Tropical Geomety

Nicolas Folinsbee

October 3, 2016

Introduction

Tropical geometry can be described as a sort of “algebraic geometry over the tropical semiring”. As polynomials over the tropical semiring are piecewise-linear functions, tropical geometry translates problems from algebraic geometry into combinatorial problems. This reduction to combinatorics has already been shown to have use in other contexts. For example, the geometry of a toric variety is entirely determined by the combinatorics of the so-called fan associated to the variety. Since the successful application of this “tropical approach” to questions of enumerative geometry by Grigory Mikhalkin ten years ago [5], tropical geometry has seen tremendous growth. In algebraic geometry, it has been used to give a seemingly much simpler proof of the Brill-Noether conjecture, originally proven by Griffiths and Harris in 1980. Due to a remarkable result from 2008, a Riemann-Roch type theorem has been shown to hold for tropical curves. This tropical Riemann-Roch, distinct from the classical version, has recently seen some use in number theory. Our attention will mostly be restricted to the study of tropical curves, which look like graphs, rather than the case of general tropical varieties which appear as polyhedral complexes. The approach taken to the study of the subject closely follows the exposition by Nathan Pflueger [1].

The Tropical Semiring

The tropical semiring is the set R ∪ {+∞} together with operations ⊕ and ⊗ where

a ⊕ b =mina, b

a ⊗ b =a + b.

This fails to be a ring as no additive inverse exists. A key feature of the tropical semiring is that for any x, we have x ⊕ x = x. In other words, all elements of the semiring are idempotent under tropical addition. Also note that +∞ ⊕ x = x for all x so that the additive inverse and multiplicative inverses are given by +∞ and 0 respectively. Consider now a polynomial

M ⊗i1 ⊗in f(x1, ..., xn) = ci1,...,in ⊗ x1 ⊗ · · · ⊗ xn i∈I

1 with coefficients in the tropical semiring and where all but finitely many of the ci1,...,in are +∞. In

⊗i1 ⊗in terms of the usual operations on R, an arbitrary monomial ci1,...,in ⊗ x1 ⊗ · · · ⊗ xn is the linear function ci1,...,in + x1i1 + ··· + xnin. From the definition of ⊕, this tells us that a tropical polynomial  will be a piecewise-linear function whose value at any point is given by min ci1,...,in +x1i1 +···+xnin}, where the minimum is taken over the finitely many monomials with ci1,...,in 6= +∞ . The points where the minimum is reached by two or more monomials are points of non-differentiability and they be of particular interest.

From an algebraic curve to a graph

Tropical curves were originally designed to model limits of curves under the log map

(C∗)2 →R2

(z1, z2) 7→(log(|z1|), log(|z2|)).

Indeed, since we have log(x1x2) = log(x1) ⊗ log(x2) and log(x1 + x2) ≈ log(x1) + log(x2) for x2 and x1 far from each other, the tropical semiring can be viewed as an approximation of logarithm arithmetic. This map is known to have nice properties, such as preserving the genus of the curve. However the so-called amoebas that the log map produces are not practical to study the behavior of a family of curves. Suppose we wish to study the image under this map of a whole smoothly varying family of polynomials p(z1, z2)t, indexed by t. Then the coefficients of this family are analytic functions of t, or in other words they form convergent power series. Ignoring issues of convergence, this family may thus be represented as a curve over the ring C[[t]].

Definition. We call n ∞ o X αi K := cit | α1 < α2 < ...with the αi rationals under a common denominator and the ci ∈ C i=1 the Puiseux series. The Puiseux series carry a natural valuation, val, where val(p) = n/m and n/m is the lowest exponent appearing in p which has a non-zero coefficient. For example, val(4t−2 −2t+t3) = −2.

It turns out that K is algebraically closed and is in fact the algebraic closure of C[[t]]. Consider a

2 curve defined over C[[t]]. Suppose it lies in AK and that its coordinates (x(t), y(t)) are convergent power series. As t goes to zero, the value of the coordinates (log(x(t)), log(y(t))) will be determined by the lowest exponent in their power series. That is, we get approximately (val(x) log |t|, val(y) log |t|). Normalizing by log |t| motivates the following definition.

Definition. For a curve C in (K∗)n, the non-archimedean ameoba of C, A(f) is defined as the closure in Rn of the map

(x1, ..., xn) → (val(x1), ..., val(xn)).

2 Due to a theorem by Kapranov, in the case where the curve is determined by a polynomial

M ⊗i1 ⊗in f(x1, ..., xn) = ci1,...,in ⊗ x1 ⊗ · · · ⊗ xn , i∈I we have

 A(V (f)) = the set where min val(ci1,...,in )+x1i1 +···+xnin} is reached by two or more monomials, where the minimum is taken over the monomials of f.

Example. Consider the polynomial f(x, y) = x + y + 1 ∈ K[x, y]. Then A(V (f)) is the set where    min x, y, 1 is achieved by two or more of the monomials. This is A(V (f)) = x = y ≤ 1 ∪ x1 =  1 ∪ x2 = 1 .

Tropical Curves

Definition. A tropical curve is a finite connected multigraph in which the edges leading to leaves are identified with the interval [0, ∞] and where the other edges are each identified with an interval [0, l],

1 where l ∈ R+ . We view leaves as the one-point compactifications of the interval [0, ∞) and call them infinite points. Other points are called finite points. Two tropical curves are said to be the same if they have the same set of infinite points and they induce the same metric space.

Definition. The genus of a tropical curve G is its genus when viewed as a topological space. Equiva- lently, the genus g of G is |E(G)| − |V (G)| + 1, where |E(G)| and |V (G)| are the size of the edge set and of the vertex set of G respectively.

We now introduce the type of function on tropical curves which we will study. They will assume a role somewhat similar to meromorphic sections of line bundles.

Definition. A rational function on the edge of a tropical curve G is a piecewise-linear function from G to R ∪  ± ∞ such that the slopes are integer valued and such that finite points are mapped to finite values. A rational function on G is a continuous function from G to R ∪  ± ∞ such that the restriction to each of its edges is a rational function.

Tropical Riemann-Roch

The tropical Riemann-Roch theorem was proven independently in 2008 by Gathmann and Kerber [6] and by Mikhalkin and Zharkov [7]. It is not known to have any analogue in the world of classical algebraic geometry which makes it one of the most powerful tools at the disposal of tropical geometers. As a testament to its power, when combined with a second method for converting algebraic curves

1Such a graph with a leaf is strictly speaking not a metric graph as infinite distances are not allowed in metric spaces.

3 to graphs, essentially by specialization on an arithmetic surface, which we will not discuss here2, it has produced a tropical proof of the Brill-Noether conjecture [3]. People familiar with the classical Riemann-Roch will be able to make clear parallels between the definitions needed to introduce the tropical theorem and the classical one. Not all these similarities should be taken at face value however as they may lead to confusion.

Definition. Let G be a tropical curve. The order of a rational function f on G at a point p, denoted ordp(f), is the sum of the slopes of the outgoing slopes of f at p.

Note that if the slope of f does not change in a neighbourhood of p, then ordp(f) = 0.

Definition. For a tropical curve G, we call the free abelian group generated by points of G the group of divisors div(G). The degree of a divisor D is defined as the sum of its coefficients. To each P rational function f on G, we may associate a divisor (f) := p∈G ordp(f). This forms a subgroup of div(G) which we will call the principal divisors and denote prin(G). We declare two divisors D and D0 linearly equivalent if D = D0 + (f) for some rational function f. We define the group P ic(G) as Div(G)/P rin(G).

Note that the degree of any principal divisor is 0. To see this, for any f on a tropical curve G, the underlying graph of G may be partitioned so that f is linear and not just piecewise-linear on each edge. Then the divisor is supported entirely on the vectices of this finite graph. Now, when taking the sum over the vertices to compute the degree, the slope of f on opposite sides of each edge will cancel themselves. This tells us for example that the linear equivalence classes of any divisor is entirely contained in a single degree.

Definition. We define the canonical divisor K of a tropical curve G as

X K := (deg(p) − 2)p. p∈G

This will be non-zero only at a finite number of points, namely at leaves and at intersections of 3 or more edges.

Definition. We call a divisor that has all non-negative coefficients effective and denote by |D| the set of effective divisors linearly equivalent to D. We have a partial order on div(G), where D1 ≥ D2 if

D1 − D1 ≥ 0. Here D1 − D1 represents the divisor obtained by taking the point wise difference of D1 and D2 and by 0 we mean the divisor that is identically zero. We definie the rank of a divisor D , denoted r as r(D) = mindeg(E): E ≥ 0, |A − E| = ∅ − 1 2See [3] for more details

4 In other words, the rank of a divisor D corresponds to the smallest number of total “elements” needing to be removed from D so that it is not linearly equivalent to any positive divisor. For example, r(0) = 0 and r((f)) = 0. However, in general, the rank of a divisor is a non-trivial thing to compute. The rank is always at most the degree of a divisor, but as the following example shows equality will in general not hold because of topological obstructions caused by non-zero genus. This brings us to the main theorem.

Theorem. For any divisor D on a tropical curve G, we have

r(D) − r(K − D) = deg(D) + 1 − g.

The tropical Riemann-Roch was used last year by Eric Katz, Josef Rabinoff, and David Zureick-Brown to give a uniform bound on the number of rational points for certain algebraic curves. [4]

Theorem. Suppose X is an algebraic curve over Q of genus at least 2. If p is a prime bigger than 2g and the Mordell − W eil rank r of X is smaller than g than the number of rational points on X is at most X(Fp) + 2r.

Example. The previous theorem tells us that the only non-trivial x ∈ Z for which x(x-1)(x-2)(x-5)(x-6) is a perfect square are 3 and 10.

Apendix

Definition. We denote the edge set and vertex set of a graph G by E(G) and V (G) respectively. By multigraph, we mean a graph that allows multiple deges between two vertices. By self-loop we mean an edge having the same vertex as both endpoints.

Definition. For a vertex v in a graph G, we denote by deg(v) the number of edges incident to v.

Definition. A valuation on a field K is a function val : K → R such that val(x+y) ≥ min{val(x), val(y)} and val(xy) = val(x) + val(y).

Definition. By metric graph, we mean a finite multigraph such that each edge has an associated postivie real number called the length of the edge. Furthermore, we regard two metric graphs are the same if they admit a common refinement (in the sense of refining a one dimensional CW-complex) with identical edge lengths.

References

[1] Nathan Pflueger, Tropical Curves, 2011. https://www.math.brown.edu/~pflueger/exposition/ TropicalCurves.pdf

5 [2] Melody Chan, Tropical Geometry, MATH 285, Harvard, spring 2013. http://www.math.harvard.

edu/~jbland/ma285y_notes.pdf

[3] Matthew Baker and David Jensen Degeneration of linear series from the tropical point of view and applications , 2015. https://arxiv.org/pdf/1504.05544v2.pdf

[4] Eric Katz, Josef Rabinoff, and David Zureick-Brown Uniform bounds for the number of rational points on curves of small Mordell-Weil rank , 2015. https://arxiv.org/pdf/1504.00694v2.pdf

[5] Grigory Mikhalkin Tropical Geometry and its Applications , 2006. https://arxiv.org/pdf/math/ 0601041v2.pdf

[6] A. Gathmann, M. Kerber, A Riemann-Roch theorem in tropical geometry, Mathematische Zeitschrift 259 (2008), 217-230.

[7] G. Mikhalkin, I. Zharkov, Tropical curves, their jacobians and theta functions, Curves and abelian varieties, 203-230, Contemp. Math. 465 (2008).

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