CHIP-FIRING GAMES JACOBIANS AND PRYM V ARIETIES YOAV LEN Abstract . We present a self-contained introduction to the theory of chip-firing games on metric

2025-05-01 0 0 554.52KB 32 页 10玖币

侵权投诉

CHIP-FIRING GAMES, JACOBIANS, AND PRYM VARIETIES

YOAV LEN

Abstract. We present a self-contained introduction to the theory of chip-ﬁring games on metric

graphs, as well as the more recent theory of tropical Prym varieties. We brieﬂy discuss the connec-

tion between these notions and their algebraic counterparts and suggest various avenues for future

research.

Contents

1. Introduction 2

Acknowledgements 3

2. Chip-ﬁring on metric graphs 3

2.1. Metric graphs 3

2.2. Divisors and chip-ﬁring 4

2.3. How to actually compute linear equivalence 5

2.4. Reduced divisors and Dhar’s burning algorithm 6

3. The rank of divisors 8

3.1. A bit of Brill–Noether theory 10

4. The structure of the tropical Jacobian 12

4.1. The Abel–Jacobi map 12

4.2. Jacobians and real tori 14

5. Double covers and Prym varieties 16

5.1. Harmonic covers 16

5.2. Prym varieties 18

5.3. Prym–Brill–Noether theory 21

6. The structure of Prym varieties 21

Appendix A. Chip-Firing Exercises 25

Appendix B. Rank exercises 27

Appendix C. Double covers and Pryms exercises 28

Appendix D. Open problems 30

References 31

arXiv:2210.14060v2 [math.AG] 31 Oct 2024

1. Introduction

These notes originate from the Cambridge summer school on combinatorial algebraic geom-

etry in September 2022. They cover chip-ﬁring games, tropical Jacobians, and tropical Prym

varieties, with a light introduction to Brill–Noether theory. The prerequisites are basic knowl-

edge in graph theory and familiarity with standard concepts such as groups, quotient spaces,

topological spaces, and metric spaces. Background in algebraic geometry is not necessary, how-

ever, familiarity with the topic would help motivate some of the notions encountered in the

course, as many of whom originate from algebraic geometry.

The theory of tropical chip-ﬁring games is the combinatorial version of the theory of divisors

on algebraic curves. The role of the curves and their divisors is played by graphs and conﬁgura-

tions of playing-chips. In its original form, the game was played on discrete graphs. However, it

is advantageous to pass to a continuous version of the game, played on metric graphs, as it better

approximates the algebraic theory and is more suitable for problems involving moduli spaces.

Furthermore, while some deﬁnitions may seem more complicated, the continuous version tends

to produce nicer and more elegant results. That should not be too surprising when comparing

with other areas of maths.

The tropical theory is not merely an analogue of the algebraic theory. A process known as

tropicalization turns an algebraic curve into a graph whose combinatorial invariants encode var-

ious geometric properties of the curve. Most notably, Baker’s specialization lemma states that

the rank of divisors may only increase under this process [Bak08], an observation at the heart of

various recent developments in the geometry of curves [JR21,FJP20,CLRW20,AP21].

The latter half of these notes is dedicated to Jacobians and Prym varieties, both of which are

abelian groups that classify divisors on curves or graphs. While the Jacobian classiﬁes divisors

on a single object, the Prym variety is associated with a double cover and classify divisors that

behave nicely with respect to the double cover. The precise deﬁnition for graphs is given in

Section 5.2 and the deﬁnition for algebraic curves is analogous.

From the perspective of the curves themselves, the Prym variety is an invariant that provides

additional data and another method for probing them. From a broader perspective, Prym vari-

eties provide a fruitful source of abelian varieties that can be parameterized and examined by

looking at curves. While Jacobians are very well understood, they only account for a 3g −3

dimensional locus in the moduli space of abelian varieties of dimension g, which is of dimension

g+1

2. Prym varieties account for a 3g-dimensional locus, which is a vast improvement. Further-

more, the Jacobian locus is contained in the closure of the Prym locus so, in a sense, we can view

Prym varieties as a generalization of Jacobians.

The fact that "there are more Pryms than Jacobians" can be used to establish structural results

on the moduli space of abelian varieties. For instance, in dimension 5, the numbers g+1

2and

3g coincide, so Prym varieties are full-dimensional in A5. A close examination of the geometry

of Prym varieties then leads to the conclusion that the space A5is uniruled [Don84, Theorem

3.3]. On a different note, the intermediate Jacobian of a smooth Fano 3-fold Xis a Prym variety

[Far11, Section 3.2], and Xis rational iff it is an actual Jacobian. Clemens and Grifﬁths showed

that this Prym variety is never a Jacobian when Xis a smooth cubic threefold, thus proving

that smooth cubic threefolds are non-rational [CG72]. The fact that intermediate Jacobians are

Prym varieties is also used by Sacca, Laza, and Voisin in [LSV17] to construct and compactify

hyperkähler manifolds.

Figure 1. Two metric graphs having the same model

These notes are organized as follows. Section 2introduces the theory of chip-ﬁring games on

metric graphs, as well as basic notions such as the Jacobian and reduced divisors. In Section 4

we delve deeper into the Jacobian and explore its structure. The section is recommended even

for readers familiar with metric chip-ﬁring, as the approach taken later when discussing Prym

varieties will mirror some of the techniques used in Section 4. Section 5.2 begins the study of

Prym varieties, followed by a more meticulous study of their structure in Section 6. Some of the

proofs throughout are left as guided exercises.

Appendices A,B, and Care devoted to exercises. All but Exercise A.8d should be solvable

using only the material encountered in these notes. Exercise A.8d requires basic familiarity with

divisors on algebraic curves. Finally, Appendix Dincludes a variety of open problems whose

solution, I believe, could be publishable. Most of them should be solvable using only the material

of this course, although one cannot know for certain before actually trying to solve them.

Acknowledgements. I warmly thank Navid Nabijou and Luca Battistella for organizing the sum-

mer school and putting together a week full of wonderful mathematical interactions. I thank Vio-

leta Lopez, Margarida Melo, Sam Payne, Thomas Saillez, Remy Smith, and Dmitry Zakharov for

helpful comments on an older draft of these notes. I also thank the participants of the summer

school for many engaging questions and discussions.

2. Chip-firing on metric graphs

2.1. Metric graphs. The fundamental objects studied throughout these notes are metric graphs:

metric spaces obtained from discrete graphs by assigning a real positive length to each edge.

When an edge has length ℓ, we can identify it with the interval of length ℓ. If a metric graph Γwas

obtained from a discrete graph G, we say that Gis a model for Γ. Note that our discrete graphs are

allowed to have loops and multiple edges. Unless stated otherwise, we assume that our graphs

are connected. The genus of a graph is the number of independent cycles, or equivalently

g(Γ) = g(G) = e−v+1, (1)

where eand vare the number of edges and vertices respectively, and fis the number of connected

components (note that the genus formula does not require the graph to be connected). The genus

does not depend on the choice of model or metric.

Example 2.1. The two metric graphs of genus 3shown in Figure 1both have the same minimal

model but different edge lengths. We can obtain additional models for them by considering

arbitrary points in the interior of edges as vertices.

If g(Γ)‰1, then Γhas a unique minimal model G0which does not include any vertices of

valency 2. In the special case where Γis just a cycle, there are inﬁnitely many minimal models

consisting of a single vertex and a loop. We will often omit the model when clear from context.

Figure 2. A divisor of degree −1on a metric graph

By abuse of notation, segments of a metric graph Γmay be referred to as edges and points as

vertices.

2.2. Divisors and chip-ﬁring. Let Γbe a metric graph. A divisor on Γis a function D:Γ→Z

with ﬁnite support. We intuitively think of a divisor as assigning a ﬁnite number of playing chips

at points of the graph, where a negative number of chips is allowed. Negative chips are referred

to as anti-chips. The degree of a divisor is the total number of chips, namely řpPΓD(p). A divisor

is called effective if it is non-negative at all points of the graph.

We stress that chips may be placed anywhere on the graph, not just the vertices.

Example 2.2. Figure 2shows a metric graph with a divisor of degree −1. We will often use

full disks to represent a positive number of chips, whereas empty circles represent anti-chips.

When the number of chips is not mentioned, a disk represents a single chip and an empty circle

represents a single anti-chip.

The set of divisors forms an abelian group via addition, denoted Div(Γ). There is a partial

ordering on this set given by D1ěDwhenever D1(p)ěD(p)at every point p. In particular,

a divisor Dis effective if and only if Dě0. The subset of Div(Γ)consisting of divisors of

degree dis denoted Divd(Γ). Note that Divd(Γ)is a group when d=0and only a torsor of

Div0(Γ)otherwise. There is a non-canonical bijection between Divd(Γ)and Div0(Γ)given by

DÞ→D−d¨p, where pis any point of Γ.

We will now deﬁne a certain equivalence relation on divisors that reﬂects the geometry of the

graph (secretly, we also want it to mimic linear equivalence from algebraic geometry). Let

φ:Γ→R

be a continuous piecewise linear function with integer slopes. Then φinduces a divisor div(φ)

where the value of div(φ)(p)at a point pis the sum of incoming slopes of φat p. In other words,

given a point pconsider all the intervals emanating from p. Since φis piecewise linear, those

intervals can be chosen small enough so that φhas a constant slope on each of them. We now

take the sum of those slopes oriented towards p.

Example 2.3. For the leftmost graph of Figure 3, let φbe the function that is constantly 0to the

left of u, has slope 1on the segment between uand v, and constant value φ(v)for any xto the

right of v. Then div(φ) = v−u.

For the graph in the middle of the ﬁgure, let ψbe the function whose value is 0on the

segment between aand b, slope 1on the segment from ato dand on the segment from bto c,

and is constant on the segment between cand d(note that the function is continuous and well

Figure 3

deﬁned because the segments between aand dand between band chave the same length). Then

div(ψ) = d+c−b−a.

Finally, for the rightmost graph, let ηbe the function whose value is 0at δ, has slope 1on

the segments leading to the outer circle, and is constant on the outer circle. Then div(η) =

α+β+γ−3δ.

A divisor of the form div φis called principal. Divisors Dand D1are said to be linearly

equivalent, denoted D»D1, if D−D1=div φfor some φ. The set of principal divisors is denoted

Prin(Γ). Note that every principal divisor has degree 0. The linear system of D, denoted |D|, is the

set of effective divisor equivalent to it, namely |D|={EPDiv(Γ)|Eě0, E »D}.

Modding out the group of divisors by linear equivalence gives rise to the Picard group of the

graph, namely,

Pic(Γ) = Div(Γ){Prin(Γ).

For any integer d, we have a group Picd(Γ) = Divd(Γ){Prin(Γ)which classiﬁes divisor classes of

degree d. In the special case where d=0, the Picard group is known as the Jacobian Jac(Γ). Note

that for a ﬁxed point pand an integer d, we have D−d¨p»D1−d¨pwhenever D»D1. As a

result, there is a bijection between the Jacobian and each Picard group Picd(Γ).

Example 2.4. Let Λbe a line segment and let D=a1v1+¨ ¨ ¨ akvkbe a divisor of degree 0, where

the viare distinct points arranged from left to right. Let φbe the piecewise linear function whose

slope is 0 to the left of v1and increases by aiat every point vi. Then div(φ) = D(the fact that

deg(D) = 0was used so the the slope of φis 0 to the right of vk). In particular, each divisor

of degree 0on this graph is principal and Jac(Λ) = {0}. Note that the only effective divisor

equivalent to Dis the 0divisor, so the linear system of Dis |D|={0}.

2.3. How to actually compute linear equivalence. As we saw in Example 2.3, on a cycle graph,

moving two chips in opposite directions at equal speed results in equivalent divisors. This

phenomena can be generalized. Suppose that a divisor D1is obtained from a divisor Dby

continuously moving chips, as long as the following condition is maintained throughout the

process:

The total momentum of the chips moving along each cycle of the graph is 0. (⋆)

Then Dand D1are linearly equivalent.

Example 2.5. Suppose that Γhas a bridge (for instance, Γcould be the graph in Figure 1) and p

and p1are points in its interior. Then p»p1. Indeed, since there is no cycle containing either p

or p1, condition ⋆is vacuously true. Similarly, if Tis a tree, then any two divisors of the same

degree are linearly equivalent.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

CHIP-FIRINGGAMES,JACOBIANS,ANDPRYMVARIETIESYOAVLENAbstract.Wepresentaself-containedintroductiontothetheoryofchip-firinggamesonmetricgraphs,aswellasthemorerecenttheoryoftropicalPrymvarieties.Webrieflydiscusstheconnec-tionbetweenthesenotionsandtheiralgebraiccounterpartsandsuggestvariousavenuesforfutur...

展开>> 收起<<

CHIP-FIRING GAMES JACOBIANS AND PRYM V ARIETIES YOAV LEN Abstract . We present a self-contained introduction to the theory of chip-firing games on metric.pdf

共32页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

CHIP-FIRING GAMES JACOBIANS AND PRYM V ARIETIES YOAV LEN Abstract . We present a self-contained introduction to the theory of chip-firing games on metric

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: