THEGRADUATESTUDENTSECTION

WHATIS… Tropical Geometry? Eric Katz Communicated by Cesar E. Silva

This note was written to answer the question, What is tropical geometry? That question can be interpreted in two ways: Would you tell me something about this research area? and Why the unusual name ‘tropi- Tropical cal geometry’? To address Figure 1. Tropical curves, such as this tropical line the second question, trop- and two tropical conics, are polyhedral complexes. geometry ical geometry is named in transforms honor of Brazilian com- science. Here, tropical geometry can be considered as alge- puter scientist Imre Simon. braic geometry over the tropical semifield (ℝ∪{∞}, ⊕, ⊗) questions about This naming is compli- with operations given by cated by the fact that he algebraic lived in São Paolo and com- 푎 ⊕ 푏 = min(푎, 푏), 푎 ⊗ 푏 = 푎 + 푏. muted across the Tropic One can then find tropical analogues of classical math- varieties into of Capricorn. Whether his ematics and define tropical polynomials, tropical hyper- work is tropical depends surfaces, and tropical varieties. For example, a degree 2 questions about on whether he preferred to polynomial in variables 푥, 푦 would be of the form do his research at home or polyhedral min(푎 + 2푥, 푎 + 푥 + 푦, 푎 + 2푦, 푎 + 푥, 푎 + 푦, 푎 ) in the office. 20 11 02 10 01 00 complexes. The main goal of for constants 푎푖푗 ∈ ℝ ∪ {∞}. The zero locus of a tropical tropical geometry is trans- polynomial is defined to be the set of points where the forming questions about minimum is achieved by at least two entries. In the above algebraic varieties into questions about polyhedral example, it would be the set of points (푥, 푦) where there complexes. A process called tropicalization attaches a are distinct indices (푖1, 푗1), (푖2, 푗2) such that polyhedral complex to an algebraic variety. The polyhe- 푎 + 푖 푥 + 푗 푦 = 푎 + 푖 푥 + 푗 푦 ≤ 푎 + 푖푥 + 푗푦 dral complex, a combinatorial object, encodes some of 푖1푗1 1 1 푖2푗2 2 2 푖푗 the geometry of the original algebraic variety. There are for all pairs (푖, 푗). These objects do not look like their clas- other ways of constructing similar polyhedral complexes, sical counterparts and instead are polyhedral complexes and the polyhedral complexes that arise can be studied of differing combinatorial types. For example, Figure 1 in their own right. We will discuss three approaches: the shows a tropical line and two tropical curves cut out by synthetic, the valuative, and the degeneration-theoretic. degree 2 polynomials in 푥 and 푦. Recall that a polyhedral complex in ℝ푛 is a union of Synthetic Approach polyhedra (that is, sets cut out by linear equations and Tropical geometry originally arose from considerations of inequalities) such that any set of polyhedra intersects in tropical algebra, itself motivated by questions in computer a common (possibly empty) face of each member. More is true about the polyhedral complexes that arise in this Eric Katz is assistant professor of mathematics at The Ohio State fashion; they are tropical varieties, which are defined University. His e-mail address is [email protected]. to be integral, weighted, balanced polyhedral complexes. For permission to reprint this article, please contact: Here integral means that their defining linear equations [email protected]. and inequalities have integer coefficients; weighted means DOI: http://dx.doi.org/10.1090/noti1507 that their top-dimensional polyhedra are assigned positive

380 Notices of the AMS Volume 64, Number 4 THEGRADUATESTUDENTSECTION integer weight (in the An algebraic torus (핂∗)푛 is the Cartesian product of above example, all finitely many copies of 핂∗ and should be thought of as weights are 1); and bal- A combinatorial analogous to (푆1)푛. A subvariety of (핂∗)푛 is the common anced means that the object encodes zero set of a system of Laurent polynomial equations in weights around a codi- the coordinates 푥1, 푥2, … , 푥푛. The tropicalization of 푋 is mension 1 face satisfy some geometry of defined to be Trop(푋) = 푣(푋), the topological closure a particular balancing of the image of 푋 under the product of valuation maps relation, which in the 1- the algebraic 푣∶ (핂∗)푛 → ℝ푛. This approach does specialize to the dimensional case says synthetic approach, a fact that we illustrate with an that the primitive in- variety. example. Consider the subvariety 푋 in (핂∗)2 defined by teger vectors along 푥 + 푦 + 1 = 0. For a point (푥, 푦) to belong to 푋, what edges emanating from must be true of its valuations? Let us express the defining a given vertex, multiplied by the weight of the edges, sum equation of 푋 in terms of the Puiseux series of 푥, 푦, and to zero. This can be thought of as a zero tension condition 1: ∞ ∞ as in physical dynamics. 푖/푁 푖/푁 0 Tropical geometry’s first major result was Grigory ∑ 푐푖푡 + ∑ 푑푖푡 + 푡 = 0. 푖=푚 Mikhalkin’s proof (2005) that the number of plane curves 푖=푙 For this equality to be satisfied, the coefficients of any of degree 푑 and genus 푔 passing through 3푑 − 1 + 푔 exponent must sum to zero. In particular, the coefficients points in general position could be computed by counting of the smallest power of 푡 must sum to 0, so there must tropical curves with multiplicity. This led to the definition be at least two such nonzero coefficients. This tells us of an abstract tropical curve as a graph equipped with that min(푣(푥), 푣(푦), 푣(1)) must be achieved by at least additional data. Much of the early development of tropical two entries. It follows that Trop(푋) is contained in the geometry involved finding tropical analogues of theorems tropical hypersurface of 푥 ⊕ 푦 ⊕ 0 (where 0 = 푣(1)). It is about algebraic curves and their enumerative geometry. a theorem of Kapranov that the reverse containment is true, in fact, for all hypersurfaces. Valuation-Theoretic Approach This valuative approach allows one to speak of tropical Another approach to tropical geometry, which was de- varieties, not just tropical hypersurfaces. The tropical scribed in an unpublished manuscript of Mikhail Kapranov varieties that arise are integral, weighted, balanced poly- from the early 1990s but dates back to work of George hedral complexes by a result of David Speyer (2005). Bergman (1971) and Robert Bieri and J. R. J. Groves (1984), Moreover, they capture some of the geometry of the is to define a tropical variety as a shadow of an algebraic original variety 푋 by reflecting 푋’s class in intersection variety. We will first discuss a more familiar, analytic theory and, under suitable smoothness conditions, some version of this approach involving logarithmic limit sets. of 푋’s cohomology. In this sense, Trop(푋) is a shadow For 푡 > 0, consider the map Log ∶ (ℂ∗)푛 → ℝ푛 given by 푡 of 푋. Computing tropicalizations of algebraic varieties (푧1, … , 푧푛) ↦ (log푡(|푧1|), … , log푡(|푧푛|)). is an interesting problem making use of Gröbner basis The image of an algebraic subvariety 푋 ⊂ (ℂ∗)푛 is called techniques. an amoeba. Under the Hausdorff limit, lim푡→∞ Log푡(푋), it becomes a piecewise linear object called a polyhedral Degeneration-Theoretic Approach ∗ 푛 fan. When one studies a family of varieties 푋푡 ⊂ (ℂ ) Tropical geometry is very closely related to the study parameterized by 푡 and considers lim푡→∞ Log푡(푋푡), one of degenerations of algebraic varieties. In the classical ∗ 푛 obtains a richer polyhedral complex. situation, one may have a family of varieties 푌푡 ⊂ (ℂ ) For the algebraic approach, let 핂 be an algebraically depending on a parameter 푡 varying in a small disc 픻 closed field equipped with a nontrivial non-Archimedean around the origin. This arises, for instance, if one is valuation 푣∶ 핂∗ → 퐺 ⊆ ℝ where 퐺 is an additive subgroup given an algebraic variety 풴 ⊂ (ℂ∗)푛 × 픻 with projection ∗ ∗ 푛 of ℝ. Here 핂 = 핂\{0} is the set of units in 핂. The 푝∶ (ℂ ) × 픻 → 픻. The varieties 푌푡 are fibers of the valuation is said to be non-Archimedean if projection restricted to 풴. One usually studies semistable 푣(푥푦) = 푣(푥) + 푣(푦), families so that 풴 is nonsingular, the fibers 푌푡 for 푡 ≠ 0 are smooth, and the central fiber 푌0 has very mild 푣(푥 + 푦) ≥ min(푣(푥), 푣(푦)). (so-called normal crossing) singularities. The irreducible An example to keep in mind is 핂 = ℂ{{푡}}, the field of components of 푌0 may intersect in a combinatorially formal Puiseux series, which is the algebraic closure of interesting fashion encoded by a polyhedral complex Γ, the field of Laurent series. Here, an element 푥 ∈ 핂 can be called the dual complex. The central fiber 푌0 is called a written as a formal power series with complex coefficients degeneration of the generic fiber 푌푡 for 푡 ≠ 0. and fractional exponents with bounded denominator: There is a purely algebraic analogue of the study ∞ of families of algebraic varieties over a disc, the study 푖/푁 푥 = ∑ 푐푖푡 , 푐푙 ≠ 0. of schemes over a valuation ring. Here, the field 핂 is 푖=푙 the algebraic analogue of the field of germs of analytic The valuation is defined by 푣(푥) = 푙/푁, the smallest functions near the origin on a punctured disc 픻∗.A exponent with nonzero coefficient. variety 푋 over 핂 is analogous to a family of varieties

April 2017 Notices of the AMS 381 THEGRADUATESTUDENTSECTION

퐸 system associated with the divisor. By a semicontinuity ℓ2 argument, this rank provides an upper bound for the dimension of the linear system on 푋 containing 퐷. With ℓ1 this bound, one is able to use combinatorial methods 퐸 퐶 ℓ1 퐶 ℓ2 to prove strong results in the theory of algebraic curves as described in the survey [1]. Recent work of Dustin Figure 2. Associated to a curve (left) is its dual graph Cartwright extends this theory to higher dimensions. (right). Research in Tropical Geometry over the punctured disc. Under suitable conditions, one Research in tropical geometry is heading in several differ- may extend 푋 to a semistable scheme 풳 over 풪, the ent directions. The foundations of tropical geometry are valuation ring of 핂. The scheme 풳 is analogous to an undergoing continual revision and are not yet settled. Ap- extension of the family over the disc. The reduction plications of tropical geometry to enumerative geometry 푋0 of 풳 to the residue field k of 풪 is analogous to are still being uncovered, many of them in the direction the central fiber of the family. Under conditions on of mirror symmetry. Tropical geometry also provides a 푋 introduced by Jenia Tevelev, Trop(푋) is very closely hands-on way of studying Berkovich spaces, a theory of related to the dual complex. The complex Trop(푋) encodes analytic geometry over complete non-Archimedean fields. some of the combinatorics of the components of 푋0. In Tropical techniques are now being employed in compu- that sense, it is not surprising that Trop(푋) would tational algebraic geometry. There have been many new reflect geometric properties of 푋. There is a certain applications of the theory of linear systems on graphs tension between algebraic geometry and combinatorics to algebraic curves. Careful use of degeneration methods in tropical geometry: if the combinatorics of Trop(푋) has led to results in Diophantine geometry, a branch are simple, the algebraic geometry of the components is of number theory. Tropical geometry also allows one to likely to be complicated and rich; if the components of the apply geometrically motivated techniques to purely com- degeneration are simple, the combinatorics of Trop(푋) binatorial objects through associated tropical varieties, are rich and capture the geometry of 푋. bringing powerful new techniques to combinatorics and One can try to make tropical geometry more intrinsic by resolving old problems. Tropical varieties have even been studying the combinatorics of degenerations of abstract studied in their own right, as they are combinatorially varieties 푋 over 핂. This approach has been developed interesting objects. furthest in the case of curves and has led to the theory of linear systems on graphs as pioneered by Matthew Further Reading Baker and Sergei Norine. We will work with an example [1] M. Baker and D. Jensen, Degeneration of linear series from borrowed from the work of Baker to illustrate this theory. the tropical point of view and applications, Proceedings 2 We examine a family of curves in ℙ parameterized by 푡: of the Simons Symposium on Nonarchimedean Geometry, consider the family 풳 of quartic curves in ℙ2 × 픻 defined Simons Symposia, Springer, 2016, 365–433. by 퐹((푋, 푌, 푍), 푡) = 0 with [2] D. Maclagan and B. Sturmfels, Introduction to Trop- 2 2 2 2 2 3 ical Geometry, Amer. Math. Soc., Providence, RI, 2015. 퐹((푋, 푌, 푍), 푡) = (푋 − 2푌 + 푍 )(푋 − 푍 ) + 푡푌 푍. MR3287221 When 푡 ≠ 0 is small, this defines a smooth plane quartic. When 푡 = 0, the curve is the union of a conic 퐶 and Editor’s Note two lines ℓ1 and ℓ2. The total space of the family is See also “What is a tropical curve?” by Grigory Mikhalkin singular but can be made nonsingular by blowing up the in the April 2007 Notices. Other related columns are “What intersection point of the two lines. This introduces a new is an amoeba?” by Oleg Viro (September 2002) and “What component of 푋0, which is a rational curve 퐸. The curve is a Gröbner basis?” by Bernd Sturmfels (November 2005). 푋0 has at worst nodal singularities, meaning that near the intersection of components, the curve locally looks like Photo Credit 푥푦 = 0. Photo of Eric Katz is courtesy of Joseph Rabinoff. In Figure 2, the central fiber of the resulting family is pictured on the left. One may form its dual graph Γ by associating a vertex to each irreducible component of 푋0 and associating an edge to each nodal singularity of 푋 . ABOUT THE AUTHOR 0 This gives the dual graph pictured on the right. ABOUT THE AUTHOR One can define divisorsText here... on the dual graph as formal Eric Katz’s mathematical interests integer combinations of vertices of Γ. There is a notion include combinatorial algebraic of specializing a divisor 풟 from the generic fiber 푋 of 풳 geometry and number theory. He to a divisor 퐷 on the dual graph Γ. In fact, one can work likes to ride bikes, run, and make out a rich combinatorial theory of divisors on graphs. outlandish statements. Here, linear equivalence of divisors is generated by chip- firing moves on graphs, which have been studied in other Name... Eric Katz contexts. One can define the rank 푟Γ(퐷) of the linear

382 Notices of the AMS Volume 64, Number 4