On Degenerations of the Projective Plane
Complementing the results of Hacking and Prokhorov, we determine explicitly all log-terminal, rational, degenerations of the projective plane that admit a non-trivial torus action.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 042, 16 pages
On Degenerations of the Projective Plane
Jürgen HAUSEN a, Katharina KIRÁLY a and Milena WROBEL b
a) Mathematisches Institut, Universität Tübingen,
Auf der Morgenstelle 10, 72076 Tübingen, Germany
E-mail: juergen.hausen@uni-tuebingen.de, kaki@math.uni-tuebingen.de
b) Institut für Mathematik, Universität Oldenburg, 26111 Oldenburg, Germany
E-mail: milena.wrobel@uni-oldenburg.de
Received December 17, 2024, in final form June 07, 2025; Published online June 12, 2025
https://doi.org/10.3842/SIGMA.2025.042
Abstract. Complementing results of Hacking and Prokhorov, we determine in an explicit
manner all log terminal, rational, degenerations of the projective plane that allow a non-
trivial torus action.
Key words: degenerations of the plane; Markov numbers; del Pezzo surfaces; torus action
2020 Mathematics Subject Classification: 14L30; 14J26; 14J10; 14D06
1 Introduction
The aim of this note is to determine explicitly all log terminal, rational degenerations of the pro-
jective plane P2 that admit a non-trivial torus action, see Theorem 5.5; note that log terminality
for a surface merely means to have at most quotient singularities. Recall from [16] that a degen-
eration of P2 is the central fiber X0 of a proper flat analytic family of surfaces over the unit disk
such that Xt
∼= P2 for all t ̸= 0. Manetti [16] characterized the log terminal degenerations of P2
as the projective algebraic complex surfaces of Picard number one with vanishing plurigenera
having at most singularities of the type 1
n2 (1, na − 1) with coprime a, n ∈ Z>0, that means
quotients of 0 ∈ C2 by the linear action of a cyclic group of order n2 with the weights (1, na−1).
The description of all normal degenerations of the projective plane involves the Markov
numbers. These are by definition the entries of the Markov triples which in turn are the triples
(x, y, z) of positive integers satisfying the diophantic equation
x2 + y2 + z2 = 3xyz.
From any Markov triple (x, y, z), one obtains new ones via mutations, that means by permuting
its entries and passing to (x, y, 3xy − z). The vertices of the Markov tree are the normalized
(i.e., ascendingly ordered) Markov triples, representing all triples obtained from a given one by
permuting its entries. In the Markov tree, two triple classes are adjacent, that means joined by
an edge, if and only if they are distinct and arise from each other by a mutation.
(1,1,1) (1,1,2) (1,2,5)
(1,5,13)
(2,5,29)
(1,13,34)
(5,13,194)
(5,29,433)
(2,29,169)
mailto:juergen.hausen@uni-tuebingen.de
mailto:kaki@math.uni-tuebingen.de
mailto:milena.wrobel@uni-oldenburg.de
https://doi.org/10.3842/SIGMA.2025.042
2 J. Hausen, K. Király and M. Wrobel
The Markov tree admits an interpretation in terms of rational projective complex surfaces.
With any vertex, represented by a Markov triple (k1, k2, l), Hacking and Prokhorov associate
the weighted projective plane P(k21 ,k
2
2 ,l
2) and they show that these are in fact all toric normal
degenerations of P2, see [9, Corollary 1.2 and Theorem 4.1]. The edges, joining adjacent Markov
triples (k1, k2, l1) and (k1, k2, l2), are reflected combinatorially by mutations of the simplices as-
sociated with the two weighted projective planes [1, 2] and geometrically by a flat one parameter
family having both of them as fibers [8, 14, 17]; moreover, [18] studies the birational geometry
behind the Markov tree. For an explicit geometric realization of a given edge joining adjacent
Markov triples (k1, k2, l1) and (k1, k2, l2), one associates with it the following surface, living in
a weighted projective space, see also [8, Example 7.7]:
X(k1, k2, l1, l2) := V
(
T1T2 + T l1
3 + T l2
4
)
⊆ P(k21 ,k
2
2 ,l2,l1)
.
By Theorem 4.5, the surface X(k1, k2, l1, l2) is well defined, quasismooth (i.e., has at most cyclic
quotient singularities), rational, del Pezzo, of Picard number one and it comes with an effective
C∗-action. Moreover, we obtain the following geometric connections between X(k1, k2, l1, l2) and
the weighted projective planes P(k21 ,k
2
2 ,l
2
i )
given by the two adjacent triples, see also the (more
general) Construction 3.4 and Propositions 3.2 and 3.6.
Theorem 1.1. Let (k1, k2, l1) and (k1, k2, l2) be adjacent Markov triples and consider the surface
X(k1, k2, l1, l2). Then we have a commutative diagram
P(k21 ,k
2
2 ,l2,l1)
[z1,z2,z
l1
4 ]←[[z1,z2,z3,z4]
uu
[z1,z2,z3,z4]7→[z1,z2,z
l2
3 ]
))
P(k21 ,k
2
2 ,l
2
1)
X(k1, k2, l1, l2)
OO
1:l21
oo
l22:1
// P(k21 ,k
2
2 ,l
2
2)
with finite coverings X(k1, k2, l1, l2) → P(k21 ,k
2
2 ,l
2
i )
of degree l2i , respectively. Moreover, there are
flat families ψi : Xi → C, where
X1 := V
(
T1T2 + ST l1
3 + T l2
4
)
⊆ P(k21 ,k
2
2 ,l2,l1)
× C,
X2 := V
(
T1T2 + T l1
3 + ST l2
4
)
⊆ P(k21 ,k
2
2 ,l2,l1)
× C
and the ψi are obtained by restricting the projection P(k21 ,k
2
2 ,l2,l1)
× C → C. For the fibers of these
families, we have
ψ−1i (s) ∼= X(k1, k2, l1, l2), s ∈ C∗, ψ−1i (0) ∼= P(k21 ,k
2
2 ,l
2
i )
.
Whereas the degenerations Xi → C are, as mentioned, well known, the coverings haven’t been
observed so far to our knowledge. Note that there may occur adjacent Markov triples (k1, k2, li)
with li = 1 for one of the i. This happens if and only if k1 and k2 are Fibonacci numbers of
subsequent odd indices. In this case, the covering is in fact an isomorphism:
X(k1, k2, l1, l2) ∼= P(k21 ,k
2
2 ,1)
.
If both li differ from one, X(k1, k2, l1, l2) is non-toric. The Fibonacci branch of the Markov
tree hosts the vertices P(1,k22 ,l
2) and the edges X(1, k2, l1, l2), where the latter surfaces also
showed up in [15, Remark 6.6]. Our first main result characterizes the surfaces X(k1, k2, l1, l2)
by their geometric properties and shows in particular that they are uniquely determined by the
underlying weighted projective planes.
Theorem 1.2. Let X be a non-toric, log terminal, rational, projective C∗-surface of Picard
number ρ(X) = 1. Then the following statements are equivalent:
On Degenerations of the Projective Plane 3
(i) The surface X is isomorphic to one of the surfaces X(k1, k2, l1, l2).
(ii) The canonical self intersection number of X is given by K2
X = 9.
Moreover, if one of these statements holds, then X is determined up to isomorphy by the num-
bers k1 and k2.
Let us refer to the representatives P(k21 ,k
2
2 ,l
2
i )
of the vertices of the Markov tree as the toric
Markov surfaces and to the representatives X(k1, k2, l1, l2) of the edges as the Markov C∗-
surfaces. Then the second main result of this note, Theorem 5.5, provides us with the following
statement on degenerations of the projective plane P2 in the sense of Manetti [16, Definition 1].
Theorem 1.3. Let X be a log terminal, rational, projective surface with a non-trivial torus
action. Then the following statements are equivalent:
(i) X is a degeneration of the projective plane.
(ii) We have ρ(X) = 1 and K2
X = 9.
(iii) X is a toric Markov surface or a Markov C∗-surface.
As a consequence of this characterization, given any log terminal, rational degeneration X of
the projective plane with precisely one singularity x ∈ X, we can say the following: if the local
Gorenstein index of x ∈ X is a Markov but not a Fibonacci number, then X does not allow
a non-trivial C∗-action.
Moreover, let us relate the second main to test configurations of the projective plane. Roughly
speaking, these are C∗-equivariant flat families ψ : X → C with general fiber ψ−1(1) ∼= P2 such
that the C∗-action on C is given by the multiplication, see [4, Definition A.1]. According
to [4, Theorem A.3], the surfaces which arise as the central fiber ψ−1(0) of a test configuration
of P2 are given up to isomorphy by the vertices of the Fibonacci branch of the Markov tree and
the edges touching one of the latter vertices. All the other vertices and edges from the Markov
tree represent degenerations of the projective plane in the sense of [16, Definition 1] which do
not occur as the central fiber of any test configuration of the projective plane.
2 Toric Markov surfaces
We provide the necessary facts on toric Markov surfaces, that means weighted projective planes
given by a squared Markov triple. The main observation of the section, Proposition 2.7, charac-
terizes the toric Markov surfaces as the toric surfaces of Picard number one and canonical self
intersection nine. The reader is assumed to be familiar with the basics of toric geometry; we
refer to [6] for the background.
Construction 2.1 (fake weighted projective spaces as toric varieties). Consider an n× (n+1)
generator matrix, that means an integral matrix
P =
[
v0 . . . vn
]
the columns vi ∈ Zn of which are parwise distinct, primitive and generate Qn as a convex cone.
For each i = 0, . . . , n, we obtain a convex, polyhedral cone
σi := cone(vj ; j = 0, . . . , n, j ̸= i).
These σi are the maximal cones of a fan Σ = Σ(P ) in Zn. The associated toric variety Z = Z(P )
is an n-dimensional fake weighted projective space.
4 J. Hausen, K. Király and M. Wrobel
The fake weighted projective spaces turn out to be precisely the Q-factorial projective toric
varieties of Picard number one. In particular, the fake weighted projective planes are exactly
the projective toric surfaces of Picard number one. We will also benefit from the following
alternative approach.
Remark 2.2. The fake weighted projective spaces Z(P ) are quotients of Cn+1\{0}. The matrix
P = (pij) from Construction 2.1 defines a homomorphism
p : Tn+1 → Tn, (t0, . . . , tn) 7→
(
tp100 · · · tp1nn , . . . , tpn0
0 · · · tpnn
n
)
,
the subgroup H := ker(p) ⊆ Tn+1 is a direct product C∗ ×G with a finite subgroup G ⊆ Tn+1
and for the induced action of H on Cn+1 we have
Z(P ) =
(
Cn+1 \ {0}
)
/H.
This provides us with homogeneous coordinates as for the classical projective space: We write
[z0, . . . , zn] ∈ Z(P ) for the H-orbit through (z0, . . . , zn) ∈ Cn+1 \ {0}.
Recall that for a point x of a normal varietyX the local class group is the factor group Cl(X,x)
of the group of all Weil divisors on X modulo those being principal on some open neighbourhood
of x. We denote by cl(x) the order of Cl(X,x).
Remark 2.3. Consider P = [v0, . . . , vn] as in Construction 2.1 and the associated Z = Z(P ).
The fake weight vector associated with P is
w = w(P ) = (w0, . . . , wn) ∈ Zn+1
>0 , wi := |det(vj ; j = 0, . . . , n, j ̸= i)|.
For the divisor class group and the local class groups of the toric fixed points z(i), having i-th
homogeneous coordinate one and all others zero, we obtain
Cl(Z) = Zn/im(P ∗) ∼= Z⊕ Γ, |Γ| = gcd(w0, . . . , wn), cl(z(i)) = wi.
Moreover, Cl(Z) ∼= Z ⊕ Γ can be identified with the character group of H ∼= C∗ × G from
Remark 2.2 via the isomorphism H ∼= SpecC[Zn/im(P ∗)].
Remark 2.4. Let Z = Z(P ) arise from Construction 2.1 and w = w(P ) as in Remark 2.3.
Then Cl(Z) is torsion free if and only if w ∈ Zn+1 is primitive. In the latter case, Z equals the
weighted projective space P(w0,...,wn).
Proposition 2.5. For a fake weighted projective plane Z = Z(P ) with fake weight vector w =
w(P ) = (w0, w1, w2), the canonical self intersection number is given by
K2
Z =
(w0 + w1 + w2)
2
w0w1w2
.
Proof. We may assume that the generator matrix P of our fake weighted projective plane Z
is of the form
P =
[
l0 l1 l2
d0 d1 d2
]
, li ̸= 0, i = 0, 1, 2.
Then we can use, for instance, [12, Remark 3.3] for the computation of the canonical self inter-
section number. ■
Definition 2.6. By a toric Markov surface we mean a surface isomorphic to a weighted pro-
jective plane P
(
k20, k
2
1, k
2
2
)
, where (k0, k1, k2) is a Markov triple.
On Degenerations of the Projective Plane 5
Proposition 2.7. Let Z be a projective toric surface of Picard number one. Then the following
statements are equivalent:
(i) Z is a toric Markov surface.
(ii) We have K2
Z = 9.
The proof relies on the following (known) elementary statement on the positive integer so-
lutions of the “squared Markov identity”; see also [7] for recent, further going work in that
direction.
Lemma 2.8. The positive integer solutions of (w0 + w1 + w2)
2 = 9w0w1w2 are precisely the
triples
(
k20, k
2
1, k
2
2
)
, where (k0, k1, k2) is a Markov triple.
Proof. Clearly, every squared Markov triple solves the equation. For the converse, we build up
the analogue of the Markov tree. Consider the involution
λ : (u0, u1, u2) 7→
(
u0, u1, 9u0u1 − 6
√
u0u1u2 + u2
)
=
(
u0, u1,
(
3
√
u0u1 −
√
u2
)2)
.
If u is a positive integer solution, then also λ(u) is one. Moreover, if the entries of u are squares,
then the entries of λ(u) are so. Starting with (1, 1, 1), we obtain
(1,1,1) (1,1,4) (1,4,25)
(1,25,169)
(4,25,841)
(1,169,1156)
(25,169,37636)
(25,841,187489)
(4,841,28561)
by successively applying λ to permutations of triples obtained so far. One directly checks that
this yields the squared triples of the Markov tree. We claim
u0 ≤ u1 ≤ u2, u2 ≥ 3 =⇒ (3
√
u0u1 −
√
u2)
2 < u2
for any positive integer solution u = (u0, u1, u2). Suppose that we have “≥” on the right hand
side. Then 9u20u
2
1 ≥ 4u0u1u2 and we obtain
(u0 + u1 + u2)
2 = 9u0u1u2 ≥ 4u22.
Consequently u0 + u1 ≥ u2. This in turn gives us 2(u0 + u1) ≥ u0 + u1 + u2 and the claim
directly follows from the estimate
3
u2
+
1
u0
≥ u0
u1u2
+
2
u2
+
u1
u0u2
=
(u0 + u1)
2
u0u1u2
≥ 1
4
(u0 + u1 + u2)
2
u0u1u2
=
9
4
.
We conclude that every positive integer solution of (w0 + w1 + w2)
2 = 9w0w1w2 arises from
(1, 1, 1) by successively applying λ to permutations of triples. ■
Proof of Proposition 2.7. Consider Z = Z(P ) as in Construction 2.1. Then, with w = w(P ),
Proposition 2.5 tells us
K2
Z =
(w0 + w1 + w2)
2
w0w1w2
.
The implication “(i) ⇒ (ii)” is a direct consequence. For the reverse direction, we additionally
use Lemma 2.8. ■
6 J. Hausen, K. Király and M. Wrobel
We take a brief look at the singularities of the toric Markov surfaces; we refer to [9, Sections 2
and 4] for a comprehensive, more general treatment. Let k, p be coprime positive integers, denote
by C
(
k2
)
⊆ C∗ the group of k2-th roots of unity and consider the action
C
(
k2
)
× C2 → C2, ζ · z =
(
ζz1, ζ
pk−1z2
)
.
Then U := C2/C
(
k2
)
is an affine toric surface and the image u ∈ U of 0 ∈ C2 is singular as soon
as k > 1. A singularity of type 1
k2
(1, pk− 1) is a surface singularity locally isomorphic to u ∈ U .
The local Gorenstein index ι(x) of a point x in a normal variety X is the order of the canonical
class in the local class group Cl(X,x).
Proposition 2.9. The fixed points z(i) ∈ Z, i = 0, 1, 2, of a toric Markov surface Z =
P
(
k20, k
2
1, k
2
2
)
are of local Gorenstein index ki and singularity type 1
k2i
(1, piki − 1).
Lemma 2.10. Consider an affine toric surface U with fixed point u ∈ U . Then u ∈ U is of type
1
k2
(1, pk − 1) if and only if cl(u) = ι(u)2.
Proof. Let u ∈ U be of type 1
k2
(1, pk − 1). Choose a, b ∈ Z such that ak − bp = 1. Consider
the affine toric surface U ′ given by the generator matrix
P ′ =
[
k k
k + b b
]
.
The corresponding homomorphism T2 → T2, t 7→
(
tk1t
k
2, t
k+b
1 tb2
)
extends to a toric morphism
π : C2 → U ′. Moreover, we obtain an isomorphism
C
(
k2
)
→ ker(π), ζ 7→
(
ζ, ζpk−1
)
.
We conclude U ′ ∼= C2/C
(
k2
)
with C
(
k2
)
acting as needed for type 1
k2
(1, pk − 1). Thus U ′ ∼= U
and, using [12, Remark 3.7], we obtain
cl(u) = |det(P ′)| = k2, ι(u) = k.
Conversely, assume cl(u) = ι(u)2. The affine toric surface U is given by a generator matrix P .
With k := ι(u), a suitable unimodular transformation turns P into
P =
[
k k
c b
]
, gcd(c, k) = gcd(b, k) = 1.
By assumption, cl(u) = |det(P )| equals ι(u)2 = k2. Thus, we may assume c = k + b. Take
a, p ∈ Z with ak − bp = 1 and p ≥ 1. Then we have an action
C
(
k2
)
× C2 → C2, ζ · z =
(
ζz1, ζ
pk−1z2
)
.
With similar arguments as above, we verify that U is the quotient C2/C
(
k2
)
for this action and
thus see that u ∈ U is of type 1
k2
(1, pk − 1). ■
Proof of Proposition 2.9. We have Cl(Z) = Z and the anticanonical class of Z is given by
wZ = k20 + k21 + k22 = 3k0k1k2 ∈ Z. Remark 2.3 tells us Cl(Z, z(i)) = Z/k2i Z. As Markov
numbers are pairwise coprime and not divisible by 3, we see that wZ is of order ki in Cl(Z, z(i)).
The assertion follows from Lemma 2.10. ■
On Degenerations of the Projective Plane 7
3 Rational projective C∗-surfaces
We first recall the necessary theory of quasismooth, rational, projective C∗-surfaces of Picard
number one; see [3, Section 5.4] and the introductory part of [10] for the general background.
Then, in Construction 3.4, we exhibit for each of our C∗-surfaces two coverings onto fake weighted
projective planes. Moreover, in Construction 3.5 and Proposition 3.6, we take an explicit look
at the toric degenerations.
A point of a rational C∗-surface X is called quasismooth if it is the image of a smooth point
of the characteristic space X̂ over X; see [10, Section 5]. It is a specific feature of a rational
C∗-surface that its singular quasismooth points are precisely its cyclic quotient singularities;
see [13, Corollary 6.12].
Construction 3.1 (Quasismooth C∗-surfaces of Picard number one). Consider an integral 3×4
matrix of the following shape:
P =
−1 −1 l1 0
−1 −1 0 l2
0 d0 d1 d2
,
1 ≤ d1 ≤ l1 ≤ l2, gcd(li, di) = 1, d0 +
d1
l1
+
d2
l2
< 0 <
d1
l1
+
d2
l2
.
Let Z(P ) denote the fake weighted projective space having P as its generator matrix, see Con-
struction 2.1. Then we obtain a surface
X(P ) := V (1 + S1 + S2) ⊆ Z(P ),
where S1, S2, S3 are the coordinates on the acting torus T3 ⊆ Z. The surface X(P ) inherits
from Z(P ) the C∗-action given on T3 ⊆ Z(P ) by
t · s = (s1, s2, ts3).
Proposition 3.2. Consider X = X(P ) in Z = Z(P ) given by Construction 3.1. Then, in
homogeneous coordinates on Z, we have the representation
X = V
(
T1T2 + T l1
3 + T l2
4
)
⊆ Z.
The C∗-surface X is projective, rational, quasismooth, del Pezzo and of Picard number one.
With any l1-th root ζ of −1, the C∗-fixed points of X are
x0 = [0, 0, ζ, 1], x1 = [0, 1, 0, 0], x2 = [1, 0, 0, 0].
The fixed point x0 is hyperbolic and x1, x2 are both elliptic. There are exactly two non-trivial
orbits C∗ · z1 and C∗ · z2 with non-trivial isotropy groups:
z1 = [−1, 1, 0, 1], |C∗z1 | = l1, z2 = [−1, 1, 1, 0], |C∗z2 | = l2.
The fake weight vector w(P ) = (w1, w2, w3, w4) of the ambient fake weighted projective space
Z = Z(P ) is given explicitly in terms of P as
w(P ) = (−l1l2d0 − l2d1 − l1d2, l2d1 + l1d2,−l2d0,−l1d0) ∈ Z4
>0.
Moreover, for the local class group orders of the three C∗-fixed points x0, x1, x2 ∈ X, we obtain
cl(x0) = −d0, cl(x1) = w2, cl(x2) = w1.
Finally, the self intersection number of the canonical divisor KX on X can be expressed as
follows:
K2
X =
(
1
w1
+
1
w2
)(
2 +
l1
l2
+
l2
l1
)
=
cl(x0)
cl(x1) cl(x2)
(l1 + l2)
2.
8 J. Hausen, K. Király and M. Wrobel
Proof of Construction 3.1 and Proposition 3.2. The assumptions on li, di made in Con-
struction 3.1 ensure that P fits into the setting of [10, Construction 4.2]. According to [10,
Proposition 4.5], the output X(P ) is a normal, rational, projective C∗-surface. Quasismooth-
ness, ρ(X) = 1 and the statements on the fixed points are covered by [10, Propositions 4.9, 4.15
and 5.1].
We are left with the canonical self intersection number. Using the general formula [12,
Proposition 7.9] for rational projective C∗-surfaces, we directly compute
K2
X =
(
1
l1
+ 1
l2
)2
d1
l1
+ d2
l2
−
(
1
l1
+ 1
l2
)2
d0 +
d1
l1
+ d2
l2
=
(
1
cl(x1)
+
1
cl(x2)
)(
2 +
l1
l2
+
l2
l1
)
,
where cl(xi) are the local class group orders of the fixed points as just determined. The assertion
then follows from cl(x1) + cl(x2) = l1l2 cl(x0). ■
Proposition 3.3. Let X be a non-toric, quasismooth, rational, projective C∗-surface of Picard
number one. Then X ∼= X(P ) with P as in Construction 3.1.
Proof. By [10, Theorem 4.18], we have X = X(P ) with a defining matrix P in the sense
of [10, Construction 4.2]. Using [10, Propositions 4.9, 4.15 and 5.1], we see that P is as in
Construction 3.1. ■
Construction 3.4 (coverings onto fake weighted projective planes). Let X=X(P ) in Z=Z(P )
arise via Construction 3.1 from a matrix
P =
−1 −1 l1 0
−1 −1 0 l2
0 d0 d1 d2
.
Let Σ denote the unique fan in Z3 having P as a generator matrix and define Σ′ ⊆ Σ to be the
subfan with the maximal cones
σ1 := cone(v1, v3, v4), σ2 := cone(v2, v3, v4), τ := cone(v1, v2).
Then the open toric subvariety Z ′ ⊆ Z given by the subfan Σ′ ⊆ Σ satisfies X ⊆ Z ′. Set
ℓ := gcd(l1, l2) and ℓi := li/ℓ. Consider
P1 :=
[
−1 −1 ℓ2
0 d0l1 ℓ2d1 + ℓ1d2
]
, P2 :=
[
−1 −1 ℓ1
0 d0l2 ℓ2d1 + ℓ1d2
]
.
These are generator matrices for fake weighted projective planes Z1 and Z2. In terms of the
fake weight vector w(P ) = (w1, w2, w3, w4) of Z = Z(P ), we have
w(P1) =
(
ℓ−1w1, ℓ
−1w2, w3
)
, w(P2) =
(
ℓ−1w1, ℓ
−1w2, w4
)
for the respective fake weight vectors. Let φi : Z
′ → Zi be the toric morphisms defined by the
linear maps Fi : Z3 → Z2 with the representing matrices
F1 :=
[
0 1 0
−d1 d1 l1
]
, F2 :=
[
1 0 0
d2 −d2 l2
]
.
Restricting to X ⊆ Z ′ gives a finite covering φ1 : X → Z1 of degree l1 and a finite covering
φ2 : X → Z2 of degree l2.
On Degenerations of the Projective Plane 9
Proof. Everything is basic toric geometry except the statement on φi : X → Zi. On the acting
tori T3 ⊆ Z ′ and T2 ⊆ Zi, the map φ2 : Z
′ → Z2 is given by
φ2(s1, s2, s3) =
(
s1, s
−d2
2 sl23
)
.
The points of X ∩ T3 are of the form ξ = (ξ1,−1 − ξ1, ξ2) with ξ1, ξ2 ∈ C∗ such that ξ1 ̸= −1.
For the image and the fibers, we obtain
φ2
(
X ∩ T3
)
=
{
η ∈ T2; η1 ̸= −1
}
, φ−12 (φ2(ξ)) =
{
(ξ1,−1− ξ1, ζξ2); ζ
l2 = 1
}
.
Consequently, φ2 is dominant, hence surjective and its general fiber contains precisely l2 points.
With the coordinate divisors C1, C2, C3 ⊆ Z1, we have
Z2 \ φ2
(
X ∩ T3
)
= C1 ∪ C2 ∪ C3 ∪ C4, C4 :=
{
η ∈ T2; η1 = −1
}
⊆ Z2.
Let Di ⊆ X be the prime divisors obtained by cutting down the coordinate divisors of Z;
see [10, Proposition 4.9]. Using surjectivity of φ2, we see
Z2 \ φ2
(
X ∩ T3
)
= φ2
(
X \ T3
)
= φ2(D1) ∪ · · · ∪ φ2(D4).
Thus, φ2 : X → Z2 must have finite fibers, proving everything we need. The map φ1 : X → Z1
can be treated in an analogous manner. ■
Construction 3.5 (degenerations to fake weighted projective planes). Consider X = X(P ) in
Z = Z(P ) as provided by Construction 3.1 and set
X1 := V
(
T1T2 + ST l1
3 + T l2
4
)
⊆ Z × C,
X2 := V
(
T1T2 + T l1
3 + ST l2
4
)
⊆ Z × C,
where the Ti are the homogeneous coordinates on Z and S is the coordinate on C. Then X1
and X2 are invariant under the respective C∗-actions on Z × C given by
ϑ · ([z1, z2, z3, z4], s) =
([
z1, z2, ϑ
−1z3, z4
]
, ϑs
)
,
ϑ · ([z1, z2, z3, z4], s) =
([
z1, z2, z3, ϑ
−1z4r], ϑs
)
.
Restricting the projection Z ×C → C yields flat families ψi : Xi → C being compatible with the
above C∗-actions and the scalar multiplication on C. Set
P̃1 :=
[
d1 d1 + l1d0 d2
l1 l1 −l2
]
, P̃2 :=
[
d2 d2 + l2d0 d1
l2 l2 −l1
]
,
and let Z̃1, Z̃2 denote the associated fake weighted projective planes. Then the central fiber
ψ−1i (0) equals Z̃i and any other fiber ψ−1i (s) is isomorphic to X.
Proof. The families ψi : Xi → C are those provided by [11, Construction 4.1] for κ = 1, 2 and
from [11, Proposition 4.6] we infer that the P̃i are the generator matrices of the central fibers.
Thus, ψ−1i (0) = Z̃i. ■
Proposition 3.6. Consider X = X(P ) in Z = Z(P ) from Construction 3.1 and the families
Xi → C from Construction 3.5. With w(P ) = (w1, w2, w3, w4), we have
w
(
P̃1
)
=
(
w1, w2,−l21d0
)
, w
(
P̃2
)
=
(
w1, w2,−l22d0
)
for the fake weight vectors of the central fibers Z̃1 and Z̃2. Moreover, the canonical self inter-
section numbers of X, Z̃1, Z̃2 satisfy
K2
X = K2
Z̃1
= K2
Z̃2
.
10 J. Hausen, K. Király and M. Wrobel
Proof. The statement on the fake weight vectors is obtained by direct computation. Also
the identity of the canonical self intersections can be directly verified, using Propositions 2.5
and 3.2. ■
Remark 3.7. The flat families ψi : Xi → C from Construction 3.5 are special equivariant test
configurations in the sense of [11, Definition 5.2] for the del Pezzo C∗-surface X. Moreover,
according to [11, Proposition 5.4], any other special equivariant test configuration of X has
limit Z̃1 or Z̃2.
4 Markov C∗-surfaces
In Construction 4.2, we associate with each pair of adjacent Markov triples a rational, projective
C∗-surface. Theorem 4.5 gathers geometric properties of these Markov C∗-surfaces, showing in
particular that they naturally represent the edges of the Markov graph. Theorem 5.4 char-
acterizes the Markov C∗-surfaces as the rational, projective C∗-surfaces of Picard number one
of canonical self intersection nine. We begin with a couple of elementary observations around
Markov triples.
Lemma 4.1. For any 0 ≤ k1 ≤ k2 ∈ R and 0 ≤ l1 ≤ l2 ∈ R, the following three conditions are
equivalent:
(i) l1l2 = k21 + k22 and l1 + l2 = 3k1k2,
(ii) l21 + k21 + k22 = 3l1k1k2 and l22 + k21 + k22 = 3l2k1k2,
(iii) l1, l2 are given in terms of k1, k2 as
l1,2 =
3k1k2 ±
√
9k21k
2
2 − 4k21 − 4k22
2
.
In particular, given any two real numbers 0 ≤ k1 ≤ k2, there exist unique real numbers 0 ≤ l1 ≤ l2
satisfying Condition (i).
Proof. Suppose that (i) holds. Then the second equation gives l2 = 3k1k2 − l1. Plugging this
into the first equation, we obtain the first equation of (ii). Similarly, considering l1 = 3k1k2− l2,
we arrive at the second equation of (ii). Solving the equations of (ii) for l1 and l2 gives (iii).
If (iii) holds, then we directly compute l1l2 = k21 + k22 and l1 + l2 = 3k1k2. ■
Lemma 4.2. Let k1 ≤ k2 and l1 ≤ l2 be positive integers. Then the following statements are
equivalent:
(i) l1l2 = k21 + k22 and l1 + l2 = 3k1k2,
(ii) l21 + k21 + k22 = 3l1k1k2 and l22 + k21 + k22 = 3l2k1k2,
(iii) (l1, k1, k2) and (k1, k2, l2) are Markov triples.
If one of (i) to (iii) holds, then gcd(l1, l2) = 1 and the triples (l1, k1, k2), (k1, k2, l2) are adjacent
and normalized up to switching (l1, k1) in the first one.
Proof. The equivalence of (i) and (ii) holds by Lemma 4.1 and the equivalence of (ii) and (iii)
is valid by the definition of a Markov triple. Now assume that one of the three conditions holds.
Then k1, k2, l2 form a Markov triple, hence are pairwise coprime and, using l2 = 3k1k2 − l1,
we see that l1 and l2 must be coprime as well. Moreover, k1 ≤ k2 and l1 ≤ l2 together with
l1 + l2 = 3k1k2 imply that the triples are normalized up to switching (l1, k1) in the first one.
Finally, l2 = 3k1k2 − l1 merely means that the triples are adjacent. ■
On Degenerations of the Projective Plane 11
Lemma 4.3. Let (l1, k1, k2) and (k1, k2, l2) be Markov triples. Then there exist integers d1, d2
such that
k22 = l2d1 + l1d2, 1 ≤ d1 ≤ l1.
The integers d1 and d2 are uniquely determined by these properties. Moreover, they satisfy
gcd(li, di) = 1.
Proof. Consider the factor ring Z/l1Z. Lemma 4.2 says that l2 and l1 are coprime. Conse-
quently, there is a multiplicative inverse c̄1 ∈ Z/l1Z of l̄2 ∈ Z/l1Z. We claim that there is
a unique d1 ∈ Z with
1 ≤ d1 ≤ l1, d̄1 = c̄1 · k̄22 ∈ Z/l1Z.
Indeed, since (l1, k1, k2) is a Markov triple, the numbers l1 and k2 are coprime. Thus, being
a product of units, d̄1 := c̄1 · k̄22 is a unit in Z/l1Z. We take the unique representative 1 ≤ d1 ≤ l1.
Now, there is a unique d2 ∈ Z with
l1d2 + l2d1 = k22,
because of l̄2 · d̄1 = l̄2 · c̄1 · k̄22 = k̄22 in Z/l1Z. To obtain gcd(l2, d2) = 1, look at the factor
ring Z/l2Z. There l̄1 admits a multiplicative inverse c̄2. This gives us d̄2 = c̄2 · k̄22 in Z/l2Z.
Hence d̄2 is a unit in Z/l2Z. ■
Construction 4.4. Let µ = ((l1, k1, k2), (k1, k2, l2)) be a pair of adjacent Markov triples, the
second triple normalized and the first one up to switching (l1, k1). Let d1, d2 ∈ Z be as provided
by Lemma 4.3 and set
P (µ) :=
−1 −1 l1 0
−1 −1 0 l2
0 −1 d1 d2
.
As in Construction 3.1, let Z(µ) be the toric threefold defined by the complete fan in Z3 hav-
ing P (µ) as its generator matrix, let S1, S2, S3 be the coordinates of the acting torus T3 ⊆ Z(µ)
and set
X(µ) := V (1 + S1 + S2) ⊆ Z(µ).
Theorem 4.5. Let X = X(µ) be as in Construction 4.4. Then X is a quasismooth, rational
C∗-surface with Cl(X) = Z and Cox ring R(X) given by
R(X) = C[T1, T2, T3, T4]/
〈
T1T2 + T l1
3 + T l2
4
〉
,
deg(T1) = k21, deg(T2) = k22, deg(T3) = l2, deg(T4) = l1.
The ambient toric threefold Z = Z(µ) is the weighted projective space P(k21 ,k
2
2 ,l2,l1)
and X is the
zero set of a homogeneous equation of degree k21 + k22 = l1l2:
X = V
(
T1T2 + T l1
3 + T l2
4
)
⊆ P(k21 ,k
2
2 ,l2,l1)
= Z.
The surface X is of Picard number one, it is non-toric if and only if l1 > 1 and the C∗-action
on X is given in homogeneous coordinates by
t · [z] =
[
t · z1, t−1 · z2, z3, z4
]
.
12 J. Hausen, K. Király and M. Wrobel
The only possible singularities of X are the elliptic fixed points, given together with their local
class group order, local Gorenstein index and singularity type by
x1 = [0, 1, 0, 0], cl(x1) = k22, ι(x1) = k2,
1
k22
(1, p2k2 − 1),
x2 = [1, 0, 0, 0], cl(x2) = k21, ι(x2) = k1,
1
k21
(1, p1k1 − 1),
with pi ∈ Z≥1 such that gcd(pi, ki) = 1. For the canonical self intersection of X, we have
K2
X = 9. Moreover, there is a commutative diagram
P(k21 ,k
2
2 ,l2,l1)
[z1,z2,z
l1
4 ]←[[z1,z2,z3,z4]
uu
[z1,z2,z3,z4]7→[z1,z2,z
l2
3 ]
))
P(k21 ,k
2
2 ,l
2
1)
X
OO
1:l21
oo
l22:1
// P(k21 ,k
2
2 ,l
2
2)
with finite coverings X(k1, k2, l1, l2) → P(k21 ,k
2
2 ,l
2
i )
of degree l2i , respectively. Finally, we obtain
flat families ψi : Xi → C, where
X1 := V
(
T1T2 + ST l1
3 + T l2
4
)
⊆ P(k21 ,k
2
2 ,l2,l1)
× C,
X2 := V
(
T1T2 + T l1
3 + ST l2
4
)
⊆ P(k21 ,k
2
2 ,l2,l1)
× C
and the ψi are given by restricting the projection P(k21 ,k
2
2 ,l2,l1)
× C → C. For the fibers of these
families, we have
ψ−1i (s) ∼= X(k1, k2, l1, l2), s ∈ C∗, ψ−1i (0) ∼= P(k21 ,k
2
2 ,l
2
i )
.
Proof of Construction 4.4 and Theorem 4.5. First, we check that P (µ) fits into Construc-
tion 3.1 with d0 = −1. By the normalizedness assumption, we have l1 ≤ l2. Lemma 4.3 delivers
1 ≤ d1 ≤ l1 and gcd(li, di) = 1. Moreover,
−1 +
d1
l1
+
d2
l2
=
−l1l2 + k22
l1l2
= − k21
l1l2
< 0 <
k22
l1l2
=
d1
l1
+
d2
l2
.
Thus, P = P (µ) is as wanted and Proposition 3.2 says that X = X(P ) is a quasismooth,
rational, projective C∗-surface. For the divisor class group and the Cox ring of X, we need to
have an exact sequence
0 // Z3 P ∗
// Z4 Q // Z // 0.
The transpose matrix P ∗ is injective and, as l1 and l2 are coprime, the columns of P ∗ generate
a primitive sublattice of Z4 and thus we have a torsion free cokernel. Moreover, we directly
check that Q · P ∗ = 0 holds with
Q =
[
k21 k22 l2 l1
]
.
Since k21 and k22 are coprime, Q maps onto Z. Consequently, [10, Propositions 4.13 and 4.16]
yield Cl(X) = Z and the desired presentation of R(X). Moreover, we can identify the ambient
toric variety as
Z(µ) = P(k21 ,k
2
2 ,l2,l1)
.
On Degenerations of the Projective Plane 13
The hyperbolic fixed point x0 is smooth by [10, Proposition 5.1]. Proposition 3.2 and Lemmas 4.2
and 4.3 give us cl(x1) and cl(x2). For the local Gorenstein indices, consider the following linear
forms
u1 =
[
d2 − d1
k22
,
d1 − d2
k22
,
3k1
k2
]
, u2 =
[
d1 − d2 + l2
k21
,
d2 − d1 + l1
k21
,−3k2
k1
]
.
Using the identities k21 = l1l2 − l2d1 − l1d2 and k22 = l2d1 + l1d2 just established, one directly
checks that the linear forms u1, u2 evaluate on the columns v1, v2, v3, v4 of P as follows:
⟨u1, v1⟩ = 0, ⟨u1, v3⟩ = 1, ⟨u1, v4⟩ = 1,
⟨u2, v2⟩ = 0, ⟨u2, v3⟩ = 1, ⟨u2, v4⟩ = 1.
We can conclude that k2u1 and k1u2 are primitive integral vectors and [12, Proposition 8.9] yields
ι(x1) = k2 and ι(x2) = k1. As X is quasismooth, x1, x2 are toric singularities and Lemma 2.10
gives us their singularity type. We obtain
K2
X =
cl(x0)
cl(x1) cl(x2)
(l1 + l2)
2 =
(
l1 + l2
k1k2
)2
= 9
for the canonical self intersection number, using Proposition 3.2 and Lemma 4.2 for the last
equation. The desired coverings from X onto the weighted projective planes are those from
Construction 3.4 followed by the obvious ones:
P(k21 ,k
2
2 ,li)
→ P(k21 ,k
2
2 ,l
2
i )
, [z1, z2, z3] 7→
[
z1, z2, z
li
3
]
.
Finally, the families Xi → C are provided by Construction 3.5 and their properties claimed in
the assertion are guaranteed by Proposition 3.6. ■
5 Proof of the main results
Here we prove the main results of this note, Theorems 5.4 and 5.5. A first observation is that log
terminal C∗-surfaces of Picard number one and canonical self intersection nine are quasismooth.
Proposition 5.1. Let X be a log terminal, rational, projective C∗-surface of Picard number one
with K2
X = 9. Then X is quasismooth.
Proof. We use the description of rational, projective C∗-surfaces of Picard number one via
defining matrices P provided by [10, Construction 4.2, Proposition 4.5]. Log terminality of X
imposes strong conditions on the upper rows of P , see [10, Proposition 5.9].
The non-quasismooth log terminal surface singularities are those of the types D or E, where
“type D” refers to the platonic triples (2, 2, n) and “type E” gathers the platonic triples (2, 3, 3),
(2, 3, 4) and (2, 3, 5) in Brieskorn’s result [5, Satz 2.10]; see also [12, Proposition 8.14]. The first
tuple of possible upper entries from [10, Proposition 5.9] is (1, y, 2, 2) and the associated defining
matrix is of the form
P =
−1 −l0 2 0
−1 −l0 0 2
0 d0 d1 1
, gcd(l0, d0) = gcd(2, d1) = 1,
d0 < 0, l0 ≥ 2,
d1
2
+
1
2
> 0,
d0
l0
+
d1
2
+
1
2
< 0.
14 J. Hausen, K. Király and M. Wrobel
The resulting C∗-surface X, built as in Construction 3.1, has [1, 0, 0, 0] ∈ X as singular point of
type D. From [12, Proposition 7.9], we infer
K2
X =
1
d1
2 + 1
2
−
2− l0 − 1
l0
d0
− 1
l20
(
d0
l0
+ d1
2 + 1
2
)
=
2
d1 + 1
+
(l0 − 1)2
d0l0
− 2
l20(d1l0 + 2d0 + l0)
.
Since the second right-hand side term is negative, we have K2
X < 4 in this case. The next tuple
of upper entries is (1, 2, y, 2), which leads to the defining matrix
P =
−1 −2 l1 0
−1 −2 0 2
0 d0 d1 1
, gcd(2, d0) = gcd(l1, d1) = 1,
d0 < 0, l1 ≥ 2,
d1
l1
+
1
2
> 0,
d0
2
+
d1
l1
+
1
2
< 0.
As before, the resulting C∗-surface X has [1, 0, 0, 0] ∈ X as a singular point of type D. This
time, the canonical self intersection is the following:
K2
X =
(
1
l1
+ 1
2
)2
d1
l1
+ 1
2
+
1
2d0
− 1
l21
(
d0 +
d1
l1
+ 1
2
) .
Note that the first term is not bounded from above, so we can’t estimate K2
X . Instead we observe
that K2
X = 9 is a quadratic equation in d0 with discriminant
∆ = 36d21 + 36d1l1 + 9l21 − 8d1 − 4l1.
The key observations are that ∆ factors as ∆ = a(9a−4) for a = 2d1+ l1 > 0 and that a(9a−4)
never is a square. Thus we can conclude K2
X ̸= 9.
These two sample cases show all the arguments for the remaining ones: either we directly
estimate K2
X < 9 or we can show that K2
X = 9 admits no integral solution. Concerning the
latter, (1, 2, y, 2) is in fact the most tricky case. ■
Remark 5.2. Log terminality is essential in Proposition 5.1. For instance, consider the rational,
projective C∗-surfaces X built from the matrices
P =
−1 −l2 + 4l − 1 l2 − l + 1 0
−1 −l2 + 4l − 1 0 2
0 −1 l−l2
2 1
, l ≥ 5,
exactly as in Construction 3.1. Then ρ(X) = 1 and K2
X = 9 by [12, Proposition 6.9]. From
[12, Corollary 8.12], we infer that X is not log terminal, thus not quasismooth.
Definition 5.3. By aMarkov C∗-surface we mean a C∗-surface isomorphic to someX(µ) arising
from Construction 4.4.
Theorem 5.4. Let X be a non-toric, log terminal, rational C∗-surface of Picard number one.
Then the following statements are equivalent:
(i) X is a non-toric Markov C∗-surface.
(ii) We have K2
X = 9.
On Degenerations of the Projective Plane 15
Moreover, if X satisfies (i) or (ii), then it is determined up to isomorphy by the local class group
orders cl(x1), cl(x2) of its elliptic fixed points.
Proof. The implication “(i) ⇒ (ii)” holds by Theorem 4.5. Let (ii) be valid. Then Proposi-
tion 5.1 shows that X is quasismooth. Thus, Proposition 3.3 allows us to assume X = X(P )
with
P =
−1 −1 l1 0
−1 −1 0 l2
0 d0 d1 d2
,
2 ≤ l1 ≤ l2, 1 ≤ d1 < l1, gcd(li, di) = 1, d0 +
d1
l1
+
d2
l2
< 0 <
d1
l1
+
d2
l2
.
Recall from Proposition 3.2 that the fake weight vector w(P ) = (w1, w2, w3, w4) of the matrix P
is given explicitly by
w(P ) = (−l1l2d0 − l2d1 − l1d2, l2d1 + l1d2,−d0l2,−d0l1) ∈ Z4
≥1.
We first show that d0 = −1 holds and that
(
w1, w2, l
2
i
)
is a squared Markov triple
(
k21, k
2
2, l
2
i
)
.
Consider the toric degenerations Z̃1 and Z̃2 of X as provided by Construction 3.5. Due to
Proposition 3.6, their fake weight vectors are w
(
P̃1
)
=
(
w1, w2,−l21d0
)
, w
(
P̃2
)
=
(
w1, w2,−l22d0
)
.
Moreover, this Proposition gives us K2
Z̃i
= K2
X = 9. Consequently, Proposition 2.7 provides us
with Markov triples (k1, k2, liδ0) such that w
(
P̃1
)
=
(
k21, k
2
2, l
2
1δ
2
0
)
, w
(
P̃2
)
=
(
k21, k
2
2, l
2
2δ
2
0
)
, where
d0 = −δ20 . To conclude the first step, we have to show δ0 = 1. Computing the anticanonical self
intersection of Z̃i according to Proposition 2.5, we obtain
9 =
(
w1 + w2 + l2i δ
2
0
)2
w1w2l2i δ
2
0
=
(
l1l2δ
2
0 + l2i δ
2
0
)2(
l2i δ
2
0 − w2
)
w2l2i δ
2
0
.
Consequently,
9
(
l4i δ
2
0w2 − l2iw
2
2
)
= δ20
(
l1l2 + l2i
)2
.
Thus, δ0 divides 3liw2. Since the entries of the squared Markov triple
(
w1, w2, l
2
i δ
2
0
)
are pairwise
coprime, we have gcd(l1, l2) = 1 and δ0 | 3li. Thus, δ0 = 1 or δ0 = 3. The latter is excluded,
because no Markov number is divisible by 3.
Having two members in common, the Markov triples (k1, k2, li), where i = 1, 2, are adjacent.
As seen above, we have k22 = l2d1+ l1d2. Thus, the uniqueness part of Lemma 4.3 shows that we
are in the setting of Construction 4.4. The equivalence of (i) and (ii) is proven. The supplement
follows directly from Theorem 4.5 and, again, the uniqueness statement of Lemma 4.3. ■
Theorem 5.5. Let X be a log terminal, rational, projective surface with a non-trivial torus
action. Then the following statements are equivalent:
(i) X is a degeneration of the projective plane.
(ii) We have ρ(X) = 1 and K2
X = 9.
(iii) X is a toric Markov surface or a Markov C∗-surface.
Proof. Assume that (i) holds. From [16, Theorem 4, Corollary 5], we infer ρ(X) = 1 and
K2
X = 9. If (ii) holds, then Proposition 2.7 and Theorem 5.4 yield (iii). Now, let (iii) be
valid. Then Proposition 2.9 and Theorem 4.5 ensure property (d) of Manetti’s Main Theo-
rem [16, p. 90], telling us that then X is a degeneration of the plane. ■
16 J. Hausen, K. Király and M. Wrobel
Remark 5.6. Note that also non-rational normal C∗-surfaces show up as degenerations of the
projective plane. To obtain a classical example, fix h ∈ C[T1, T2, T3] of degree d with V (h) ⊆ P2
smooth and consider X := V (T4S − h) ⊆ P(1, 1, 1, d)×C, where S is the coordinate on C. The
projection X → C to the last coordinate has fibers Xs
∼= P2 over s ∈ C∗ and the central fiber X0
is the cone over the smooth curve V (h) ⊆ P2, acted on by C∗ via t · [z] = [z1, z2, z3, tz4].
Acknowledgements
We would like to thank Yuchen Liu for fruitful discussions stimulating this work. Moreover, we
are grateful to Andrea Petracci for valuable information about existing articles on the algebraic
geometry around the Markov numbers. Finally, we are indebted to the referees for carefully going
through our manuscript, valuable suggestions and pointing to a gap in the proof of a former,
incorrect version of the second main theorem.
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1 Introduction
2 Toric Markov surfaces
3 Rational projective C*-surfaces
4 Markov C^*-surfaces
5 Proof of the main results
References
|
| id | nasplib_isofts_kiev_ua-123456789-213535 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:50:24Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| spelling | Hausen, Jürgen Király, Katharina Wrobel, Milena 2026-02-18T11:27:44Z 2025 On Degenerations of the Projective Plane. Jürgen Hausen, Katharina Király and Milena Wrobel. SIGMA 21 (2025), 042, 16 pages 1815-0659 2020 Mathematics Subject Classification: 14L30; 14J26; 14J10; 14D06 arXiv:2405.04862 https://nasplib.isofts.kiev.ua/handle/123456789/213535 https://doi.org/10.3842/SIGMA.2025.042 Complementing the results of Hacking and Prokhorov, we determine explicitly all log-terminal, rational, degenerations of the projective plane that admit a non-trivial torus action. We would like to thank Yuchen Liu for the fruitful discussions that stimulated this work. Moreover, we are grateful to Andrea Petracci for valuable information about existing articles on algebraic geometry around the Markov numbers. Finally, we are indebted to the referees for carefully going through our manuscript, for valuable suggestions, and for pointing out a gap in the proof of a former, incorrect version of the second main theorem. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Degenerations of the Projective Plane Article published earlier |
| spellingShingle | On Degenerations of the Projective Plane Hausen, Jürgen Király, Katharina Wrobel, Milena |
| title | On Degenerations of the Projective Plane |
| title_full | On Degenerations of the Projective Plane |
| title_fullStr | On Degenerations of the Projective Plane |
| title_full_unstemmed | On Degenerations of the Projective Plane |
| title_short | On Degenerations of the Projective Plane |
| title_sort | on degenerations of the projective plane |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213535 |
| work_keys_str_mv | AT hausenjurgen ondegenerationsoftheprojectiveplane AT kiralykatharina ondegenerationsoftheprojectiveplane AT wrobelmilena ondegenerationsoftheprojectiveplane |