \(\mathrm{Trop}(X)\) to the algebraic geometric properties of \(X\).

It has been observed in [W. Gubler, Invent. Math. 169, No. 2, 321–376 (2007; Zbl 1153.14029)] and [M. Einsiedler, M. Kapranov and D. Lind, J. Reine Angew. Math. 601, 139–157 (2006; Zbl 1115.14051)] that \(\mathrm{Trop}(X)\) naturally arises as a projection of the Berkovich analytic space \(X^{\mathrm{an}}\) associated to \(X\) to \(N_\mathbb{R}\).

In his guide Gubler works out many aspects of this theory in their natural generality and provides the reader with accessible foundational knowledge of the non-Archimedean approach to tropical geometry.

After a concise introduction to non-Archimedean analytic geometry he constructs the tropicalization map \(\mathrm{trop}:T^{\mathrm{an}}\rightarrow N_\mathbb{R}\) that is given by associating to a seminorm \(| .|_x\) on the coordinate ring \(K[M]\) of \(T\) the element \(m\mapsto -\log| \chi^m|_x\) in \(N_\mathbb{R}=Hom(M,\mathbb{R})\). The tropical variety \(\mathrm{Trop}(X)\) associated to \(X\subseteq T\) is then defined as projection of \(X^{\mathrm{an}}\) to \(N_\mathbb{R}\) under the tropicalization map.

Furthermore Gubler uses the theory of models of algebraic varieties over valuations rings to give an extrinsic geometric definition of the initial degenerations associated to \(X\). From the point of view of commutative algebra these initial degenerations can be computed using computer algebra systems and form one of the crucial techniques used when explicitly computing with tropical varieties (see [D. Maclagan and B. Sturmfels, Introduction to tropical geometry. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1321.14048)]).

In a next step Gubler generalizes the theory of toric schemes over discrete valuation rings (see [G. Kempf et al., Toroidal embeddings. I. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0271.14017)]) to toric schemes over arbitrary valuation rings of rank one.

Based on these foundations, he gives a generalization of Tevelev’s theory of compactifications of subschemes of tori in toric varieties, so called tropical compactifications (see [J. Tevelev, Am. J. Math. 129, No. 4, 1087–1104 (2007; Zbl 1154.14039)]), to compactifications in arbitrary toric schemes over valuation rings \(R\) of rank one. This finally leads to a general theory of tropical multiplicities encompassing all earlier approaches. The crucial difficulty in this development is that the valuation ring \(R\) may not be Noetherian and, when dealing with the geometry of models over \(R\), one has to leave the familiar realm of Noetherian schemes.

The author of this review has profited immensely from reading this survey article and strongly recommends anyone with even only a remote interest in the relationship between tropical and non-Archimedean geometry to read it.

For the entire collection see [Zbl 1266.14003].

It has been observed in [W. Gubler, Invent. Math. 169, No. 2, 321–376 (2007; Zbl 1153.14029)] and [M. Einsiedler, M. Kapranov and D. Lind, J. Reine Angew. Math. 601, 139–157 (2006; Zbl 1115.14051)] that \(\mathrm{Trop}(X)\) naturally arises as a projection of the Berkovich analytic space \(X^{\mathrm{an}}\) associated to \(X\) to \(N_\mathbb{R}\).

In his guide Gubler works out many aspects of this theory in their natural generality and provides the reader with accessible foundational knowledge of the non-Archimedean approach to tropical geometry.

After a concise introduction to non-Archimedean analytic geometry he constructs the tropicalization map \(\mathrm{trop}:T^{\mathrm{an}}\rightarrow N_\mathbb{R}\) that is given by associating to a seminorm \(| .|_x\) on the coordinate ring \(K[M]\) of \(T\) the element \(m\mapsto -\log| \chi^m|_x\) in \(N_\mathbb{R}=Hom(M,\mathbb{R})\). The tropical variety \(\mathrm{Trop}(X)\) associated to \(X\subseteq T\) is then defined as projection of \(X^{\mathrm{an}}\) to \(N_\mathbb{R}\) under the tropicalization map.

Furthermore Gubler uses the theory of models of algebraic varieties over valuations rings to give an extrinsic geometric definition of the initial degenerations associated to \(X\). From the point of view of commutative algebra these initial degenerations can be computed using computer algebra systems and form one of the crucial techniques used when explicitly computing with tropical varieties (see [D. Maclagan and B. Sturmfels, Introduction to tropical geometry. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1321.14048)]).

In a next step Gubler generalizes the theory of toric schemes over discrete valuation rings (see [G. Kempf et al., Toroidal embeddings. I. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0271.14017)]) to toric schemes over arbitrary valuation rings of rank one.

Based on these foundations, he gives a generalization of Tevelev’s theory of compactifications of subschemes of tori in toric varieties, so called tropical compactifications (see [J. Tevelev, Am. J. Math. 129, No. 4, 1087–1104 (2007; Zbl 1154.14039)]), to compactifications in arbitrary toric schemes over valuation rings \(R\) of rank one. This finally leads to a general theory of tropical multiplicities encompassing all earlier approaches. The crucial difficulty in this development is that the valuation ring \(R\) may not be Noetherian and, when dealing with the geometry of models over \(R\), one has to leave the familiar realm of Noetherian schemes.

The author of this review has profited immensely from reading this survey article and strongly recommends anyone with even only a remote interest in the relationship between tropical and non-Archimedean geometry to read it.

For the entire collection see [Zbl 1266.14003].

Reviewer: Martin Ulirsch (Providence)