Lattice polytopes with a view toward algebraic...

Lattice polytopes &Toric geometry

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

Unimodular polytopes

EKM’s

High index

Lattice polytopes with a view towardalgebraic geometry

Benjamin Nill(Case Western Reserve University → Stockholm University)

MEGA Frankfurt 2013

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

What is Toric Geometry?

XΣ∼= XΣ′ ⇐⇒ affine lattice automorphism of N maps cones

in Σ to cones of Σ′

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Why Toric Geometry?

I Shopping mall for algebraic geometers looking for:I examplesI experimentsI databasesI conjecturesI intuitionI proof ideas

I Hammer for geometric combinatorialists to nail proofs(and to provide new viewpoints and interesting headaches!)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Why Toric Geometry?

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Why Toric Geometry?

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Toric Geometry is MEGA!

Toric varieties allow for

I explicit combinatorial description

I often complete classification results

I effective computation of invariants

This talk:

LATTICE POLYTOPES AND TORIC GEOMETRY

Focus on

Classifications and invariants of toric Fano varieties

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

This talk:

Focus on

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

This talk:

Focus on

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Polytopes enter the picture ...

DefinitionA polytope Q is called lattice polytope, if its vertices are latticepoints.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Different lattice polytopes may define same toric variety!

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Different lattice polytopes may define same toric variety!

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

... Fano polytopes ...

DefinitionA polytope Q is called Fano polytope, if

I the origin is in the interior

I every vertex is a primitive lattice point

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

... and toric Fano varieties!

DefinitionProjective X is Fano variety, if −KX is ample Q-Cartier divisor.

Correspondence:

toric Fano varieties XΣQ←→ Fano polytopes Q

Reality check: Is XΣ Fano?

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Correspondence:

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Correspondence:

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Correspondence:

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Correspondence:

Reality check: Is XΣ Fano? No!

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

The M-side of things

Let M be the dual lattice of N.

DefinitionThe dual polytope of a Fano polytope Q is defined as

Q∗ := {y : 〈x , y〉 ≥ −1 ∀ x ∈ Q}

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

DefinitionThe dual polytope of a Fano polytope Q is defined as

Q∗ := {y : 〈x , y〉 ≥ −1 ∀ x ∈ Q}

Vertices of Q∗ don’t have to be lattice points anymore!

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Correspondence

Q is reflexive ←→ XΣQis Gorenstein

(Q∗ is lattice polytope) ←→ (−KX is Cartier divisor)

In this case, H0(XΣQ,O(−KX )) has Q∗ ∩M as C-basis.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

What’s the point about Fano varieties?I of fundamental importance in the Minimal Model Program

I have beautiful algebro-geometric properties

I they are MEGA!

Conjecture (Alexeev-Batyrev-Borisov-· · · )

For fixed n ∈ N, ε > 0, there are finitely many families ofn-dimensional Fano varieties with ε-logterminal singularities.

Known to hold for

I Fano manifolds

I toric Fano varieties

Complete classification exists for Fano manifolds up to dimension 3

Let’s have a look at the toric case ...

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

I they are MEGA!

Known to hold for

I Fano manifolds

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

I they are MEGA!

Known to hold for

I Fano manifolds

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

I they are MEGA!

Known to hold for

I Fano manifolds

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

I they are MEGA!

Known to hold for

I Fano manifolds

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

THE TABLE of toric Fano varieties X of dim. n

n canonical Gorenstein smooth2 16 16 53 674,688 4,319 184 473,800,776 1245 8666 7,6227 72,2568 749,8929 8,229,721

due to Kasprzyk Kreuzer, Skarke Watanabe, Watanabe; Batyrev; Sato; Kreuzer, Nill; Øbro

−→ Graded Ring Database (GRDB)−→ Homepage of Maximilian Kreuzer (†)−→ Homepage of Andreas Paffenholz

Current development [Coates, Corti, Galkin, Golyshev, Kasprzyk]

Many (all?) Fano manifolds may be reconstructed from ‘mirror’Laurent polynomials found using these polytopes

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

THE TABLE of toric Fano varieties X of dim. n

n canonical Gorenstein smooth2 16 16 53 674,688 4,319 184 473,800,776 1245 8666 7,6227 72,2568 749,8929 8,229,721

−→ Graded Ring Database (GRDB)−→ Homepage of Maximilian Kreuzer (†)−→ Homepage of Andreas Paffenholz

Current development [Coates, Corti, Galkin, Golyshev, Kasprzyk]

Many (all?) Fano manifolds may be reconstructed from ‘mirror’Laurent polynomials found using these polytopes

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Excursion: Normal form algorithm

LOTS of Fano polytopes, how to check, if they are isomorphic?

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Solution: Normal form for list of vertices of lattice polytope Q:[1 0 0 −1 0 0 00 1 0 0 1 −1 00 0 1 1 −1 0 −1

]⇒ NF (Q) =

[1 0 1 0 −1 −1 00 1 −1 0 1 1 −10 0 0 1 −1 0 0

Keyword: Hermite-normal form (of permutations of the columns)

Max Kreuzer(1960-2010)

Implementation in PALP; MAGMA, SAGE [Grinis, Kasprzyk ’13]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

]⇒ NF (Q) =

[1 0 1 0 −1 −1 00 1 −1 0 1 1 −10 0 0 1 −1 0 0

]Keyword: Hermite-normal form (of permutations of the columns)

Implementation in PALP

; MAGMA, SAGE [Grinis, Kasprzyk ’13]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

]⇒ NF (Q) =

[1 0 1 0 −1 −1 00 1 −1 0 1 1 −10 0 0 1 −1 0 0

]Keyword: Hermite-normal form (of permutations of the columns)

Implementation in PALP; MAGMA, SAGE [Grinis, Kasprzyk ’13]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Two classification approaches in a nutshell

Growing [Kasprzyk]Step 1: Classifying minimal Fano’s (of certain type)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Two classification approaches in a nutshell

Growing [Kasprzyk]Step 1: Classifying minimal Fano’s (of certain type)

Step 2: Recursively add vertices

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Two classification approaches

Duality [Kreuzer, Skarke]

Q reflexive ⇔ Q∗ reflexive

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Two classification approaches

Duality [Kreuzer, Skarke]

Q reflexive ⇔ Q∗ reflexive

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Volume bounds on canonical polytopes

DefinitionA Fano polytope Q is called canonical polytope, if the origin isthe only lattice point in the interior.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Theorem [Hensley ’83; Lagarias, Ziegler ’91; Pikhurko ’01]

The volume of n-dimensional canonical polytopes is at most

14n 2n+1

Theorem [Averkov, Krumpelmann, Nill ’13]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Theorem [Hensley ’83; Lagarias, Ziegler ’91; Pikhurko ’01]

14n 2n+1

Theorem [Averkov, Krumpelmann, Nill ’13]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Sharp bounds on the maximal volume

I n = 2: 4.5, attained only by

I n = 3 [Kasprzyk ’08]: 12, attained by two canonical tetrahedra

I n ≥ 4: Conjecture [Zaks, Perles, Wills ’82; Hensley ’83]:

2(sn − 1)2/n!,

s1 = 2, s2 = 3, s3 = 7, s4 = 43, . . . , sk := s1 · · · sk−1+1.

Bound attained only for canonical simplex

Qn := conv(0, s1e1, . . . , sn−1en−1, 2(sn − 1)en)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

2(sn − 1)2/n!,

s1 = 2, s2 = 3, s3 = 7, s4 = 43, . . . , sk := s1 · · · sk−1+1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

2(sn − 1)2/n!,

s1 = 2, s2 = 3, s3 = 7, s4 = 43, . . . , sk := s1 · · · sk−1+1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Theorem [Averkov, Krumpelmann, Nill ’13]Conjecture holds for canonical simplices.

Applications

I [Kasprzyk ’13] 35,947 4-dimensional canonical simplices

I [Kasprzyk, N. ’13] Conjecture holds for reflexive polytopes

Challenging problems:

I Conjecture still open for canonical polytopes

I Sharp bounds on number of lattice points?!

I Proof sketch

I Translation into algebraic geometry

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Theorem [Averkov, Krumpelmann, Nill ’13]Conjecture holds for canonical simplices.

Applications

I [Kasprzyk ’13] 35,947 4-dimensional canonical simplices

I [Kasprzyk, N. ’13] Conjecture holds for reflexive polytopes

Challenging problems:

I Conjecture still open for canonical polytopes

I Sharp bounds on number of lattice points?!

I Proof sketch

I Translation into algebraic geometry

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Proof of volume bound

Let Q be simplex with origin only interior lattice point.Let β0 ≥ · · · ≥ βn > 0 be the barycentric coordinates of 0:

n∑i=0

βivi = 0,n∑

βi = 1

Lemma [Pikhurko ’01]

n! vol(Q) ≤ 1

β0 · · ·βn−1

Theorem [Averkov ’11]Let Q be lattice simplex. Then

for j = 0, . . . , n − 1 we have

β0 · · ·βj ≤ βj+1 + · · ·+ βn

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

n∑i=0

βivi = 0,n∑

βi = 1

n! vol(Q) ≤ 1

β0 · · ·βn−1

for j = 0, . . . , n − 1 we have

β0 · · ·βj ≤ βj+1 + · · ·+ βn

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

n∑i=0

βivi = 0,n∑

βi = 1

n! vol(Q) ≤ 1

β0 · · ·βn−1

for j = 0, . . . , n − 1 we have

β0 · · ·βj ≤ βj+1 + · · ·+ βn

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Lemma [N.’07]Let x0 ≥ · · · ≥ xn > 0 such that x0 + · · ·+ xn = 1 and

x0 · · · xj ≤ xj+1 + · · ·+ xn

for j = 0, . . . , n − 1.

If n ≥ 4, then

x0 · · · xn−1 ≥1

2(sn − 1)2

Proof finishQ canonical simplex =⇒

n! vol(Q) ≤ 1

β0 · · ·βn−1≤ 2(sn − 1)2

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

x0 · · · xj ≤ xj+1 + · · ·+ xn

for j = 0, . . . , n − 1. If n ≥ 4, then

x0 · · · xn−1 ≥1

2(sn − 1)2

n! vol(Q) ≤ 1

β0 · · ·βn−1≤ 2(sn − 1)2

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

x0 · · · xj ≤ xj+1 + · · ·+ xn

for j = 0, . . . , n − 1. If n ≥ 4, then

x0 · · · xn−1 ≥1

2(sn − 1)2

n! vol(Q) ≤ 1

β0 · · ·βn−1≤ 2(sn − 1)2

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Why does the sequence 2, 3, 7, 43, . . . appear?

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

≤ β1

− ε

+ β2 + β3 + β4

− ε)

≤ β2 + β3 + β4

= 1− 12 − β1

− ε)

· β2 ≤ β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

contradiction, since

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Assume

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

− ε

+ β2 + β3 + β4

− ε)

≤ β2 + β3 + β4

= 1− 12 − β1

− ε)

· β2 ≤ β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

β0 + ε + β1 − ε + β2 + β3 + β4 = 1

β0 + ε > β1 − ε > β2 > β3 > β4 > 0

β0 + ε < β1 − ε + β2 + β3 + β4

(β0 + ε) · (β1 − ε) ≤ β2 + β3 + β4

= 1− 12 − β1

(β0 + ε) · (β1 − ε) · β2 ≤ β3 + β4

(β0 + ε) · (β1 − ε) · β2 · β3 ≤ β4

minimal

(β0 + ε) · (β1 − ε) · β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

β0 + ε + β1 − ε + β2 + β3 + β4 = 1

β0 + ε > β1 − ε > β2 > β3 > β4 > 0

β0 + ε < β1 − ε + β2 + β3 + β4

(β0 + ε) · (β1 − ε) ≤ β2 + β3 + β4

= 1− 12 − β1

(β0 + ε) · (β1 − ε) · β2 ≤ β3 + β4

(β0 + ε) · (β1 − ε) · β2 · β3 ≤ β4

minimal

(β0 + ε) · (β1 − ε) · β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

− ε

+ β2 + β3 + β4

− ε)

≤ β2 + β3 + β4

= 1− 12 − β1

− ε)

· β2 ≤ β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

− ε

+ β2 + β3 + β4

− ε)

= β2 + β3 + β4

= 1− 12 − β1

− ε)

· β2 = β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

− ε

+ β2 + β3 + β4 = 1− β0

− ε)

= β2 + β3 + β4

= 1− 12 − β1

− ε)

· β2 = β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

− ε

+ β2 + β3 + β4

− ε)

= β2 + β3 + β4

= 1− 12 − β1

− ε)

· β2 = β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

− ε

+ β2 + β3 + β4

− ε)

= β2 + β3 + β4 = 1− 12 − β1

− ε)

· β2 = β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

− ε

+ β2 + β3 + β4 = 1

− ε

> β2 > β3 > β4 > 0

− ε

+ β2 + β3 + β4

− ε)

= β2 + β3 + β4

= 1− 12 − β1

− ε)

· β2 = β3 + β4

− ε)

· β2 · β3 ≤ β4

with minimal

− ε)

· β2 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

− ε

+ 17 + β3 + β4 = 1

− ε

> 17 > β3 > β4 > 0

− ε

+ 17 + β3 + β4

− ε)

= 17 + β3 + β4

= 1− 12 − β1

− ε)

· 17 = β3 + β4

− ε)

· 17 · β3 ≤ β4

with minimal

− ε)

· 17 · β3

(β0 + ε)(β1 − ε) = β0β1 − ε(β0 − β1)− ε2 < β0β1.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

What about algebraic geometry?

(−KX )n = n! vol(Q∗).

Proof arguments imply (with a slight tweak)

CorollaryLet X := XΣQ

toric Fano variety for canonical simplex Q.

n n = 2 n = 3 n ≥ 4

(−KX )n ≤ 9 72 2(sn − 1)2

equality P2 P(3, 1, 1, 1), P(

2(sn−1)s1

, . . . , 2(sd−1)sn−1

, 1, 1)

P(6, 4, 1, 1)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

(−KX )n = n! vol(Q∗).

n n = 2 n = 3 n ≥ 4

(−KX )n ≤ 9 72 2(sn − 1)2

equality P2 P(3, 1, 1, 1), P(

2(sn−1)s1

, . . . , 2(sd−1)sn−1

, 1, 1)

P(6, 4, 1, 1)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

(−KX )n = n! vol(Q∗).

n n = 2 n = 3 n ≥ 4

(−KX )n ≤ 9 72 2(sn − 1)2

equality P2 P(3, 1, 1, 1), P(

2(sn−1)s1

, . . . , 2(sd−1)sn−1

, 1, 1)

P(6, 4, 1, 1)

P(q0, . . . , qn) is weighted projective space, given as quotient of(C∗)n+1 via

(λ0, . . . , λn) · (x0, . . . , xn) := (λq0

0 x0, . . . , λqnn xn).

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Back on track: Toric Fano manifolds

DefinitionA Fano polytope Q is called unimodular polytope, if the vertexset of every facet of Q forms a lattice basis.

Correspondence:

toric Fano manifolds XΣQ←→ unimodular polytopes Q

Up to lattice isomorphisms only ONE unimodular simplex:

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Correspondence:

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Correspondence:

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Picard number of toric Fano manifolds

Picard number is most important invariant:

ρX = rank Pic(X ) = |Verts(Q)| − n

ρX = 1 ⇐⇒ X ∼= Pn

Sharp bounds on Picard numberρX ≤ 2n2 − n [Voskresenskii, Klyachko ’84]

ρX ≤ 2 + 2√

(n2 − 1)(2n − 1) [Debarre ’03]

ρX ≤ 2n OPTIMAL! [Casagrande ’05]

Mikkel Øbro’s breakthrough thesis ’07

I Short proof of Casagrande’s bound

I Introduced total ordering of unimodular polytopes

I Gave direct, fast enumeration algorithm

Still open: Sharp upper bound on χ(X ) = |facets(Q)|

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

ρX ≤ 2 + 2√

(n2 − 1)(2n − 1) [Debarre ’03]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

ρX ≤ 2 + 2√

(n2 − 1)(2n − 1) [Debarre ’03]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

ρX ≤ 2 + 2√

(n2 − 1)(2n − 1) [Debarre ’03]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

ρX ≤ 2 + 2√

(n2 − 1)(2n − 1) [Debarre ’03]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

ρX ≤ 2 + 2√

(n2 − 1)(2n − 1) [Debarre ’03]

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Toric Fano manifolds with large Picard number

Let S2 be P2 blown-up in two torus-invariant points:

X toric Fano manifold with ρX = 2n − k

I k=0: [N. 05; Casagrande ’07] n even, S2 × · · · × S2

I k=1: [Øbro ’07] 1 type, if n is even; 2 types, if n is odd

I k=2: [Assarf, Joswig, Paffenholz ’13] 11 types, if n ≥ 6 iseven; 5 types, if n ≥ 6 is odd

Conjecture [Assarf, Joswig, Paffenholz ’13]If k ≤ n/3, then

X ∼= X ′ × S2 × · · · × S2

for X ’ ≤ 3k + 1-dimensional toric Fano manifold.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

X ∼= X ′ × S2 × · · · × S2

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

X ∼= X ′ × S2 × · · · × S2

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Excursion: Einstein-Kahler-Manifolds (EKM)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Minkowski’s lattice point theoremIf K is n-dimensional convex body such that

1. origin is only interior lattice point,

2. K is centrally-symmetric with respect to origin,

then vol(K ) ≤ 2n.

Criterion [Wang, Zhu ’03]

2. holds for dual of unimodular polytope Q ⇐⇒ XΣQis EKM

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Ehrhart’s conjecture [’64]If K is n-dimensional convex body such that

2. origin is barycenter of K ,

then vol(K ) ≤ (n + 1)n/n!,

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

then vol(K ) ≤ (n + 1)n/n!, with equality if and only if

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

[N., Paffenholz ’09]:Found counterexample to conjecture on EKM’s in Øbro’s databasein dimension 6.

Theorem [Berman, Berndtsson ’12]

I Pn has largest anticanonical degree among toric EKM’s.

I Ehrhart’s conjecture holds for duals of Fano polytopes.

−→ [N., Paffenholz, ’12]:‘On the equality case in Ehrhart’s volume conjecture’

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

[N., Paffenholz ’09]:Found counterexample to conjecture on EKM’s in Øbro’s databasein dimension 6.

Theorem [Berman, Berndtsson ’12]

I Pn has largest anticanonical degree among toric EKM’s.

I Ehrhart’s conjecture holds for duals of Fano polytopes.

−→ [N., Paffenholz, ’12]:‘On the equality case in Ehrhart’s volume conjecture’

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

What about higher dimensions?

n canonical Gorenstein smooth2 16 16 53 674,688 4,319 184 473,800,776 1245 Guess: ∼ 1018 8666 7,6227 72,2568 749,8929 8,229,721

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

What about higher dimensions?

n canonical Gorenstein smooth2 16 16 53 674,688 4,319 184 473,800,776 1245 Guess: ∼ 1018 8666 7,6227 72,2568 749,8929 8,229,721

DefinitionIndex of Gorenstein toric Fano variety X :

iX := max(r : −KX/r Cartier divisor)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

The index

Index of XΣQis maximal r such that exists such P.

Example: iP2 = 3.

DefinitionP lattice polytope.

I P is called Gorenstein polytope of index r , if rP is reflexive(up to affine-lattice isomorphisms)

I P is called smooth Gorenstein polytope, if rP is dual ofunimodular polytope (up to affine-lattice isomorphisms)

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

The index

Example: iP2 = 3.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

The index

Example: iP2 = 3.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

A little step further into higher dimensions

Toric Fano manifolds with large index [Lorenz, N. ’13]Algorithm for smooth Gorenstein polytopes of large index yields

n \ix 1 2 3 4 5 6 7 8 9 10 11 12 13 142 3 1 13 15 2 14 118 4 1 15 853 11 1 16 7, 590 27 3 1 17 72, 167 83 4 1 18 749, 620 256 12 2 1 19 8, 228, 801 891 23 4 1 1

10 ∗ ∗ 63 6 2 1 111 ∗ ∗ ∗ 13 3 1 112 ∗ ∗ ∗ ∗ 6 2 1 113 ∗ ∗ ∗ ∗ ∗ 3 1 1

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

A pattern explained ...

Let iX > n+33 .

Mukai-Conjecture [Casagrande ’05]

ρX (iX − 1) ≤ n

impliesρX ≤ 2

[Kleinschmidt ’88] implies

I X ∼= Pn (iX = n + 1) or

I X ∼= P n2 × P n

2 (iX = n+22 , n even) or

X ∼= PPn+1−r (O(a1)⊕ · · · ⊕ O(at)⊕ Or−t),

for a1 + · · ·+ at = n + 2− 2r > 0.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Let iX > n+33 .

ρX (iX − 1) ≤ n

impliesρX ≤ 2

I X ∼= Pn (iX = n + 1) or

I X ∼= P n2 × P n

X ∼= PPn+1−r (O(a1)⊕ · · · ⊕ O(at)⊕ Or−t),

for a1 + · · ·+ at = n + 2− 2r > 0.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Let iX > n+33 .

ρX (iX − 1) ≤ n

impliesρX ≤ 2

I X ∼= Pn (iX = n + 1) or

I X ∼= P n2 × P n

X ∼= PPn+1−r (O(a1)⊕ · · · ⊕ O(at)⊕ Or−t),

for a1 + · · ·+ at = n + 2− 2r > 0.

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

... and there’s more to this than meets the eye

LetX ∼= PPn+1−r (O(a1)⊕ · · · ⊕ O(at)⊕ Or−t),

for a1 + · · ·+ at = n + 2− 2r .

Then the smooth Gorenstein polytope P is the

\blackbox{Cayley polytope associated to}

(n + 1− 2r)-dimensional

generic Calabi-Yau complete intersection Y

of r hypersurfaces of degrees

a1 + 1, . . . , at + 1, 1, . . . , 1

in Pn+1−r .

Isomorphism type of Y only depends on a1, . . . , at , so only finitelymany choices of such Y for fixed dim(Y ).

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

for a1 + · · ·+ at = n + 2− 2r .

a1 + 1, . . . , at + 1, 1, . . . , 1

in Pn+1−r .

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

for a1 + · · ·+ at = n + 2− 2r .

a1 + 1, . . . , at + 1, 1, . . . , 1

in Pn+1−r .

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

THE TABLE of the ‘Stringy Landscape’

Hodge numbers of (so far found) Calabi-Yau 3-folds[Ashmore, He ’11, Jurke ’13]

More than 90% found as Calabi-Yau complete intersections inGorenstein toric Fano varieties [Batyrev-Borisov construction].

Benjamin Nill

Fano polytopes

Normal form

Two approaches

Volume bounds

EKM’s

High index

Publications (with more references)I Nill, Kasprzyk: Fano polytopes, in: Strings, Gauge Fields,

and the Geometry Behind - The Legacy of MaximilianKreuzer, World Scientific, 2012

I Coates, Corti, Galkin, Golyshev, Kasprzyk: Fano SearchBlog, http://coates.ma.ic.ac.uk/fanosearch/

I Grinis, Kasprzyk: Normal forms of convex lattice polytopes,arXiv:1301.6641

I Nill: Volume and lattice points of reflexive simplices, DCG37, 301–320, 2007.

I Øbro: Classification of smooth Fano polytopes, thesis, 2007

I Assarf, Joswig, Paffenholz: Smooth Fano Polytopes WithMany Vertices, arXiv:1209.3186

I Nill, Paffenholz: On the equality case in Ehrhart’s volumeconjecture, arXiv:1205.1270

I Nill, Lorenz: On smooth Gorenstein polytopes,arXiv:1303.2138

Lattice polytopes with a view toward algebraic...

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