arXiv:2310.00245v1 [math.AG] 30 Sep 2023
SINGULARITIES AND CLUSTERS.
V.V.FOCK
Dedicated to Norbert A’Campo on occasion of his 80-th birthday.
Abstract. The aim of this note is to describe a geometric relation between
simple plane curve singularities classified by simply laced Cartan matrices
and cluster varieties of finite type also classified by the simply laced Cartan
matrices. We construct certain varieties of configurations of flags out of Dynkin
diagrams and out of singularities and show that they coincide if the Dynkin
diagram corresponds to the singularity.
The correspondence between singularities and cluster varieties was first observed
by A.Fomin, P.Pilyavsky, D.Thurston, and E.Shustin in their remarkable paper [2].
Starting from certain real forms of deformations of the singularities introduced by
N.A’Campo [1] and S.Gussein-Zade [3] they produced a cluster variety and showed
that different resolutions of the same singularity give the same cluster variety. In
this note we try to make this correspondence more geometrical and less mysterious.
In particular we show that there is a map from the base of a versal deformation
of the singularity to the corresponding cluster variety. For this purpose we generalize a construction by R.Nevanlinna [4] brought to our attention by B.Shapiro
[5]. R.Nevanlinna studied the map from the space of the differential operators of
the form D = ∂ 2 /∂z 2 + P (x), where P is a polynomial, to the collections of points
in a projective line P 1 . These points just correspond to solutions of the equation
Dψ = 0 vanishing at infinity along different rays on the complex plane and viewed
as lines in the two-dimensional space of global solutions. On the other hand, the
symbol of this operator p2 + P (x) is just an equation for the versal deformation of
a plane curve singularity of type A.
Our construction is in a sense a generalization of this one for singularities of
other types. Namely, a versal family of a planar singularity can be considered as a
family of Lagrangian curves in the plane (C2 , dp dx). Observe that this family can
be locally parameterized by the cohomology class of the form pdx. On the other
hand the family of equations defining these curves can be transformed into a family
of differential operators having the equations as their symbol. The space of Stokes
data at infinity of the differential operator is the corresponding cluster variety.
The correspondence between symbols and operators is of course not canonical. First of all, we can change change a representation of the symplectic plane
(C2 , dp dx) as a cotangent bundle to a complex line C in different ways. All these
ways give different Stokes data, but, as one can verify in the examples, they are
equivalent as cluster varieties. Sometimes the birational equivalence of the corresponding configuration space is not so trivial and can be considered as a generalization of the Gale duality.
2000 Mathematics Subject Classification. 14H20, 13F60, 34M40.
1
2
V.V.FOCK
On the other hand the correspondence between the space of differential operators
and its symbol is not canonical either. We conjecture that this map becomes
canonical in the tropical limit and the cluster coordinates in this limit are periods
of the Lagrangian curves given by the versal family of the singular one.
I am strongly indebted to F.Chapoton for reading paper and making very crucial
remarks.
1. Recall: Configurations corresponding to planar bipartite graphs
Let a be a collection of points of a projective space P (V ) vector space V . Denote
by hai a projective subspace generated by the points from a. Recall that a collection
a of k points is called free if dim hai = k − 1 and a circuit if dim hai = k − 2. For
example, two points form a circuit if they coincide, three points form a circuit if
they are colinear four points form a circuit if they are coplanar etc.
Let Γ be a bipartite graph with the set of white vertices W , black vertices B and
edges E. For simplicity we assume here that any two vertices are connected by no
more than one edge. We say that an association of a point pw in a projective space
to every white vertex w ∈ W corresponds to Γ if for every black vertex b ∈ B the
points corresponding to its white neighbors form a circuit. We also require that the
collection of points does not belong to a proper projective subspace.
Denote by XΓ the set of configurations corresponding to Γ considered up to the
action of the projective group P GL(V ). We call the dimension of the configuration
the dimension of the projective space P (V ). We say that the graph Γ is minimal if removal of any black vertex increases the dimension of the corresponding
configuration.
A discrete connection on a graph Γ is an association of one dimensional vector
spaces to vertices and association to edges of isomorphisms of the spaces corresponding to its endpoints. Given a basis in each of the one-dimensional subspaces, the discrete connection becomes an association of nonzero numbers to edges. These numbers can be organized into a matrix Mbw with columns and rows enumerated by the
black and white vertices, respectively and with zeroes for pairs of vertices not connected by an edge. Changing the bases amounts to the multiplication of this matrix
by invertible diagonal matrices from the left and from the right. Given a closed path
of the graph, the monodromy of the connection around this graph is a composition of
the maps corresponding to the edges. In terms of the connection matrix if the path
passes consecutively through the vertices b0 , w0 , . . . , bk , wk , b0 then the monodromy
is given by a Laurent monomial in the matrix entries Mbw11 (Mbw21 )−1 · · · (Mbw0k )−1 . If
the graph is planar a graph connection is uniquely determined by the monodromies
around its faces.
The set of graph connections on a graph Γ is in a bijection with the set of
configurations corresponding to Γ. Indeed, given the matrix Mbw representing the
connection we can consider it as a map M : CB → CW . The image of the standard
basis of CW in the projectivized cokernel P (CW /Im M ) form the desired configuration. Conversely, given a configuration of points in a projective space P (V ), for
every white vertex w choose a vector p̃w ∈ V representing pw . Given a black vertex
b the chosen
corresponding to its white neighbors satisfy a nontrivial linear
P vectors
relation
Mbw p̃w = 0. The matrix Mbw forms the desired graph connection.
SINGULARITIES AND CLUSTERS.
3
Call two configurations equivalent if one can construct one from another by
adding and removing points which can be constructed from the remaining points.
The graphs corresponding to equivalent configurations are also called equivalent.
There are four equivalences generating many (and conjecturally all) others.
Namely
..
. .
..
. ←→ .. ..
.
Two-valent black vertex means that two points corresponding to its white
neighbors coincide, and of course one of them can be removed.
.
..
. .
1′ . ..
. ←→ .. ..
Two-valent white vertex means that the circuits corresponding to its black
neighbors have one common point, and obviously their union (with one of
the points de-doubled) is also a circuit.
···
···
d
d′
X′ ′
.. a X Y c ..
.. a′
c ..
←→
2. .
.
.
.
Y′
b
b′
···
···
Let a,b,c,d be the circuits corresponding to black vertices and X and Y
points corresponding to the white ones. Let Y ′ = hb \ {X, Y }i ∩ (XY ). The
intersection is a point since obviously the line (XY ) belongs to the subspace
hbi and the subspace hb \ {X, Y }i is of codimension 1 in it. Similarly let
X ′ = hd \ {X, Y }i ∩ (XY ). We can now define a′ = a ∪ {X ′ , Y ′ } \ {X},
b′ = b ∪ {Y ′ } \ {X, Y }, c′ = c ∪ {X ′ , Y ′ } \ {Y }, d′ = d ∪ {X ′ } \ {X, Y }.
···
···
1.
2′ .
..
.
.
..
. ←→ ..
..
.
···
···
This fragment of the graph corresponds to four collinear points. Any triple
of such point is a circuit. Any two of such circuits define such configurations
and can be exchanged by the remaining two preserving planarity of the
graph.
2. From Dynkin diagram to bipartite graph.
For any planar graph (in our examples they will be Dynkin diagrams) one can
associate a planar bipartite graph dual to the original graph (See example shown
on Fig. 1a, 2a and 3a.) Namely, surround every vertex of valency k > 2 by a
2k-gon in a way that its every second side crosses the respective edge of the original
graph. For a 1-valent vertex surround it with a square. Then identify the sides of
the polygons crossing the same edges of the original graph. Finally color vertices in
black and white and add one edge with a white vertex to the every black 2-valent
vertex drawn around vertices of valency 1.
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V.V.FOCK
3. Stokes data.
∂
) be a differential operator with polynomial coefficients of order n.
Let D(z, ∂z
Let D(x, p) be its symbol and ∆ be its Newton polygon, namely the convex hull
of the points with coordinates (a, b) ∈ Z2 for each monomial term xa pb of the
polynomial D(x, p).
Consider the algebraic curve C = {(x, p) ∈ (C∗ )2 |D(x, p) = 0}. This curve has
the genus equal to the number of integer points strictly inside the polygon ∆ and
can be compactified by adding points, called compactification points, corresponding
to sides of the polygon. By a side we mean a segment of the boundary of ∆ between
two adjacent points of Z2 . If we orient the boundary of the polygon counterclockwise
each side s correspond to indivisible vector (a, b) ∈ Z2 . The sum of such vectors
obviously vanishes. At the compactification points corresponding to s the functions
x and p have a zero of order −bs and as , respectively. Therefore in the vicinity of
such point we have xas pbs = O(1) and hence p ∼ As x−as /bs for some constants As .
One should however realize that the correspondence between sides and compactification points is not canonical but is defined up to permutations of sides
corresponding to equal vectors.
The same polygon defines a curve in C2 . The sides of the polygon with a > 0 and
b ≤ 0 correspond to compactifcation points with both coordinates tending to a finite
constant and thus belonging to the curve in C2 . The remaining compactification
points are called points at infinity.
Consider now the equation D(x, p) = 0 as a family of equations for the indeterminate p depending on x as a parameter. Our aim is to determine the asymptotic
behavior of its roots when x → ∞. This behavior of x corresponds to the sides
of the polygon with bs > 0 (and thus going upward on a picture where the p-axis
points up). Each side corresponds to bs roots with asymptotic p ∼ As x−as /bs . The
total number of roots is thus equal to the height of the polygon, which is just the
degree of the polynomial D(x, p) with respect to p.
Now proceed to the study of the asymptotic behaviour of the differential equation
∂
)ψ = 0. In a simply connected domain where the coefficient of the n-th
D(x, ∂x
derivative vanishes nowhere, it has an n-dimensional space of solutions. However
in a sufficiently small vicinity of infinity the differential equation defines an ndimensional local system. Indeed, fix a number R such that the coefficient of the
highest derivative of D does not vanish for |x| > R, a number ε ∈]0, π/2[ and an
angle α ∈ R/2πZ. The domain {x ∈ C| | arg x − α| < ε, |x| > R} satisfies the
above conditions. Define Vα as the space of solutions of the differential equation
in this domain. The space Vα does depend neither of ε nor of R and does not
depend on α locally. Namely if |α1 − α2 | < ε the corresponding domains intersect,
their intersection is simply connected and therefore the spaces Vα1 and Vα2 can be
identified. Hence the family Vα form an n-dimensional local system over the circle
R/2πZ.
R
Every solution of the equation has asymptotic behavior ψ(x) ∼ e p(x)dx for a
root p(x) and therefore for generic angle α the space Vα is filtered by the rates of
R reiα
growth of the functions ℜe
p(x)dx with r → +∞.
Consider a sufficiently large R and mark the points As (Reiα )(bs −as )/bs on the
complex plane. (For a given s there are bs such points.) These points are ordered
according to their projection on the real axis and the interval between the i-th
SINGULARITIES AND CLUSTERS.
5
and the i + 1-st points correspond to the subspace of Vα of dimension i. As α
runs around the circle, the points rotate around the origin with the angular speed
(bs − as )/bs and the order of their projections changes. We call the collection of
points corresponding to generic α with their angular speeds indicated for each point
the growth diagram. When the i-th and the i + 1-th projections pass through each
other the i-th dimensional subspace of Vα changes.
The local system and the sequence of flags in its fibers constitute the Stokes data
of the differential operator D at infinity. If the coefficient of the highest derivative
in the operator D is constant, the local system is trivial and the Stokes data at
infinity amounts to the collection of flags in a fixed vector space.
For nontrivial local systems, taking the universal cover of the circle we can
consider a finite sequence of flags in a local system as a infinite quasi-periodic one
in a fixed vector space. Recall that the sequence is called quasi-periodic if its shift
by a period coincides with the action of an element of GL(n).
Recall that pairs of complete flags in an n-dimensional space up to diagonal
action of GL(n) are in bijection with the permutation group Sn . The standard
generators si of this group correspond to the pairs of flags different only in the
subspaces of dimension i.
Therefore a sequence of flags in a local system on a circle such that adjacent
flags differ in one subspace can be encoded by an infinite periodic word in the
same generators s1 , . . . , sn−1 . We will denote such words as [w], where w is a
period of the infinite word. For example a word [(s2 s1 )m ] corresponds to a 3dimensional local system with 2m flags such that the subspaces of dimension 1 and
2 change alternatively. Such sequence is equivalent to the quasiperiodic sequence
of 1-dimensional subspaces since the 2-dimensional subspaces can be restored from
the 1-dimensional ones. In the projective space this sequence corresponds to a
quasi-periodic set of points with the period of length m.
Observe that if two words are related by braid relations they correspond to
equivalent sequences of flags.
The word [(s2 s21 )m ] corresponds to a sequence of flags where one dimensional
subspace changes twice after each change of the 2-dimensional subspace. In the
projective space it corresponds to a quasi-periodic broken line with a marked point
on each side.
As another example consider a growth diagram given by a regular n-gon rotating
around its center with angular velocity m/n. We will show that it corresponds to
an m-periodic sequence of points in P n−1 . Indeed, in this case all even subspaces
change at once and then all odd subspaces change at once. The corresponding
word is therefore w = [(wodd weven )m ] where wodd = s1 s3 · · · and weven = s2 s4 · · ·
are the products of the odd and even generators, respectively. We claim that such
configuration is equivalent just to the sequence of 1-dimensional subspaces with
quasi-period m. Indeed, given a sequence of flags corresponding to the word w one
can construct the sequence of 1-dimensional subspaces just discarding all subspaces
of higher dimension. On the other hand given a sequence of 1-dimensional subspaces
{p̃i |i ∈ Z/nZ}, one can construct a sequence of 2m flags
F2i+1 = {p̃i ⊂ p̃i + p̃i+1 ⊂ p̃i−1 + p̃i + p̃i+1 ⊂ p̃i−1 + p̃i + p̃i+1 + p̃i+2 ⊂ · · · }
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V.V.FOCK
and
F2i = {p̃i ⊂ p̃i−1 + p̃i ⊂ p̃i−1 + p̃i + p̃i+1 ⊂ p̃i−2 + p̃i−1 + p̃i + p̃i+1 ⊂ · · · }
and observe that F2i differ from F2i+1 in odd dimensional terms and from F2i−1 in
even dimensional terms.
4. Example: An .
The Dynkin diagram of type An is just a chain of n vertices and the corresponding
bipartite graph shown on Fig.1a consists of n + 3 white and n + 1 black vertices. It
(n + 3)/2
(n + 3)/2
b
a
c
Figure 1.
corresponds just to n + 3 collinear points.
The versal deformation of a singularity of type An is represented by the polynomial
D(x, p) = p2 + xn+1 + a1 + a2 x + · · · + an xn−1 .
The corresponding Newton polygon is a right triangle with sides 2 and n (shown
on Fig.1b for n = 4) with one or two sides for n odd or even, respectively, going
upwards. It corresponds to a curve of genus ⌊(n − 1)/2⌋ with one or two points
at infinity, respectively with homology of rank n. The growth diagram (shown on
Fig.1c) consists of two points, which are opposite if n is even, rotating with the angular speed (n+3)/2. It corresponds to the word [sn+3
] and thus the configurations
1
of n + 3 points in P 1 .
Interchanging p and x we get a differential equation of order n + 1, the growth
diagram consists of points forming a regular n + 1-gon rotating about its center with angular velocity (n + 3)/(n + 1) which corresponds to the cyclic word
[(wodd weven )n+3 ] and thus to n + 3 points in P n . These two configuration space are
known to be isomorphic since one can trace a unique rational normal curve through
this point thus obtaining (n + 3) points in P 1 .
5. Example: D4 .
Consider the Dynkin diagram of the type D4 and construct a bipartite graph
shown on Fig. 2a. It corresponds to three groups of collinear points C, 1, 2, A,
A, 3, 4, B and B, 5, 6, A and the dimension of the configuration is 2. The configuration is obviously equivalent to the configuration of 6 points 1, 2, 3, 4, 5, 6 ∈ P 2 as
shown on Fig.2b. This configuration of points corresponds to a cyclic word [(s2 s31 )3 ]
The versal deformation of the singularity D4 reads as
D(p, x) = p3 + x2 p + a1 + a2 p + a3 p2 + a4 x
SINGULARITIES AND CLUSTERS.
7
2
C
3
1
4
B
6
1
A
2
5
2
2
C
A
5
a
6
4 B
3
b
c
d
Figure 2.
with the Newton polygon is a quadrilateral shown on the Fig.2c. There are three
sides of the polygon directed upward: (−1, 1) with multiplicity 2 and (1, 1) with
multiplicity 1. It corresponds to a curve of genus 1 with three points at infinity,
and with homology of rank 4. The growth diagram consist of two points rotating
with angular speed 2 and one point closer to the center which does not move.
It gives a word [(s2 s21 )4 ]. Therefore the space Stokes data can be considered as
the configuration space of quadrilaterals with marked points on each side. Such
configurations are equivalent to the configurations of six points — four points on
the sides and two opposite vertices of the quadrilateral. On the other hand one can
deduce this equivalence from the equality of the words [(s2 s21 )4 ] and [(s2 s31 )3 ] in the
braid group.
Consider another form of the same singularity just with the variables p and x
interchanged.
D(p, x) = x3 + xp2 + a1 + a2 x + a3 x2 + a4 p
The Newton polygon is just the reflection of the original one, but the corresponding
differential equation is of order 2 and the local system is nontrivial since the coefficient at the highest derivative vanish at the origin. The growth diagram consists
of two points rotating with the angular speed 2 and thus the Stokes data amounts
to the configuration of four lines in a two-dimensional local system on a circle.
Remarkably these two configuration space turns out to be birationally isomorphic. The only isomorphism I know is given by describing both as cluster varieties,
and I don’t know any geometric way to describe it.
6. Example: E8 .
Consider the Dynkin diagram of the type E8 and construct a bipartite graph
shown on Fig. 3a. On the same picture we show the corresponding bipartite
graph. This diagram corresponds to configurations of 13 points corresponding to
white vertices with black vertices corresponding to collinear triples of points. It
implies that there are three groups of collinear points C, 1, 2, 3, 4, 5, A, A, 6, 7, 8, B
and B, 9, 10, C. Such configuration can be realized in two dimensional projective
space P 2 as a triangle with 2,3 and 5 points on its sides, respectively as shown on
Fig. 3b. Observe that this configuration space is birationally isomorphic to the
space of unrestricted 8-tuples of points 1, 2, 7, 8, 9, 10, X, Y in P2 . Indeed , as it
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V.V.FOCK
is clear from the picture the points A, B, C, 3, 4, 5, 6, 7 can be reconstructed out of
1, 2, 7, 8, 9, 10, X, Y and vise versa.
4
7
5
6
3
A
8
10
2
2
3
1
B
C
1
Y
9
4
C
5
X
9
6
A
10
7
8
B
b
a
8/3
8/3
8/3
c
d
Figure 3.
The versal deformation of the singularity E8 is
D(x, p) = x5 + p3 + a1 + a2 x + a3 x2 + a4 x3 + a5 p + a6 xp + a7 x2 p + a8 x3 p
the Newton polygon is shown of Fig.3c. It has one side (−5, 3) directed upward
with multiplicity 1. Thus the curve has genus 4 with one point at infinity with
the rank of homology group 8. The growth diagram shown on Fig.3d consists of
three points with angle 2π/3 between them rotating with the angular speed 8/3. It
corresponds to the periodic word [(s1 s2 )8 ] and thus the Stokes data amounts to a
configuration of 8 points in P 2 .
Exchanging x and p we get on the growth diagram 5 points in a vertices of a
regular pentagon rotating about its center with the angular speed 8/5. It corresponds to a sequence of 8 points in P 4 . The two configuration space are birationally
isomorphic via Gale duality.
7. Other cases.
We leave the detailed consideration of singularities of other types as an exercise.
Dn : The Dynkin diagram corresponds to a configuration of triangles in the
projective plane P 2 with 2, 2 and n − 2 points on its respective sides.
The corresponding singularity xp2 + xn+1 corresponds to a configuration
of n lines in a two-dimensional local system on a circle. The differential
operator corresponding to pn+1 + x2 p corresponds to configurations of flags
in P n .
SINGULARITIES AND CLUSTERS.
9
E6 : The Dynkin diagram as well as the singularity p3 + x4 correspond to a
configuration of triangles in P 2 with 2,3 and 3 points on the sides, respectively. The singularity p3 + x4 corresponds to the word [(s1 s2 )7 ], i.e., to
configurations of 7-tuples of points of P 2 . It is easy to see that the two
configuration spaces are equivalent. The singularity p4 + x3 corresponds to
configurations of 7-tuples points in P 3 .
E7 : The Dynkin diagram corresponds to a configuration of triangles in P 2 with
2,3 and 4 points on the sides, respectively. The singularity p3 + x3 p corresponds to configurations of pentagons in P 2 with one marked point on
each side, which is equivalent to the space of configurations given by the
Dynkin diagram. The singularity xp3 + x3 corresponds to a configuration
space of 21-periodic sequences flags in P 3 which is too complicated to be
described here.
References
[1] N.A’Campo, Le groupe de monodromie du déploiement des singularités isolées de courbes
planes I, Math. Ann. 213(1975), 1–32.
[2] S.Fomin,
P.Pylyavskyy,
E.Shustin,
D.Thurston. Morsifications and mutations.
J.Lond.Math.Soc. 105(2022)4 pp
[3] S.Gusein-Zade, Dynkin diagrams of the singularities of functions of two variables, Funkcional.
Anal. i Prilozhen. 8(1974), no. 4, 23–30.
[4] R.Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten. Acta
Math.58,295–373 (1932)
[5] B.Shapiro, On Evgrafov–Fedoryuk’s theory and quadratic differentials, Anal. Math. Phys.
5(2015),171–181.
Université de Strasbourg