Complete intersection constructions of toricdegenerations of Fano varieties
Thomas Prince
Imperial College London

Tuesday 2nd August, 2016

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

1 / 31

Overview

1

Motivation

2

The Construction

3

Algorithms for discovering Fano 4-folds

4

Data and examples

5

Mirror Symmetry

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

2 / 31Goal
In this talk I will to describe a construction which takes as input a reflexive
polytope P (or anti-canonically polarised toric Fano variety XP ) andreturns a collection of pairs (P̃, φ) consisting of a polytope P̃ and a map
φ : Ambient(P̃) → Zr which determines a toric complete intersection ofXP̃ isomorphic to XP .

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

3 / 31Goal
In this talk I will to describe a construction which takes as input a reflexive
polytope P (or anti-canonically polarised toric Fano variety XP ) andreturns a collection of pairs (P̃, φ) consisting of a polytope P̃ and a map
φ : Ambient(P̃) → Zr which determines a toric complete intersection ofXP̃ isomorphic to XP .

Applications
Generate possibly previously unknown Fano 4-folds from reflexive
polytopes by studying general elements of the linear system defining
XP in XP̃ .
Use Mirror Symmetry techniques to identify which of these Fano
manifolds appear on existing lists.
Provide a testing ground for extending results on Mirror Symmetry for(orbifold) del Pezzo surfaces to higher dimensions.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

3 / 31History and references I
This construction grew out of a project pursued jointly with Coates,
Kasprzyk, written up into two articles.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

4 / 31History and references I
This construction grew out of a project pursued jointly with Coates,
Kasprzyk, written up into two articles.

Four-dimensional Fano complete intersections
Here we searched all possible four-dimensional Fano complete intersections
X , a section of L = ⊕i Li in Y such that:
Y is smooth.
Y is Fano (−KY is ample).
The line bundles Li are nef.
P
− (KY + i Li ) is ample (so X is Fano by adjunction).
We classified such varieties X by their Quantum period sequence, andfound 527 Fano manifolds not appearing on a previous list.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

4 / 31History and references II

The process for computing the Quantum period sequence uses Mirror
Symmetry to replace the Fano manifold with a mirror-dual Laurent
polynomial, via a technique going back to Givental/Hori–Vafa. We asked
if there was a systematic way of inverting this process. That is, can werecover a Fano manifold from a candidate mirror?

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

5 / 31History and references II

The process for computing the Quantum period sequence uses Mirror
Symmetry to replace the Fano manifold with a mirror-dual Laurent
polynomial, via a technique going back to Givental/Hori–Vafa. We asked
if there was a systematic way of inverting this process. That is, can werecover a Fano manifold from a candidate mirror?

Laurent Inversion
In this article we describe such a technique, which we call Laurent
Inversion, and demonstrate its use by exhibiting a Fano manifold not in
the previous 527 or other list. This talk is essentially the sum of my recent
efforts to better understand this technique.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

5 / 31

Dual perspectives on polytopes
To fix conventions we start small.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

6 / 31

Dual perspectives on polytopes
To fix conventions we start small. An integral polytope P in a lattice N
can be recorded in two different (dual) ways:

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

6 / 31

Dual perspectives on polytopes
To fix conventions we start small. An integral polytope P in a lattice N
can be recorded in two different (dual) ways:
As the convex hull of its vertices (or integral points):
P = conv(v |v ∈ Verts(P)).
As the intersection of a (possibly redundant) collection of inequalities:
P = ∩i {m|hui , mi ≥ ai }.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

6 / 31

Dual perspectives on polytopes
To fix conventions we start small. An integral polytope P in a lattice N
can be recorded in two different (dual) ways:
As the convex hull of its vertices (or integral points):
P = conv(v |v ∈ Verts(P)).
As the intersection of a (possibly redundant) collection of inequalities:
P = ∩i {m|hui , mi ≥ ai }.

RemarkWriting NQ for N ⊗Z Q, given a reflexive polytope P ⊂ NQ we define
Q := P ◦ to be its polar polytope in M := hom(N, Z).

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

6 / 31

Dual perspectives on polytopes
To fix conventions we start small. An integral polytope P in a lattice N
can be recorded in two different (dual) ways:
As the convex hull of its vertices (or integral points):
P = conv(v |v ∈ Verts(P)).
As the intersection of a (possibly redundant) collection of inequalities:
P = ∩i {m|hui , mi ≥ ai }.

RemarkWriting NQ for N ⊗Z Q, given a reflexive polytope P ⊂ NQ we define
Q := P ◦ to be its polar polytope in M := hom(N, Z).

Conventions on varieties
For (Fano) P ⊂ NQ the toric variety XP is defined by its spanning fan, forQ ⊂ MQ the toric variety XQ is defined by its normal fan.
Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

6 / 31

Scaffolding: the N-side
The main idea of the construction is simple: rather than writing XP as aconvex hull of points, we take the convex hull of a collection of polytopes.
This collection forms additional data which allows us to build an ambient
space for XP .

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

7 / 31

Scaffolding: the N-side
The main idea of the construction is simple: rather than writing XP as aconvex hull of points, we take the convex hull of a collection of polytopes.
This collection forms additional data which allows us to build an ambient
space for XP .

Input dataFix a pair (P, T ) where P ⊂ NQ is a reflexive polytope and T is a tuple ofintegral polytopes T ∈ T we will refer to T as the type.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

7 / 31

Scaffolding: the N-side
The main idea of the construction is simple: rather than writing XP as aconvex hull of points, we take the convex hull of a collection of polytopes.
This collection forms additional data which allows us to build an ambient
space for XP .

Input dataFix a pair (P, T ) where P ⊂ NQ is a reflexive polytope and T is a tuple ofintegral polytopes T ∈ T we will refer to T as the type.

Definition
A scaffolding S is a finite collection of elements
S = (v , −a) ∈ Ñ := N ⊕ ZT such thatX
conv(v +aT T |S ∈ S) = P
T ∈T

where aT is the T component of a ∈ ZT .
Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

7 / 31

Scaffolding: the N-side
RemarkThe construction works for any type polytopes, but I will generally assume
that Verts(T ) = T ∩ N for all T . It is reasonable to restrict to the case T
is a simplex.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

8 / 31

Scaffolding: the N-side
RemarkThe construction works for any type polytopes, but I will generally assume
that Verts(T ) = T ∩ N for all T . It is reasonable to restrict to the case T
is a simplex.
We introduce some terminology for scaffoldings.
A strut is an element S ∈ S.
The rank of S is the number of struts.
The codimension of (S, T ) is the number of type polytopes.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

8 / 31

Scaffolding: the N-side
RemarkThe construction works for any type polytopes, but I will generally assume
that Verts(T ) = T ∩ N for all T . It is reasonable to restrict to the case T
is a simplex.
We introduce some terminology for scaffoldings.
A strut is an element S ∈ S.
The rank of S is the number of struts.
The codimension of (S, T ) is the number of type polytopes.

Rays of the ambient space
Define Σ̃P,S (1) to be the collection of rays in ÑQ generated by the integral
pointsS ∪ {(v , eT )|v ∈ Verts(T )}where eT is the indicator of T in ZT .
Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

8 / 31

Examples of scaffoldings

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

9 / 31

Examples of scaffoldings

The rays given by our construction
fit into the matrix

−1 0
0 1 0 0
ρ :=  0 −1 0 0 1 0
−1 −1 −1 1 1 1

Thomas Prince (Imperial College London)

The rays given by this codimension
2 example fit into

0 −1 1 0 0 0
 0 −1 0 0 1 0
ρ := 
−1 −1 1 1 0 0
−1 −1 0 0 1 1Complete Intersections

Tuesday 2nd August, 2016

9 / 31

Examples of scaffoldings

The rays given by our construction
fit into the matrix

−1 0
0 1 0 0
ρ :=  0 −1 0 0 1 0
−1 −1 −1 1 1 1

The rays given by this codimension
2 example fit into

0 −1 1 0 0 0
 0 −1 0 0 1 0
ρ := 
−1 −1 1 1 0 0
−1 −1 0 0 1 1

These examples will recover classical constructions of a degree 6 del Pezzo
surface. In these examples we also see the ray matrix is (nearly) in echleon
form, a property we see again later.
Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

9 / 31

Scaffolding: the M-side
To produce a fan (rather than just the rays) we construct a polytope Q̃ in
M̃ := hom(Ñ, Z) and define Σ̃P,S to be its normal fan.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201610 / 31

Scaffolding: the M-side
To produce a fan (rather than just the rays) we construct a polytope Q̃ in
M̃ := hom(Ñ, Z) and define Σ̃P,S to be its normal fan.

Defining Q̃
Recall the set of integral points we defined in Ñ
R := S ∪ {(v , eT )|v ∈ Verts(T )}Define a function φ : R → Z by setting(−1
φ(r ) =
0

r ∈S
r∈
/S

Let Q̃ := {(u, b) ∈ M̃ : hr , (u, b)i ≥ φ(r )}Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201610 / 31

Scaffolding: the M-side
Define Y := XQ̃ via the normal fan Σ̃ of Q̃. There are divisors DT definedfor each T ∈ T .

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201611 / 31

Scaffolding: the M-side
Define Y := XQ̃ via the normal fan Σ̃ of Q̃. There are divisors DT definedfor each T ∈ T .

The linear system
DefineDT :=Xh(v , eT )iv ∈Verts T

Observe that each standard basis vector eT? in M̃ is a functional on Ñ
evaluating to 1 on each ray of {(v , eT )|v ∈ Verts(T )} and non-positively
on all other rays.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201611 / 31

Scaffolding: the M-side
Define Y := XQ̃ via the normal fan Σ̃ of Q̃. There are divisors DT definedfor each T ∈ T .

The linear system
DefineDT :=Xh(v , eT )iv ∈Verts T

Observe that each standard basis vector eT? in M̃ is a functional on Ñ
evaluating to 1 on each ray of {(v , eT )|v ∈ Verts(T )} and non-positively
on all other rays.
P
This definition of Y ensures that −KY − T DT is an ample divisor. For
this to be a useful definition however there are a number of things tocheck.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201611 / 31

Scaffolding: the M-side
At this stage there a number of (elementary) technical lemmas required tomake sure the definitions make sense.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201612 / 31

Scaffolding: the M-side
At this stage there a number of (elementary) technical lemmas required tomake sure the definitions make sense.

Lemma
All rays generated by elements of R appear in Σ̃.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201612 / 31

Scaffolding: the M-side
At this stage there a number of (elementary) technical lemmas required tomake sure the definitions make sense.

Lemma
All rays generated by elements of R appear in Σ̃.

Lemma
The toric complete intersection defined by the collection {eT? : T ∈ T } isisomorphic to XP .

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201612 / 31

Scaffolding: the M-side
At this stage there a number of (elementary) technical lemmas required tomake sure the definitions make sense.

Lemma
All rays generated by elements of R appear in Σ̃.

Lemma
The toric complete intersection defined by the collection {eT? : T ∈ T } isisomorphic to XP .
We will omit all the details of the proofs, but the proofs follow from a
reinterpretation of the scaffolding on the M-side, and we will explain thisidea.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201612 / 31

PL functions and M-side struts
Let Q be the polar polytope of P, recall this is integral if P is reflexive.
For each T ∈ T there is a decomposition QT of Q given by the normal
fan of T .

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201613 / 31

PL functions and M-side struts
Let Q be the polar polytope of P, recall this is integral if P is reflexive.
For each T ∈ T there is a decomposition QT of Q given by the normal
fan of T . We define a map from MQ to M̃Q

The PL map ιDefine the map 
ι(u) = u, T 7→ −minv ∈Verts(T )hu, v i∈ M ⊕ ZT

Restricting ι to Q we have a PL embedding of Q into the boundary of Q̃.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201613 / 31

PL functions and M-side struts
Let Q be the polar polytope of P, recall this is integral if P is reflexive.
For each T ∈ T there is a decomposition QT of Q given by the normal
fan of T . We define a map from MQ to M̃Q

The PL map ιDefine the map 
ι(u) = u, T 7→ −minv ∈Verts(T )hu, v i∈ M ⊕ ZT

Restricting ι to Q we have a PL embedding of Q into the boundary of Q̃.

Supporting hyperplanes
Given a facet F of Q, we say an affine hyperplane H in M̃ supports F if
ι(F ) ⊂ H ∩ ι(Q) and H ∩ ι(int Q) = ∅.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201613 / 31

PL functions and M-side struts
Our main result establishes the correspondence between the N and M side
versions of the scaffolding.

Duality result
There is a bijection between affine hyperplanes H supporting F and strutsof type T meeting v ∈ Verts P, where F is the facet of Q dual to v .

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201614 / 31

Special cases
There are two obvious scaffoldings that can be found for any polytope.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201615 / 31

Special cases
There are two obvious scaffoldings that can be found for any polytope.

Point struts
Take any type T and form the trivial scaffolding S indexed by Verts P{(v , 0)|v ∈ Verts(P)} ∈ N ⊕ ZT
The ambient space is a non-compact toric variety in which the toric variety
XP degenerates to a collection of toric varieties, this is referred to as the
Mumford degeneration by Gross–Siebert and forms an important
ingredient in their procedure.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201615 / 31

Special cases
There are two obvious scaffoldings that can be found for any polytope.

Point struts
Take any type T and form the trivial scaffolding S indexed by Verts P{(v , 0)|v ∈ Verts(P)} ∈ N ⊕ ZT
The ambient space is a non-compact toric variety in which the toric variety
XP degenerates to a collection of toric varieties, this is referred to as the
Mumford degeneration by Gross–Siebert and forms an important
ingredient in their procedure.

Single strut
At the other extreme take T to be the singleton set consisting of P. Then
there is a scaffolding S consisting of the single element (0, −1) ∈ N ⊕ Z.
This ambient space is simply the projective cone over XP .
Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201615 / 31

Designing algorithms to find Fano 4-folds

There are 473,800,776 four dimensional reflexive polytopes. Ideally wewould design a comprehensive search of this list for all types of scaffolding
and recover a wide class of Fano 4-folds, and a good lower bound on the
total number of Fano 4-folds. However, there are obvious computationaldifficulties.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201616 / 31

Designing algorithms to find Fano 4-folds

There are 473,800,776 four dimensional reflexive polytopes. Ideally wewould design a comprehensive search of this list for all types of scaffolding
and recover a wide class of Fano 4-folds, and a good lower bound on the
total number of Fano 4-folds. However, there are obvious computationaldifficulties.
In this section we briefly illustrate some simple ways of identifying
polytopes which certain scaffoldings are likely to apply to and how tocheck these ambient spaces.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201616 / 31

Techniques for finding line segment scaffoldings
We pursue two ideas for finding novel scaffoldings on reflexive 4-topes.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201617 / 31

Techniques for finding line segment scaffoldings
We pursue two ideas for finding novel scaffoldings on reflexive 4-topes.

Singularities in a hyperplane
For each vertex v of Q := P ◦ say v is smooth if the tangent cone at v is.
Suppose that the codimension of the linear space spanned by vertices
which are not smooth is equal to one. Then there is a primitive dual
vector f ∈ N and we use the line segment [0, f ] to scaffold P.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201617 / 31

Techniques for finding line segment scaffoldings
We pursue two ideas for finding novel scaffoldings on reflexive 4-topes.

Singularities in a hyperplane
For each vertex v of Q := P ◦ say v is smooth if the tangent cone at v is.
Suppose that the codimension of the linear space spanned by vertices
which are not smooth is equal to one. Then there is a primitive dual
vector f ∈ N and we use the line segment [0, f ] to scaffold P.

Lines of nodes
Suppose there is a 3-cone in the spanning fan of P which is a conifold,that is, the cone over a square at height one. Geometrically there is a C?
of ODPs in the four-fold XP . Suppose all singularities of XP lie in theclosure of this C? . Scaffolding this with horizontal or vertical line segments
produces an ambient space which may smooth this line of nodes.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201617 / 31

Finding smoothings
We do not list an algorithm for checking whether these ambient spaces
define smoothings, but indicate its main features.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201618 / 31

Finding smoothings
We do not list an algorithm for checking whether these ambient spaces
define smoothings, but indicate its main features.

Quasismoothness
Recall that the toric variety Y = XQ̃ is a GIT quotient of a vector space V
by a complex torus T. We check smoothness of the complete intersection
in V \Z where Z is the vanishing locus of the irrelevant ideal. Even if wepass this test we need to study the interesections with the singularities of
the ambient.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201618 / 31

Finding smoothings
We do not list an algorithm for checking whether these ambient spaces
define smoothings, but indicate its main features.

Quasismoothness
Recall that the toric variety Y = XQ̃ is a GIT quotient of a vector space V
by a complex torus T. We check smoothness of the complete intersection
in V \Z where Z is the vanishing locus of the irrelevant ideal. Even if wepass this test we need to study the interesections with the singularities of
the ambient.

Avoiding singularities
We also check this in the cone V , restricting the equations to certain
strata and checking if the complete intersection equations give zeroes
which would give singularities.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201618 / 31

Full-dimensional scaffoldings
At the other extreme from line segment scaffolding we can study types Tsuch that each T is a standard simplex, the T lie in pairwise orthogonal
subspaces and their dimensions sum to dim P = dim N. We call such
scaffoldings full.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201619 / 31

Full-dimensional scaffoldings
At the other extreme from line segment scaffolding we can study types Tsuch that each T is a standard simplex, the T lie in pairwise orthogonal
subspaces and their dimensions sum to dim P = dim N. We call such
scaffoldings full.

Echleon form
For a scaffolding as above the ray matrix ρ is (after an easy change ofvariables) in echelon form. Since the kernel of the ray map is the GIT
weight data we can read this information from the scaffolding.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201619 / 31

Full-dimensional scaffoldings
At the other extreme from line segment scaffolding we can study types Tsuch that each T is a standard simplex, the T lie in pairwise orthogonal
subspaces and their dimensions sum to dim P = dim N. We call such
scaffoldings full.

Echleon form
For a scaffolding as above the ray matrix ρ is (after an easy change ofvariables) in echelon form. Since the kernel of the ray map is the GIT
weight data we can read this information from the scaffolding.
This was the original setting of this construction in CKP14. The ray
matrix exactly encodes the scaffolding, and since the rank of the ambient
space is the number of struts iterating over matrices encoding a small
number of struts ought to produce interesting examples.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201619 / 31

Reading off the weight matrix
It is easy to read off the weight matrix for a full scaffolding. Note the typeof a full scaffolding is equivalent to a partition of a basis B of N, BT .

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

20 / 31

Reading off the weight matrix
It is easy to read off the weight matrix for a full scaffolding. Note the typeof a full scaffolding is equivalent to a partition of a basis B of N, BT .

StrutsFor each strut (v , a) ∈ S define

vT :=

X

he ? , v ie, and bT := aT +e∈BT

Thomas Prince (Imperial College London)X

he ? , v ie∈BTComplete Intersections

Tuesday 2nd August, 2016

20 / 31

Reading off the weight matrix
It is easy to read off the weight matrix for a full scaffolding. Note the typeof a full scaffolding is equivalent to a partition of a basis B of N, BT .

StrutsFor each strut (v , a) ∈ S define

vT :=

X

he ? , v ie, and bT := aT +

e∈BT

X

he ? , v ie∈BTWeight matrix
Fix a strut S = (v , a) ∈ S. This defines a row RS of the weight matrix.
RS is the element ofM
ZS ⊕Zdim T +1 ∼= ZS∪B∪T
T ∈T

given by concatenating eS with vectors (−vT , bT ) for each T .
Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

20 / 31The weight matrix, example
The weight matrix has the general form
D = I|S| A
Where the columns of A are further divided into blocks of length dim T + 1
for each T ∈ T . As a daft example, we can consider precisely those Fano
triangles covered by one dilated and translated standard simplex.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

21 / 31The weight matrix, example
The weight matrix has the general form
D = I|S| A
Where the columns of A are further divided into blocks of length dim T + 1
for each T ∈ T . As a daft example, we can consider precisely those Fano
triangles covered by one dilated and translated standard simplex.

Weight data
The ambient space is always Y = P(1, a, b, c) for some a, b, c ∈ Z>0 .
Well-formedness is equivalent to the primitivity of the vertices of thetriangle. the linear system we consider is L = O(a + b + c) and thus wehave that −KY − L = O(1).

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 2016

21 / 31

Data and Examples

We’ll divide outcomes into surface, threefold and fourfold cases. Startingwith the ten del Pezzo surfaces we have the following easy result:

Proposition
All of the 10 del Pezzo surfaces can be recovered by scaffolding. Everyreflexive polytope admits a scaffolding producing a general section which isone of the eight del Pezzo surfaces with very ample anti-canonical bundle.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201622 / 31

Data and Examples

We’ll divide outcomes into surface, threefold and fourfold cases. Startingwith the ten del Pezzo surfaces we have the following easy result:

Proposition
All of the 10 del Pezzo surfaces can be recovered by scaffolding. Everyreflexive polytope admits a scaffolding producing a general section which isone of the eight del Pezzo surfaces with very ample anti-canonical bundle.
We clearly have no such result in 3 or 4 dimensions. In this section wereview the known constructions for these dimensions as well as for log delPezzo surfaces.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201622 / 31del Pezzo surfaces
Continuing our example from earlier we can verify that the scaffoldings on
the hexagon produce
P2 × P2 as a section of O(1, 1) ⊕ O(1, 1) or,
P1 × P1 × P1 as a section of O(1, 1, 1).

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201623 / 31del Pezzo surfaces
Continuing our example from earlier we can verify that the scaffoldings on
the hexagon produce
P2 × P2 as a section of O(1, 1) ⊕ O(1, 1) or,
P1 × P1 × P1 as a section of O(1, 1, 1).
We can also find dP5 as the blowup of the intersection of two quadrics.

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201623 / 31del Pezzo surfaces
Continuing our example from earlier we can verify that the scaffoldings on
the hexagon produce
P2 × P2 as a section of O(1, 1) ⊕ O(1, 1) or,
P1 × P1 × P1 as a section of O(1, 1, 1).
We can also find dP5 as the blowup of the intersection of two quadrics.

This scaffolding produces the ambient space P2 × P1 with linear system
O(2, 1). A general section is a graph over P2 except at the vanishing of apair of quadrics, at which the surface contains a line.
Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201623 / 31log del Pezzo surfaces
As part of the classification of del Pezzo surfaces with 31 (1, 1) singularities
we used scaffoldings to construct a number of models, summarized in thefollowing result.

Proposition
Of the 26 del Pezzo surfaces with at most 13 (1, 1) singularities which admita toric degeneration exactly 24 can be recovered via a scaffolding
construction.
For example, the log del Pezzo surface X2,5/3 (2 31 (1, 1) singularities,
degree 5/3) has a rank two ambient space (left).

Thomas Prince (Imperial College London)Complete Intersections

Tuesday 2nd August, 201624 / 31

Threefolds
There are 98 Fano threefolds with very ample anti-canonical bundle. In
Quantum Periods for 3-Dimensional Fano Manifolds
Coates–Corti–Galkin–Kasprzyk computed the Quantum Period of allsmooth Fano 3-folds. Of these, 86 are either smooth toric or havecomplete intersection constructions. All of these are examples of (possibly
trivial) scaffoldings.
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Threefold example
For example, consider the blowup of P3 in the intersection of two cubics.
We can write this as a section of O(1, 3) in P1 × P3 .

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Threefold example
For example, consider the blowup of P3 in the intersection of two cubics.
We can write this as a section of O(1, 3) in P1 × P3 .


1 0 0 0 0 1
D :=0 1 1 1 1 0
Where the order indicates the identity block. We can read off the struts as
v = (0, 0, 0), a = 1,v = (−1, −1, −1), a = 3.

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Threefold example
For example, consider the blowup of P3 in the intersection of two cubics.
We can write this as a section of O(1, 3) in P1 × P3 .


1 0 0 0 0 1
D :=0 1 1 1 1 0
Where the order indicates the identity block. We can read off the struts as
v = (0, 0, 0), a = 1,v = (−1, −1, −1), a = 3.
This produces a scaffolding by a pair of tetrahedra.

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Fourfolds: CKP14In CKP14 we found 527 examples of Fano index 1 Fano 4-folds which were
not smooth toric varieties or products.

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Fourfolds: CKP14In CKP14 we found 527 examples of Fano index 1 Fano 4-folds which were
not smooth toric varieties or products.

Scaffoldings for 527 Fano fourfolds
All the 527 Fano 4-folds found as in CKP14 are examples of thescaffolding
P construction. This follows from the fact that we insist that
−KY − i Li is ample for our complete intersections in that work.

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Fourfolds: CKP14In CKP14 we found 527 examples of Fano index 1 Fano 4-folds which were
not smooth toric varieties or products.

Scaffoldings for 527 Fano fourfolds
All the 527 Fano 4-folds found as in CKP14 are examples of thescaffolding
P construction. This follows from the fact that we insist that
−KY − i Li is ample for our complete intersections in that work.
Of course, for the smooth toric Fano 4-folds the trivial scaffolding suffices
to reproduce the variety.

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Fourfolds: generating new examples
Applying the algorithms mentioned above to the first 5000 polytopes in
the Kreuzer–Skarke list we recover an additional 19 Fano smooth varieties
which are different from the ones previously recovered. There are anumber of ways we can find 4-folds not in the list 527 from CKP14.

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Fourfolds: generating new examples
Applying the algorithms mentioned above to the first 5000 polytopes in
the Kreuzer–Skarke list we recover an additional 19 Fano smooth varieties
which are different from the ones previously recovered. There are anumber of ways we can find 4-folds not in the list 527 from CKP14.
Y is notP
Q-factorial ambient. Alternatively, resolving Y ,−KY − i Li becomes strictly nef.
Y is not Fano. The scaffolding construction will also produce Y notsemi-Fano.
Y has orbifold (simplicial) singularities.

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Fourfolds: generating new examples
Applying the algorithms mentioned above to the first 5000 polytopes in
the Kreuzer–Skarke list we recover an additional 19 Fano smooth varieties
which are different from the ones previously recovered. There are anumber of ways we can find 4-folds not in the list 527 from CKP14.
Y is notP
Q-factorial ambient. Alternatively, resolving Y ,−KY − i Li becomes strictly nef.
Y is not Fano. The scaffolding construction will also produce Y notsemi-Fano.
Y has orbifold (simplicial) singularities.

Lines of nodes
We find 936 polytopes in the first 5000 of the reflexive 4-topes whose
singular set is a line of ODPs. Of these the scaffolding construction
produces a smoothing in 162 of these cases. Of these 15 are ‘new’.
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Laurent Polynomials
Mirror Symmetry appears in a central, though rather superficial way in ourconstruction. In particular, from a given scaffolding S on P one canconstruct a Laurent polynomial fS , whose Newton polytope is precisely P.

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Laurent Polynomials
Mirror Symmetry appears in a central, though rather superficial way in ourconstruction. In particular, from a given scaffolding S on P one canconstruct a Laurent polynomial fS , whose Newton polytope is precisely P.

Laurent polynomial
For each strut S ∈ S, define the strut polynomial
!aT
Y X
v
m
fS := z
z
T ∈T

m∈T

we then define a Laurent polynomial fS by summing over these,
X
fS :=
fS
S∈S

By construction Newt(fS ) = P.
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Laurent Polynomials

In fact, this is the mirror-dual to X as a complete intersection given by
Givental/Hori–Vafa, indeed, by construction we know that.

Proposition
The Hori–Vafa construction applied to the toric variety Y := XP̃ and linear
system L outputs an affine toric variety X 0 and superpotential W : X 0 → C.
There is a birational map φ from (C? )n to X 0 such that φ? W = fS .

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Laurent Polynomials

In fact, this is the mirror-dual to X as a complete intersection given by
Givental/Hori–Vafa, indeed, by construction we know that.

Proposition
The Hori–Vafa construction applied to the toric variety Y := XP̃ and linear
system L outputs an affine toric variety X 0 and superpotential W : X 0 → C.
There is a birational map φ from (C? )n to X 0 such that φ? W = fS .
Of course this proposition only makes sense when a Mirror Theorem canbe applied,
P so Y should be an orbifold and care should be taken when
−KY − i Li is not ample.

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Tuesday 2nd August, 201630 / 31The End

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