科技部補助專題研究計畫成果報告
期末報告
熱帶曲線上因子之秩的計算
計 畫 類 別 ： 個別型計畫
計 畫 編 號 ： MOST 102-2115-M-004-001-
執 行 期 間 ： 102 年 08 月 01 日至 103 年 09 月 30 日
執 行 單 位 ： 國立政治大學應用數學學系
計 畫 主 持 人 ： 蔡炎龍
計畫參與人員： 碩士班研究生-兼任助理人員：黃明怡
報 告 附 件 ： 移地研究心得報告
處 理 方 式 ：
1.公開資訊：本計畫可公開查詢
2.「本研究」是否已有嚴重損及公共利益之發現：否
3.「本報告」是否建議提供政府單位施政參考：否
中 華 民 國 103 年 12 月 29 日
中 文 摘 要 ： 本報告主要是研究離散型熱帶因子, 特別是熱帶曲線上的熱
帶因子之秩的計算。我們主要結果可以將一個熱帶因子的秩
的所有可能解果降至幾個數字。在許多情況中, 我們甚至可
以直接得到答案。另外, 我們猜測連續型的熱帶因子也會有
相同的結果。
中文關鍵詞： 熱帶因子, 熱帶幾何, 秩
英 文 摘 要 ： In this report, we survey the discrete tropical
divisor theory, and focus on the computation of the
rank of a tropical divisor on a tropical curve. Our
main result can reduced the possible numbers for rank
of a divisor into just a few numbers. In many cases,
we can even get exact number for the rank of a
tropical divisor. We conjecture that
英文關鍵詞： Tropical Divisor, Tropical Geometry, Rank
國科會專題研究成果報告:
熱帶曲線上因子之秩的計算
計畫編號: NSC 1022115M004 001
-計畫主持人: 蔡炎龍 (政治大學應用數學系)
Chapter 1
報告內容
1.1 Introduction
Tropical geometry rise much attention these years. One major reason is that Mikhalkin [?] successfully developed and applied techniques of tropical geometry to calculate the Gromov-Witten invariants in CP2, which had been
proved previously by Kontsevich [?], and Caporaso and Harris [?] with much deep methods of algebraic geometry. We refer to Gathmann’s article [?] for basic concepts and applications of tropical geometry.
As in the classical algebraic geometry, we can define divisors for tropical curves. There are at least two types of definitions, one is called the “discrete” version of divisors, and the other is “continues” version of divisors. For sim-plicity, we only discuss discrete version of divisors of a tropical curve, yet many results can be easily applied to the continuous situation.
The rank of a divisor D for a tropical curve is the tropical counterpart of the dimension of the vector space of meromorphic functions satisfying Div(f ) + D is effective. We have tropical analogous of the Riemann-Roch theorem. Baker and Norine [?] introduced a version of the Riemann-Roch theorem for graphs. Gathmann and Kerber [?], and Mikhalkin and Zharkov [?] used the result to prove the Riemann-Roch theorem for tropical curves. Roughly speaking, they extended Baker and Norine’s result to metric graphs. After that, Amini and Caporaso [?] extended the Riemann-Roch theorem to weighted tropical curves. All versions of the Riemann-Roch theorem for tropical curves of course give us an important tool to calculate the ranks of divisors of tropical curves. Besides that, Hladký, Král, and Norine [?] give an algorithm to calculate the ranks. We will take different approach from their method, which we shall explain in the following sections.
1.2 Basic Definitions and Results
Let Γ be a tropical curve. Γ is naturally corresponding to a finite graph
G = (E, V ) which we will explain more in details latter. For any graph G, we
define (discrete) divisors are formal sum ofZ-linear combination of the vertices. That is, a divisor D on G (Γ) is of the form:
D =∑
v∈V
av· v,
where av ∈ Z. We say a divisor D on the curve Γ is exactly a divisor on the
corresponding graph G. The set of all divisors on G (Γ) is denoted by Div(G) or Div(Γ). The degree of a divisor is the sum of all coefficients.
A meromorphic function on G is simply a function
f : V → Z.
That is, f∈ Hom(V, Z) and we denote the set Hom(V, Z) by M(G).
Each f ∈ M(G) is corresponding to a divisor D(f ) = ∑ v∈V δv(f )· v, where δv(f ) = ∑ e=wv∈Ev (f (v)− f(w)).
A divisor of this form is called a principal divisor. Two divisors D1, D2 are
equivalent (D1∼ D2) if they are differed by a principal divisor. That is, there
is f ∈ M(G) such that
D1− D2= D(f ).
An effective divisor E is a divisor that coefficients are all nonnegative, and we use E≥ 0 to indicate it is an effective divisor. For a divisor D ∈ Div(G), we
define the linear system associated to D to be the set
|D| = {E ∈ Div(G) | E ≥ 0, E ∼ D}.
Finally, we define the rank of a given divisor D. The rank of a divisor
D∈ Div(G) is defined as the following.
rank D = max{s | |D − E| ̸= ∅ for all E ≥ 0 and deg E = s}.
For a graph G, we define the canonical divisor
K = ∑
v∈V (G)
(deg(v)− 2)(v).
Baker and Norine [?] gave a version of tropical Riemann-Roch Theorem. rank(D)− rank(K − D) = deg(D) + g − 1. (1.1) The rank(D) is what we are primely interested. In the following section, we shall use examples to show the ideas of our project.
1.3 Illustrated Example
In this section, we give examples that basically explain what we did. Let G = (E, V ) be a graph where E is the collection of edges and V is the collection of vertices.
Let Γ be a tropical line, we emphasize the vertex at center by making it a large point, and denoted the vertex by v0.
..
v0
Removing the rays pointing to the infinity, we get exactly one point (the vertex v0.
..
v0
We can “verify” the tropical Riemann-Roch theorem. Let D be the divisor
D = 3· (v0).
Since a meromorphic function f on G is simply a function from V (G) ={v0}
to Z, there is a c ∈ Z such that f(v0) = c. Thus we can find the divisor
corresponding to f :
(f ) = 0· (v0).
The rank of D is either
max{n | for all E, deg(E) = n, E ≥ 0, we have |D − E| ̸= ∅},
or
min{m|there is E ≥ 0, such that deg(E) = m, |D − E| = ∅ − 1}。
The only divisor E such that deg(E) = 3 is E = 3· v0. Therefore, D− E =
0· v0= (f ). We conclude that rank(D)≥ 3。
The only divisor E such that deg(E) = 4 is E = 4· v0. Clearly,|D − E| = ∅,
thus we have
rank(D) = 3. It is easy to check that the canonical divisor is
K =−2 · v0.
Then K− D = −5 · (v0), so|K − D| = ∅. That is,
The left hand side of the tropical Riemann-Roch ?? is
r(D)− r(K − D) = 3 − (−1) = 4.
Since deg(D) = 3, and the genus g =|E(G)| − |V (G)| + 1 = 0, so the right hand side of the tropical Riemann-Roch theorem is
deg(D)− g + 1 = 4.
Thus, we verify the tropical Riemann-Roch theorem for a tropical line. In general, calculating the rank would be not so easy. We have to use the tropical Riemann-Roch to find a good estimation.
1.4 Basic Settings
Let Γ be a tropical curve. We remove the rays of the tropical curve. Then we have the corresponding finite graph G. Define
Div(Γ) := Div(G).
The graph G is called the graph corresponding to the tropical curve Γ. What we mean by a tropical divisor D is actually a divisor on the graph G.
1.5 Rank Theorem
We state our main theorem here.Main Theorem. Let Γ be a tropical curve and let D be a divisor on Γ. (a) If deg D < 0 then rank(D) =−1.
(b) If deg D≥ 0 then deg D−g ≤ rank(D) ≤ deg D.
Proof. Part (a) is easy. Since deg D < 0, by definition|D − E| is empty for all
effective divisor E on Γ. Thus, rank D =−1.
For part (b), we consider two cases: |K − D| = ∅ and |K − D| ̸= ∅.
If|K − D| = ∅, we have rank(K − D) = −1 by definition. By the tropical
Riemann-Roch theorem, we obtain
rank(D)− (−1) = deg D − g + 1, thus rank(D) = deg D− g. In particular,
deg D−g≤ rank(D) ≤ deg D
holds.
Now, if|K − D| ̸= ∅. Let E be an arbitrary effective divisor on Γ of degree
deg(D− E) = deg D − deg E,
=−1
Hence|D − E| = ∅. Therefore, rank D is at most deg D.
Note that|K − D| ̸= ∅, so rank(K − D) ≥ 0. By the tropical Riemann-Roch
theorem, we have
rank(D)≥ deg D − g + 1.
Hence,
deg D−g≤ rank(D) ≤ deg D.
Remark 1. Let D ∈ Div(Γ) such that deg D ≥ 0. In the proof of our Main
Theorem, we can get an even better inequality for the cases|K −D| ̸= 0, namely deg D− g + 1 ≤ rank(D) ≤ deg D.
1.6 Applications to Our Theorem
Our main theorem from previous section gives us a range for the rank of a divisor D. Therefore, we only need to check a few possible numbers to see which one is the correct number for rank(D). Sometimes, we even get the rank immediately such as the following example.
Example 1. Let Γ be a tropical curve of genus 1. Let D = 3·v1−2·v2+ 5·v3∈
Div(Γ). We have deg D = 3− 2 + 5 = 6 ≥ 0. By the Remark ??, we have
deg D− g + 1 ≤ rank(D) ≤ deg D,
but
deg D− g + 1 = deg D − 1 + 1 = deg D = 6.
Chapter 2
成果自評
We survey the theory of tropical discrete divisors. The goal is to quickly find the rank of given divisors, and we did find a very good bound for the rank of a divisor. Our main theorem provide a quick way to get the rank of a divisor. Sometime it require to check a few possible numbers, sometimes it reduced to only one possible. We plan to publish these results in the near future.
Finally, we discuss some plans for future research projects. First, what we did here is for discrete tropical divisors, and naturally, we would like to see if our theories can be applied to continuous tropical divisors. We are very confident that is the case.
Moreover, for some interesting geometry objects, like elliptic curves, it is very interesting to see what kind of information the tropical divisor theory can provide. Of course, the first step is to give a “good” definition of tropical version of the objects we are interested in, and then check if they have the same properties as the classical objects.
Bibliography
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[2] Matthew Baker and Serguei Norine. Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math., 215(2):766–788, 2007.
[3] L. Caporaso and J. Harris. Counting plane curves of any genus. Invent.
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[6] Jan Hladký, Daniel Král’, and Serguei Norine. Rank of divisors on tropical curves, 2010.
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