bounds on lattice polygons and the classification of toric log del...

Bounds on lattice polygons and the classification

of toric log Del Pezzo surfaces of small index

Benjamin Nill(joint work with Alexander Kasprzyk & Maximilian Kreuzer)

arXiv:0810.2207

AMS Meeting at SFSU 2009

Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 1 / 27

I. Generalities on lattice polytopes


Bounding(?!) the volume in terms of interior lattice points

N ∼= Zd lattice, NR := N ⊗Z R ∼= Rd ,

K ⊆ NR d-dimensional convex body.





Let i := | int(P) ∩ N |.

i = 0:





Let i := | int(P) ∩ N |.

i = 0: vol(K ) unbounded.

i ≥ 1:





Let i := | int(P) ∩ N |.


i ≥ 1: (Minkowski ’10) K centrally-symmetric w.r.t. 0

=⇒ vol(K ) ≤ 2d i+12 .





Let i := | int(P) ∩ N |.


i ≥ 1: vol(K ) unbounded.


Unfortunately, central-symmetry is essential ...


Unfortunately, central-symmetry is essential ...

not a lattice point


... but not for lattice polytopes

Let P ⊆ NR d-dimensional lattice polytope.



Let P ⊆ NR d-dimensional lattice polytope(P convex hull of finitely many lattice points).




Theorem (Hensley ’83)

There exists a function f depending only on d and i such that

| int(P) ∩ N | = i ≥ 1 =⇒ vol(P) ≤ f (d , i).




Theorem (Lagarias, Ziegler ’91)

There exists a function g depending only on d , i , k such that

| int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ g(d , i , k).


Sharp upper bounds?

Theorem (Lagarias, Ziegler ’91)

| int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ ikd((7(ik + 1))d2d+1.


Sharp upper bounds?

Theorem (Pikhurko ’01)

| int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ (8dk)d (8k + 7)d22d+1i .

Asymptotically, these doubly-exponential bounds are good.


Sharp upper bounds?



Asymptotically, these doubly-exponential bounds are good.However, sharp bounds only known for d = 1 and special cases for d = 2.


Sharp upper bounds?




Conjecture (Zaks, Perles, Wills ’82)

The correct order of the maximal volume is given by explicit latticesimplices.


Sharp upper bounds?




Conjecture (Zaks, Perles, Wills ’82)

The correct order of the maximal volume is given by explicit latticesimplices.

(N. ’07): They have maximal volume among reflexive simplices with d ≥ 3(special case of i = 1 = k).


II. Focus on lattice polygons


IP-polygons

Let N ∼= Z2, P ⊆ NR lattice polygon.


IP-polygons


Definition

P is called IP-polygon, if 0 ∈ int(P).


IP-polygons


Definition

P is called IP-polygon, if 0 ∈ int(P).

F(P) denotes set of facets (= edges).

Definition

Let F ∈ F(P). Then the local index ℓF is the lattice distance of 0 from F .


F


F

ℓF = 3


F


F

ℓF = 2


Three numerical invariantsLet P be an IP-polygon.

Definition

The index of P :ℓP := lcm(ℓF : F ∈ F(Q)).



Definition


The maximal local index of P :

mP := max(ℓF : F ∈ F(Q)).



Definition




The order of P :

oP := min(k ∈ N≥1 : int(P/k) ∩ N = {0}).



Definition




The order of P :

oP := min(k ∈ N≥1 : int(P/k) ∩ N = {0}).

Observation

oP ≤ mP ≤ ℓP .Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 11 / 27

Example:


Example: mP = 3 < ℓP = 2 · 3 = 6


Example: oP = 2 < mP = 3 < ℓP = 2 · 3 = 6

P/2


Goal: Bounding vol(P) in terms of oP

oP = min(k ∈ N≥1 : | int(P/k) ∩ N | = 1).



oP = min(k ∈ N≥1 : | int(P) ∩ (kN) | = 1).

Corollary to (Lagarias, Ziegler ’91)

P IP-polygon ⇒ vol(P) = O(o18Q ).



oP = min(k ∈ N≥1 : | int(P) ∩ (kN) | = 1).



[Pikhurko ’01] P IP-triangle ⇒ vol(P) = O(o5Q

).


Goal: Bounding vol(P) in terms of oP difficult!

oP = min(k ∈ N≥1 : | int(P) ∩ (kN) | = 1).



[Pikhurko ’01] P IP-triangle ⇒ vol(P) = O(o5Q

).



(0,−1)

(oP , 0)(−oP , 0)

(0, oP )



(0,−1)

(oP , 0)(−oP , 0)

(0, oP )

Conjectural correct asymptotics: oP(oP + 1)2 = O(o3P).



(0,−1)

(oP , 0)(−oP , 0)

(0, oP )

Conjectural correct asymptotics: oP(oP + 1)2 = O(o3P).

Still open!


Modest goal: Bounding vol(P) in terms of mP



Proposition (Kasprzyk, Kreuzer, N. 08)

P IP-polygon ⇒ vol(P) ≤ 2m3P + 4m2

P .





P .

Asymptotically correct: in example oP = mP = ℓP .


III. Motivation from toric geometry


LDP-polygons

Let P ⊆ NR be an IP-polygon.

Definition

P is LDP-polygon, if all vertices of P are primitive.


LDP-polygons


Definition


Let XP be the toric variety associated to the fan spanned by the faces of P .


LDP-polygons


Definition


Let XP be the toric variety associated to the fan spanned by the faces of P .

One-to-one correspondence

LDP-polygons P ←→ toric log Del Pezzo surfaces XP

Log Del Pezzo surfaces

(normal complex variety, log-terminal singularities,−KX ample Q-Cartier.)

studied intensively by Nikulin et.al.


The index - revisitedLet N = Z2, M = Hom(N, Z), 〈·, ·〉 : M ×N → Z.



Definition

For F ∈ F(P), we define ηF ∈ M as the primitive outer normal(〈ηF , x〉 = ℓF for all x ∈ F ).



Definition


Therefore, the dual polytope

P∗ = {y ∈ MR : 〈y , x〉 ≥ −1}

has ηF/ℓF as its vertices.



Definition



P∗ = {y ∈ MR : 〈y , x〉 ≥ −1}

has ηF/ℓF as its vertices. ℓP = 1 ⇐⇒ P∗ lattice polytope ⇐⇒ P reflexive.



Definition



P∗ = {y ∈ MR : 〈y , x〉 ≥ −1}


Observation

ℓP = min(k ∈ N≥1 : kP∗ lattice polytope)



Definition



P∗ = {y ∈ MR : 〈y , x〉 ≥ −1}


Observation

ℓP = min(k ∈ N≥1 : kP∗ lattice polytope)

= min(k ∈ N≥1 : k KXPCartier divisor) = ℓXP

.


Classification results

Boundedness

There exist only finitely many toric log Del Pezzo surfaces (up toisomorphisms) of given index ℓ.



Boundedness

There exist only finitely many toric log Del Pezzo surfaces (up toisomorphisms) of given index ℓ.

(Alexeev, Nikulin 06; Nakayama ’06):Complete lists for general log Del Pezzo surfaces with ℓ ≤ 2.


Classification resultsLDP-polygons P of index ℓP = 1 (⇔ oP = 1):



Proposition (Kasprzyk, Kreuzer, N.)

Letn(ℓ) := # toric log Del Pezzo surfaces with index ℓ.

Then

ℓ 1 2 3 4 5 6 7 8

n(ℓ) 16 30 99 91 250 379 429 307

ℓ 9 10 11 12 13 14 15 16

n(ℓ) 690 916 939 1279 1142 1545 4312 1030



Proposition (Kasprzyk, Kreuzer, N.)

Letn(ℓ) := # toric log Del Pezzo surfaces with index ℓ.

Then

ℓ 1 2 3 4 5 6 7 8

n(ℓ) 16 30 99 91 250 379 429 307

ℓ 9 10 11 12 13 14 15 16

n(ℓ) 690 916 939 1279 1142 1545 4312 1030

Note ’local minima’ for large prime powers.


IV. Sketch of proof of volume bound


Step 1: Enough to bound # of boundary lattice points

vol(P) = 12

∑F∈F(P) vol(conv(F ,0))



vol(P) = 12


= 12

∑F∈F(P) ℓF (|F ∩ N | − 1)



vol(P) = 12


= 12

∑F∈F(P) ℓF (|F ∩ N | − 1)

≤ mP

2 |∂P ∩ N |.


Step 2: Enough to bound # of lattice points on one facet

Øbro’s special facets: Let F be an edge of P such that

p :=∑

x∈∂P∩N

x ∈ R≥0F .



Øbro’s special facets: Let F be an edge of P such that

p :=∑

x∈∂P∩N

x ∈ R≥0F .

p


Step 2: Enough to bound # of lattice points on one facetØbro’s special facets: Let F be an edge of P such that

p :=∑

x∈∂P∩N

x ∈ R≥0F .

p

F



p

F



} }

p

F

G R



} }

p

F

G R

Number of boundary lattice points in G bounded by

2ℓF ≤ 2mP .



} }

p

F

G R

Number of boundary lattice points in R bounded by

∑

x∈R

(−〈ηF , x〉)



} }

p

F

G R


∑

x∈R

(−〈ηF , x〉) =∑

x 6∈R

〈ηF , x〉



} }

p

F

G R


∑

x∈R

(−〈ηF , x〉) =∑

x 6∈R

〈ηF , x〉 ≤ ℓF |F ∩ N |+ ℓF (ℓF − 1).



} }

p

F

G R


∑

x∈R

(−〈ηF , x〉) =∑

x 6∈R

〈ηF , x〉 ≤ mP |F ∩ N |+ mP(mP − 1).


Step 3: # of lattice points on F bounded by mP

Lemma

|F ∩ N | ≤ 2(ℓF + 1)oP + 1.


Step 3: # of lattice points on F bounded by mP

Lemma

|F ∩ N | ≤ 2(ℓF + 1)oP + 1.

Upper bound only attained (for ℓF ≤ oP) by

(0,−1)

(oP , 0)(−oP , 0)

(0, lF )


Improvements on the volume bound



P .




P LDP-polygon ⇒ vol(P) ≤ 2m3P + 4m2

P −mP .

For LDP-polygons:

No equality (with ℓF = oP = mP) in Lemma of Step 3!




P LDP-polygon and mP ≥ 3 prime⇒ vol(P) ≤ 2m3P + 2m2

P −mP .

For LDP-polygons:


Bound on |F ∩ N | can be improved, if ℓF = mF .





P −mP .

For LDP-polygons:



Proof uses particular property of LDP-polygons:





P −mP .

For LDP-polygons:



Proof uses particular property of LDP-polygons:

x ∈ int(R≥0(F )) ∩ N, 〈ηF , x〉 = 1 =⇒ P ⊆ F − R≥0x .


Finally, a simple(?) open question

Maximal IP-triangle is not an LDP-polygon:

(0,−1)

(mP , 0)(−mP , 0)

(0,mP ) (mP(mP + 1),mP )(−mP(mP + 1),mP )




(0,−1)

(mP , 0)(−mP , 0)

(0,mP ) (mP(mP + 1),mP )(−mP(mP + 1),mP )

Question:

Do LDP-polygons P exist such that vol(P) = O(ℓ3P) for ℓP >> 0 ?




(0,−1)

(mP , 0)(−mP , 0)

(0,mP ) (mP(mP + 1),mP )(−mP(mP + 1),mP )

Question:

Do LDP-polygons P exist such that vol(P) = O(ℓ3P) for ℓP >> 0 ?

Can construct LDP-triangles P with ℓP >> 0 and vol(P) = O(ℓ3/2P

).


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