Classification of Minimal Polygons with Specified
Singularity Content
Daniel Cavey(Joint work with Edwin Kutas)
University of Nottingham

2018

Fano Varieties
A projective manifold, or a variety X over C comes equipped witha notion of curvature. We use this to place X into one of threeclasses:General type

Calabi – Yau

Fano

Negative Curvature
Infinitely Many

Flat
Unknown

Positive Curvature
Finitely Many

It is natural to ask for a classification of smooth Fano varieties indimension n.

Fano Varieties

Smooth varieties (manifolds) are classified up to dimension 3.
Dimension
12

Number of Smooth
Fano Varieties
1103105

ExamplesP1P2 , P1 × P1 andblow-up of P2 in ≤ 8 points
P3 , etc.

The 2-dimensional classification is due to del Pezzo in 1880s andthe 3-dimensional classification due to Mori–Mukai in 1980s. We
do not know much in dimension greater or equal than 4.

Fano Polytopes
Let N be a lattice. A polytope P in NR = N ⊗ R is a set of theformnXoX
P=λu u : λu > 0 andλu = 1 ,u∈Su∈S

where S ⊂ NR is a finite set of points. A Fano polytope is afull-dimensional convex polytope such that
I the vertices V(P) ∈ N are all primitive.
I the origin lies in the strict interior of P.

When N is a rank-two lattice P is known as a Fano polygon. We
consider polytopes up to GL(N)-equivalence.
∼=

1 −1
via the change of basis
on N.
0 1

Fano Polytopes
The span of each face E of a Fano polygon P, by which we meanR≥0 E , defines a cone. We obtain a fan in NR corresponding to P.
←→This determines a toric del Pezzo surface XP .
←→ P2Many properties of XP have combinatorial analogues in the Fanopolygon P; examples include the singularities and the anticanonicaldegree (−KXP )2 .

Mirror Symmetry
Smooth Fano Variety

Mirror Symmetry

Laurent Polynomial
1
f = x + y + z + xyz

Newt(f )deformation

Toric Geometry

Toric Fano Variety

Fano Polytope

Mirror Symmetry

Given a Fano manifold X , mirror symmetry tells us (conjecturally)
how to associate a Laurent polynomial f which is said to be mirror
dual to X .
I Choice of mirror dual is not necessarily unique.
I Can transform fvia a mutation (a special birationaltransformation φ : (C× )n → (C× )n ) to obtain another
Laurent polynomial gI g also mirror dual to X .
I This notion of a mutation is captured when we move to theFano polytopes Newt(f ) and Newt(g ).

Mutation of Polygons
Let P ⊂ NR be a polygon, and E be an edge of P. Consider theprimitive inward pointing normal ωE ∈ M = Hom(N, Z) of thisedge. This vector can be thought of as a grading function on thepolygon P. For h ∈ Z, defineωh (P) = conv{v ∈ N ∩ P : ωE (v ) = h}.

E-1210Mutation of Polygons
Choose vE to be a primitive vector of the lattice N such thatωE (vE ) = 0. Set F = conv{0, vE }. For all h < 0, suppose that
there exists Gh ⊂ NR such that{v ∈ V(P) : ωE (v ) = h} ⊆ Gh + |h|F ⊆ ωh (P).
Then we define the mutation of P given by ωE , F and Gh to be[[mut(ωE ,F ) (P) = convGh ∪(ωh (P) + hF ) ⊂ NRh<0h≥0Choosing vE =(1,−1)
and G−1 ={(0,1)}E−−−−−−−−−−−−→21-1-10012

Mutation of Polygons

Lemma
Let E be an edge of a Fano polygon P with primitive inner normalvector ωE ∈ M. Then P admits a mutation with respect to ω if
and only if |E ∩ N| − 1 ≥ |ω(E )|.
We can use mutations to define an equivalence relation on the setof Fano polygons.

Cyclic Quotient Singularities
A quotient singularity R1 (a, b) is given by the action of µR on C2 by(x, y ) 7→ (a x, b y ) where  is an R th root of unity, and considering
Z = Spec(C[x, y ]µR ). The germ of the origin is the singularity.
For example consider a 21 (1, 1) singularity. Let G = Z/2Z and = −1. We consider the action of G on C2 described by−1 · (x, y ) = (−x, −y ).
We have
Z = Spec C[x, y ]G
= Spec C[x 2 , xy , y 2 ]= Spec C[u, v , w ]/(uw − v 2 )= V(uw − v 2 ) ⊂ C3 .

Cyclic Quotient Singularities
A quotient singularity R1 (a, b) is cyclic if:
gcd(R, a) = gcd(R, b) = 1.
Set:
k = gcd(a + b, R).
So:
a + b = kc and R = kr .
We can write the cyclic quotient singularity as
1(1, kc − 1).
krCyclic Quotient Singularities

We have two types of singularities here:
I A cyclic quotient singularity 1 (1, kc − 1) is a T-singularity if
kr
r | k.
I

A T-singularity admits a qG-smoothing.
(Kollar–Shepherd-Barron)I A cyclic quotient singularity 1 (1, kc − 1) is an R-singularity if
kr

k < r.
IAn R-singularity is rigid under qG-deformation.
(Kollar–Shepherd-Barron)

Singularity Content
Consider an arbitrary cyclic quotient singularity σ = kr1 (1, kc − 1).
By the Euclidean Algorithm there exists unique non-negative
integers n and k0 such that:k = nr + k0 .
If k0 > 0, then σ is qG-deformation equivalent to a k10 r (1, k0 c − 1)
cyclic quotient singularity. The residue of σ is given by:(∅, if k0 = 0res(σ) =1k0 r (1, k0 c − 1) , otherwise.
The singularity content of σ is given by the pair:SC(σ) = n, res(σ) .

Singularity Content
Consider a cone C corresponding to an edge E of a polygon:lh

By the Euclidean algorithm:
l = hn + r .
We divide C into separate sub-cones C0 , · · · , Cn where C1 , · · · , Cn
(known as T-cones) have lattice length h, and C0 has lattice lengthr and is known as an R-cone.

Singularity Content
Each cone corresponds to a cyclic quotient singularity of thecorresponding toric variety. This allows us to define the singularity
content of an edge E .
Let P ⊂ NR be a polygon.
I Label the edges of P in clockwise order E1 , · · · Ek .
I Each edge Ei corresponds to a cyclic quotient singularity σicorresponding to this cone.
I Let SC(Ei ) = ni , res(σi ) .
I We define the singularity content of P to be:

SC(P) =kX
i=1where B = {res(σ1 ), · · · , res(σk )}.

ni , B ,

Singularity ContentConsider the following polygon P.
E0E2
E1I E0 and E1 both give T-cones.
I E2 gives an R-cone representing a 1 (1, 1) singularity.
5
1So P has singularity content 2, { 5 (1, 1)}

Singularity content is an invariant under mutation!

Classification of Fano Polygons

Conjecture A: There exists a bijective correspondence between
the set of mutation-equivalence classes of Fano polygons and theset of qG-deformation equivalence classes of locally qG-rigid TG
del Pezzo surfaces with cyclic quotient singularities.
Recent results from Corti, Heuberger, Kasprzyk, Nill, Princecertainly support this conjecture.

Classification of Fano Polygons
I There are precisely 10 mutation-equivalence classes of Fanopolygons with singularity content (n, ∅). They are in bijective
correspondence with the 10 families of smooth del Pezzo
surfaces.
I There are precisely 26 qG-deformation families of del Pezzo
surfaces with m ≥ 1 singular points of type 31 (1, 1) admittinga toric degeneration. They are in bijective correspondencewith 26 mutation-equivalence classes of Fano polygons withsingularity content (n, {m × 31 (1, 1)}), m ≥ 1.
←→ P1 × P1

Classification of Fano Polygons

Kutas and C. have designed an efficient algorithm to build on thework of Kasprzyk, Nill and Prince.
Input: Singularity Content (n, B).
Output: Representative of every mutation-equivalence class ofFano polygons with singularity content (n, B).
Assuming Conjecture A holds, this is equivalent to a classificationof locally qG-rigid del Pezzo surfaces admitting a toricdegeneration.

Classification of Fano Polygons

As a corollary to this algorithm we have the followingclassifications:
I There are precisely 14 mutation-equivalence classes of Fanopolygons
with singularity contentn, {m1 × 13 (1, 1), m2 × 61 (1, 1)} with m1 ≥ 0, m2 > 0.
I There are precisely 12 mutation-equivalence classes of Fanopolygons with singularity content n, {m × 15 (1, 1)} with
m > 0.

Classification of Fano Polytopes

1.1

1.6

1.11

1.2

1.7

1.3

1.4

1.8

1.12

1.5

1.9

1.13

1.10

1.14

Figure 1: Minimal Representatives of Mutation-Equivalence Classes ofFanoPolygons with Singularity Contentn, {m1 × 13 (1, 1), m2 × 16 (1, 1)} where m1 ≥ 0, m2 > 0.