UNFOCUSED NOTES ON THE MARKOFF EQUATION AND T-SINGULARITIES MARKUS PERLING Abstract. We consider minimal resolutions of the singularities for weighted projective planes of type

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UNFOCUSED NOTES ON THE MARKOFF EQUATION AND T-SINGULARITIES

MARKUS PERLING

Abstract. We consider minimal resolutions of the singularities for weighted projective planes of type

P(e2, f2, g2), where e, f, g satisfy the Markoﬀ equation e2+f2+g2= 3efg. We give a complete

classiﬁcation of such resolutions in terms of continued fractions similar to classical work of Frobenius.

In particular, we investigate the behaviour of resolutions under mutations and describe a Cantor set

emerging as limits of continued fractions.

1. Introduction

The main aim of these notes is to describe the minimal resolutions of singularities for weighted

projective planes P(e2, f2, g2), where e, f, g are integral solutions of the Markoﬀ equation:

e2+f2+g2= 3efg.

It has been observed in [HP10] that these singularities are T-singularities in the sense of Wahl [Wah81].

Such singularities have been connected to exceptional vector bundles on rational surfaces by results of

Hacking-Prokhorov [HP10] and Hacking [Hac13]. This aspect has also been investigated in our earlier

work [HP08], [Per18]. We think that the complete understanding of these resolutions and the fractal

structure emerging via the associated Markoﬀ tree are of independent interest, but our hope is that this

also may lead to further insights into the global structure of exceptional collections.

Consider a T-singularity of type 1

n2(1, nk−1), i.e. a cyclic quotient singularity of order n2with weights

1, nk −1. The minimal resolution of such a singularity can be desribed via Hirzebruch-Jung continued

fractions Ja1, . . . , atK=a1−1/(a2−1/(a3− ···(at−1−1/at)···)). These continued fractions have

been classiﬁed for T-singularities in [KS88] (see also Section 6). Our ﬁrst basic tool will be a simpliﬁed

representation of these Hirzebruch-Jung continued fractions in terms of integer sequences c1, . . . , cs(see

Section 6), which is based on the Koll´ar–Shepherd-Barron classiﬁcation. It turns out that the cican be

recovered from the regular continued fraction expansion n2/(nk −1) = [cs+ 1, cs−1, . . . , c2, c1+x, c1+

2−x, c2, . . . , cs−1, cs+ 1], where x= 2 if sis even and x= 0 if sis odd (Theorem 7.8).

Now, for any Markoﬀ triple (e, f, g) with 1 < e, f < g,P(e2, f2, g2) has the three T-singularities

e2(1, ewe−1), 1

f2(1, fwf−1) and 1

g2(1, gwg−1), respectively, where we call we, wf, wgthe T-weights

of this triple. When passing from one Markoﬀ triple to the next, say (e, g, E) with E= 3f g −e, we will

show in Section 7 how to construct the continued fraction of E2/(EwE−1) from those of 1

e2(1, ewe−1),

f2(1, fwf−1) and 1

g2(1, gwg−1), respectively, and thereby obtain a complete classiﬁcation of the minimal

resolutions of T-singularities associated to Markoﬀ triples in terms of their mutations along the Markoﬀ

tree.

A strongly related question is the distribution of quotients g/wg≈g2/(gwg−1) and whether limits

of such quotients exists when mutating Markoﬀ triples downward paths in the Markoﬀ tree. This is

very closely related to classical work by Frobenius [Fro68]. In loc. cit., Frobenius derives a complete

description of continued fraction expansions of quotients g/rg, where rgis associated to a triple that

contains gsuch that r2

g≡ −1 mod g. Using Frobenius’ description, it is straightforward to see that limits

limi→∞ rgi/giexist, for any sequence githat follows a path downward the Markoﬀ tree (Proposition 4.3).

It turns out that the set of these limit points forms a Cantor set. We will show that this set has measure

and Hausdorﬀ dimension zero (Theorems 4.9 and 4.12). Moreover, we will characterize this Cantor set by

its natural intervals and we will show that the boundary points of these intervals are quadratic numbers,

which are explicitly described in Proposition 4.5.

The quotients wg/g are related to the rg/g simply by the aﬃne transform wg/g = 3rg/g −1 for every

g(Corollary 5.4). Therefore, their limit points via the Markoﬀ tree exist as well and the resulting Cantor

set is an aﬃne transform of the one associated to the rg/g. Their dimension and measure therefore

2020 Mathematics Subject Classiﬁcation. Primary: 14J17, 11J06; Secondary: 28A78, 28A80, 14F08.

arXiv:2210.12982v1 [math.AG] 24 Oct 2022

2 MARKUS PERLING

coincide (Theorem 7.11). The boundary points are quadratic as well and their explicit formulas and

periodic continued fraction expansions are computed in Secton 7.

We point out that the quotients rg/g have shown up in earlier work as the slopes of exceptional vector

bundles on P2, see [Rud89], [DL85], and their fractal nature has already been observed in [DL85].

Overview. In Section 2 we ﬁx some general conventions regarding the Markoﬀ tree that we will follow

for the rest of these notes. In Section 3, we collect all fundamental facts and computations related to

Markoﬀ triples that will be needed in the subsequent sections. This includes some well-known formulas,

but also some new formulas for which we could not ﬁnd any reference in the literature. Also, we review

Frobenius’ results on continued fractions related to Markoﬀ triples. In Section 4 we use the results of

Section 3 to analyze the Cantor set that emerges from the quotients rg/g. In Section 5, we introduce

T-weights and perform some calculations similar as in Section 3. In Section 6, we analyze the continued

fraction expansions of T-singularities. We then apply these results in the ﬁnal Section 7 in order to

classify the minimal resolutions for T-singularities associated to the Markoﬀ equation and to describe

the Cantor set that emerges from the quotients wg/g.

In Appendix A we collect some basic facts about continued fractions and continuants that we will need

throughout the text. In Appendix B we introduce a new class of polynomials related to T-singularities.

These are not needed in the main body of these notes, but they may be of independent interest. In

Appendix C, we collect some material that we need in order to deal with the Hausdorﬀ dimension. In

Appendix D, we display the ﬁrst few levels of the Markoﬀ tree, both only the maximal elements and the

triples. We found it quite useful to be able to occasionally pinpoint a speciﬁc example and its location

in the tree in one peek. Appendix E has a computer generated list of the ﬁrst 300 Markoﬀ numbers

together with their weights and T-weights. If needed, the table can be extracted from this documents

X source. In Appendix F we present some simplistic numerical experiments regarding the growth of

Markoﬀ numbers and the uniqueness conjecture.

2. The Markoff Tree

In this section we will state some general facts and ﬁx our general conventions regarding the Markoﬀ

tree (we refer to [Aig13] for an overview). We are exclusively interested in positive integral solutions

(e, f, g) of the Markoﬀ equation e2+f2+g2= 3efg, which we will call Markoﬀ triples. If for any given

Markoﬀ triple (e, f, g) we denote E= 3fg −e,F= 3eg −f, and G= 3ef −g, then we obtain three

more Markoﬀ triples (E, f, g), (e, F, g), and (e, f, G). We call these substitutions mutations of the triple

(e, f, g). By rearranging the formulas 3eg =f+F, 3fg =e+E, and 3ef =g+G, it becomes obvious that

mutations are reversible. All Markoﬀ triples can be obtained by mutation, starting from the fundamental

solution (1,1,1). So we can consider the set of all Markoﬀ triples as vertices of a trivalent acyclic graph

whose edges are labeled by the mutations, where the fundamental solution is the natural root. By the

symmetry of the Markoﬀ equation, we have a natural action of the symmetric group S3on the set of

Markoﬀ triples. This action has an almost everywhere trivial stabilizer with exceptions (1,1,1) and the

permutations of (1,2,1). These triples, which are the only ones where e, f, g are not pairwise distinct,

are called singular solutions of the Markoﬀ equation. All other Markoﬀ triples are called regular. The

set of regular Markoﬀ triples up to permutation forms a binary tree, whose root is represented by the

triple (1,5,2). Figure 1 shows the ﬁrst few levels of this tree. As in Figure 1, we will use from now on

(1,5,2)

(1,13,5)

(1,34,13)

(1,89,34) (34,1325,13)

(13,194,5)

(13,7561,194) (194,2897,5)

(5,29,2)

(5,433,29)

(5,6466,433) (433,37666,29)

(29,169,2)

(29,14701,169) (169,985,2)

Figure 1. A part of the Markoﬀ tree.

the convention that every regular Markoﬀ triple will be represented by (e, g, f), such that 0 < e, f < g.

Because g < E, F , mutation of eor fdoes mean that we descend downwards the Markoﬀ tree, where we

MARKOFF EQUATION AND T-SINGULARITIES 3

ﬁx the direction as indicated in Figure 2. We will in call the triple (e, F, g) the left mutation and (g, E, f)

(e, g, f)

(e, F, g) (g, E, f )

Figure 2. Left and right mutation.

the right mutation of (e, f, g), respectively. With this convention, if we go upwards by mutating g, then

we will either end up with the triple (e, f, G) (if e<f) or (G, e, f) (if e>f). By requiring that the root

is represented by (1,5,2), the ordering of every representative is thus uniquely ﬁxed. In particular, if we

traverse any level below the root from left to right then we ﬁnd that alternatingly e < f or e > f , with

1 = e<f for the leftmost vertex and e>f= 2 for the rightmost vertex.

It follows that, given a triple (e, g, f), then starting from this triple, there are precisely four branches

where at least one of e, f, g is preserved, as indicated by Figure 3.

(e, g, f)

(e, ·, g)

(e, ·,·)

(·,·, g)

(g, ·, f)

(g, ·,·)

(·,·, f)

Figure 3. How the elements in a Markoﬀ triple propagate downwards.

The two outmost branches of the Markoﬀ tree are of the form

(1, F2n+1, F2n−1) for n≥2 and (P2n−1, P2n+1,2) for n≥1,

where Fiand Pidenote the Fibonacci and Pell numbers. Note that 5 = F5=P3. We call these the

Fibonacci and Pell branches, respectively.

3. The Markoff tree and finite continued fractions `

a la Frobenius

In this section we give an expository overview around some aspects of Frobenius’ classical account

[Fro68] on the Markoﬀ equation which are related to ﬁnite continued fractions. We also refer to [Cas57,

Chapter II], [CF89], [Aig13], [Reu19]. We will paraphrase some of the material to suit our needs and

complement it by a few observations of our own which, though probably well-known to specialists, we

couldn’t locate in the literature.

3.1. Weights and coweights. Denote (e, g, f) a regular Markoﬀ triple. Then we set:

re:= (0 if e= 1,

f−1gmod eelse,

rf:= g−1emod f,

rg:= e−1fmod g.

Note that the choice re= 0 for e= 1 is made for consistency, as we will see below. Then the Markoﬀ

equation implies that

e≡ −1 mod e(for e > 1), r2

f≡ −1 mod f, and r2

g≡ −1 mod g.

4 MARKUS PERLING

We choose to call the triple (re, rg, rf) the weights associated to the triple (e, g, f). Moreover, we deﬁne

the coweights (se, sg, sf) via the following equations:

e=−1 + see, r2

f=−1 + sff, r2

g=−1 + sgg.

Note that in particular se= 1 for e= 1. The weights and coweights satisfy a number of nice equations,

in particular

grf−frg=e,

erg−gre=f,(1)

erf−fre=G

(see [Cas57, §II.3] or [Fro68, p. 602]).

Remark: In [Rud89] the weights have been identiﬁed with the ﬁrst Chern classes of the vector bundles

in an exceptional collection on P2and the ratios re/e, rg/g, rf/f with their slopes.

Lemma 3.1: Let (e, g, f)be a regular Markoﬀ triple.

(i) The weights and coweights transform under mutation as follows:

(e,g,f)

(re,rg,rf)

(se,sg,sf)

(e,F,g)

(re,rF,rg)

(se,sF,sg)

(g,E,f)

(rg,rE,rf)

(sg,sE,sf)

where

rE= 3frg−re, sE= 3f sg−se, rF= 3erg−rf,and sF= 3esg−sf.

(ii) The slopes of a regular Markoﬀ triple are strictly increasing:

e<rg

g<rf

(see also [Rud89, Prop. 3.1]).

(iii) We have re< e −re,rf< f −rf, and rg< g −rgfor all e, g and all f > 2.

Proof. (i) We only consider the left mutation; the right mutation follows analogously. With F= 3eg −f

and using the Markoﬀ equation we get g−1Fmod e=−f−1gmod e=g−1fmod e=re. Similarly, we

have F−1emod g=−f−1emod g=e−1fmod g=rg. Hence, reand rg(and thus seand sg) don’t

change under left mutation. To determine rF, we write ﬁrst

g= 3ef −G= 3ef +fre−erf= 3e(erg−gre) + f re−erf=e(3erg−rf)−F re,

where we use (1) twice. On the other hand, using (1) for the triple (e, F, g), we have

g=erF−F re.

Comparing terms then yields rF= 3erg−rf. Similarly, the equality sF= 3esg−sffollows via evaluation

of r2

F=−1 + F sFby direct computation, using (1) and the Markoﬀ equation.

(ii) The statement is trivially true for the triple (1,5,2). Then we use (i) and induction over the

Markoﬀ tree.

(iii) The statement is obviously true for g= 5. By (ii) we get for left- and right mutation:

e<rF

F<rg

g<1

2and rg

g<rE

E<rf

f<1

and by induction over the Markoﬀ tree, the assertion follows for all gand therefore for all e > 1 and

f > 2. For e= 1 the assertion is trivially true. 

Remark: Lemma 3.1 (i) can also be inferred from Cohn’s matrices [Coh55] (see [Aig13, §4.3]).

MARKOFF EQUATION AND T-SINGULARITIES 5

The slopes indeed contain some partial information about a triple’s position in the Markoﬀ tree.

Corollary 3.2:

(i) Consider the n-th level of the Markoﬀ tree (where the 0-th level given by the triple (1,5,2)) and

denote m1, . . . , m2nthe maximal elements of the Markoﬀ triples on this level, enumerated from left

to right, with weights rm1, . . . , rm2n. Then the slopes rmi/miare strictly increasing from left to

right:

0<rm1

<··· <rm2n

m2n

(ii) Let (m1, m2, m3)be a regular solution of the Markoﬀ equation with 2< m1< m2< m3and set

i=m−1

i+1mi+2 mod mifor i= 1,2,3(where we read the indices modulo 3). Then this solution

shows up as a left mutation in the Markoﬀ tree (i.e. as regular Markoﬀ triple (m1, m3, m2)) iﬀ one

(and therefore all) of the inequalities r0

i< mi−r0

iholds.

Figure 4 shows the Markoﬀ triples, their weights and their coweights of the ﬁrst four levels of the

Markoﬀ tree. In the Fibonacci branch, any triple is of the form (e, g, f) = (1, F2n+1, F2n−1) for n≥2

(1, 5, 2 )

(0, 2, 1 )

(1, 1, 1 )

(1, 13, 5 )

(0, 5, 2 )

(1, 2, 1 )

(1, 34, 13 )

(0, 13, 5 )

(1, 5, 2 )

(1, 89, 34 )

(0, 34, 13 )

(1, 13, 5 )

(34, 1325, 13 )

(13, 507, 5 )

( 5, 194, 2 )

(13, 194, 5 )

( 5, 75, 2 )

( 2, 29, 1 )

(13, 7561, 194 )

( 5, 2923, 75 )

( 2, 1130, 29 )

(194, 2897, 5 )

( 75, 1120, 2 )

( 29, 433, 1 )

(5, 29, 2 )

(2, 12, 1 )

(1, 5, 1 )

(5, 433, 29 )

(2, 179, 12 )

(1, 74, 5 )

(5, 6466, 433 )

(2, 2673, 179 )

(1, 1105, 74 )

(433, 37666, 29 )

(179, 15571, 12 )

( 74, 6437, 5 )

(29, 169, 2 )

(12, 70, 1 )

( 5, 29, 1 )

(29, 14701, 169 )

(12, 6089, 70 )

( 5, 2522, 29 )

(169, 985, 2 )

( 70, 408, 1 )

( 29, 169, 1 )

Figure 4. The ﬁrst four levels of the Markoﬀ tree with triples (e, g, f), (re, rg, rf), (se, sg, sf).

and correspondingly, we have

(re, rg, rf) = (0, F2n−1, F2n−3) and (se, sg, sf) = (1, F2n−3, F2n−5).

Similarly, in the Pell branch, any triple is of the form (e, g, f) = (P2n−1, P2n+1,2) for n≥1 with

(re, rg, rf)=(P2n−2, P2n,1) and (se, sg, sf)=(P2n−3, P2n−1,1).

Deﬁnition 3.3: For any real numbers x, y ≥2/3 we set

(i) ∆x:= 9x2−4

(ii) ∆x,y := 1

2(xp∆y+y√∆x).

For a regular Markoﬀ triple (e, g, f ), the inequalities 2ef < g < 3ef hold trivially. The following

lemma sharpens these inequalities.

Lemma 3.4: Let (e, g, f)be a regular Markoﬀ triple.

(i) The following inequalities hold:

∆e,f −ef

g< g < ∆e,f .

In particular,

g=b∆e,f c.

(ii) The ﬁrst inequality can be sharpened to:

∆e,f −2

3g<∆e,f −2

9ef < g.

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UNFOCUSEDNOTESONTHEMARKOFFEQUATIONANDT-SINGULARITIESMARKUSPERLINGAbstract.WeconsiderminimalresolutionsofthesingularitiesforweightedprojectiveplanesoftypeP(e2;f2;g2),wheree;f;gsatisfytheMarkoequatione2+f2+g2=3efg.Wegiveacompleteclassicationofsuchresolutionsintermsofcontinuedfractionssimilartoclassi...

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UNFOCUSED NOTES ON THE MARKOFF EQUATION AND T-SINGULARITIES MARKUS PERLING Abstract. We consider minimal resolutions of the singularities for weighted projective planes of type

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