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  • 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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 2 / 27

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

    N ∼= Zd lattice, NR := N ⊗Z R ∼= R d ,

    K ⊆ NR d-dimensional convex body.

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 3 / 27

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

    N ∼= Zd lattice, NR := N ⊗Z R ∼= R d ,

    K ⊆ NR d-dimensional convex body.

    Let i := | int(P) ∩ N |.

    i = 0:

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 3 / 27

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

    N ∼= Zd lattice, NR := N ⊗Z R ∼= R d ,

    K ⊆ NR d-dimensional convex body.

    Let i := | int(P) ∩ N |.

    i = 0: vol(K ) unbounded.

    i ≥ 1:

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 3 / 27

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

    N ∼= Zd lattice, NR := N ⊗Z R ∼= R d ,

    K ⊆ NR d-dimensional convex body.

    Let i := | int(P) ∩ N |.

    i = 0: vol(K ) unbounded.

    i ≥ 1: (Minkowski ’10) K centrally-symmetric w.r.t. 0 =⇒ vol(K ) ≤ 2d i+12 .

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 3 / 27

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

    N ∼= Zd lattice, NR := N ⊗Z R ∼= R d ,

    K ⊆ NR d-dimensional convex body.

    Let i := | int(P) ∩ N |.

    i = 0: vol(K ) unbounded.

    i ≥ 1: vol(K ) unbounded.

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 3 / 27

  • Unfortunately, central-symmetry is essential ...

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 4 / 27

  • Unfortunately, central-symmetry is essential ...

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 4 / 27

  • Unfortunately, central-symmetry is essential ...

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 4 / 27

  • Unfortunately, central-symmetry is essential ...

    not a lattice point

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 4 / 27

  • ... but not for lattice polytopes

    Let P ⊆ NR d-dimensional lattice polytope.

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 5 / 27

  • ... but not for lattice polytopes

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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 5 / 27

  • ... but not for lattice polytopes

    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).

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 5 / 27

  • ... but not for lattice polytopes

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

    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).

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 5 / 27

  • Sharp upper bounds?

    Theorem (Lagarias, Ziegler ’91)

    | int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ ikd((7(ik + 1))d2 d+1

    .

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 6 / 27

  • Sharp upper bounds?

    Theorem (Pikhurko ’01)

    | int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ (8dk)d (8k + 7)d2 2d+1

    i .

    Asymptotically, these doubly-exponential bounds are good.

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 6 / 27

  • Sharp upper bounds?

    Theorem (Pikhurko ’01)

    | int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ (8dk)d (8k + 7)d2 2d+1

    i .

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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 6 / 27

  • Sharp upper bounds?

    Theorem (Pikhurko ’01)

    | int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ (8dk)d (8k + 7)d2 2d+1

    i .

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

    Conjecture (Zaks, Perles, Wills ’82)

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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 6 / 27

  • Sharp upper bounds?

    Theorem (Pikhurko ’01)

    | int(P) ∩ (kN) | = i ≥ 1 =⇒ vol(P) ≤ (8dk)d (8k + 7)d2 2d+1

    i .

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

    Conjecture (Zaks, Perles, Wills ’82)

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

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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 6 / 27

  • II. Focus on lattice polygons

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 7 / 27

  • IP-polygons

    Let N ∼= Z2, P ⊆ NR lattice polygon.

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 8 / 27

  • IP-polygons

    Let N ∼= Z2, P ⊆ NR lattice polygon.

    Definition

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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 8 / 27

  • IP-polygons

    Let N ∼= Z2, P ⊆ NR lattice polygon.

    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 .

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 8 / 27

  • Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 9 / 27

  • F

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 9 / 27

  • F

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 9 / 27

  • F

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 9 / 27

  • F

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 9 / 27

  • F

    ℓF = 3

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 9 / 27

  • Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 10 / 27

  • F

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 10 / 27

  • F

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 10 / 27

  • F

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 10 / 27

  • F

    ℓF = 2

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 10 / 27

  • Three numerical invariants Let P be an IP-polygon.

    Definition

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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 11 / 27

  • Three numerical invariants Let P be an IP-polygon.

    Definition

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

    The maximal local index of P :

    mP := max(ℓF : F ∈ F(Q)).

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 11 / 27

  • Three numerical invariants Let P be an IP-polygon.

    Definition

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

    The maximal local index of P :

    mP := max(ℓF : F ∈ F(Q)).

    The order of P :

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

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 11 / 27

  • Three numerical invariants Let P be an IP-polygon.

    Definition

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

    The maximal local index of P :

    mP := max(ℓF : F ∈ F(Q)).

    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:

    Benjamin Nill (FU Berlin) Lattice polygons and toric log Del Pezzos 12 / 27

  • Example: mP = 3 &lt

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