Lattice Simplices of Bounded Degree Einstein Workshop on Lattice Polytopes 2016 Johannes Hofscheier Otto-von-Guericke-Universit¨ at Magdeburg joint with Akihiro Higashitani Tuesday 13 December 2016
Motivation LLS LSD Deg. 2 Appl. Basic Definitions Lattice polytope : P = conv( v 1 , . . . , v n ) ⊆ R d where v i ∈ Z d . � � � kP ∩ Z d � � � ( k ∈ Z ≥ 0 ). Ehrhart polynomial : L P ( k ) := h ∗ h ∗ -polynomial : � k ≥ 0 L P ( k ) t k = P ( t ) (1 − t ) d +1 for h ∗ P ∈ Z ≥ 0 [ t ] . degree of P : deg( P ) = deg h ∗ P ( t ) . Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Motivation Question A What are the polynomials which can be interpreted as the h ∗ -polynomial of a lattice polytope? Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Linear Polynomials P = [0 , a ] ⊆ R ( a ∈ Z ) L P (0) = 1 Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Linear Polynomials P = [0 , a ] ⊆ R ( a ∈ Z ) L P (0) = 1 , L P (1) = a + 1 Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Linear Polynomials P = [0 , a ] ⊆ R ( a ∈ Z ) L P (0) = 1 , L P (1) = a + 1 , L P (2) = 2 a + 1 Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Linear Polynomials P = [0 , a ] ⊆ R ( a ∈ Z ) L P (0) = 1 , L P (1) = a + 1 , L P (2) = 2 a + 1 , L P (3) = 3 a + 1 , . . . Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Linear Polynomials P = [0 , a ] ⊆ R ( a ∈ Z ) L P (0) = 1 , L P (1) = a + 1 , L P (2) = 2 a + 1 , L P (3) = 3 a + 1 , . . . ⇒ L P ( k ) = ka + 1 � � � ( ka + 1) t k = a kt k + t k = a t 1 (1 − t ) 2 + 1 − t k ≥ 0 k ≥ 0 k ≥ 0 = ( a − 1) t +1 ⇒ h ∗ P ( t ) = ( a − 1) t + 1 . (1 − t ) 2 Degree 1 All lin. polynomials 1 + at ∈ Z 2 ≥ 0 [ t ] can be interpreted as h ∗ -vectors. Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Quadratic Polynomials Degree 2 [Henk & Tagami, Treutlein] All polynomials 1 + a 1 t + a 2 t 2 ∈ Z ≥ 0 [ t ] with � 7 if a 2 = 1 a 1 ≤ 3 a 2 + 3 if a 2 ≥ 2 can be interpreted as h ∗ -polynomials. (Need polytopes up to dimension 3 .) Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Simpler Question Question B What are the h ∗ -polynomials coming from lattice simplices ? Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Question B: Degree 1 All lin. polynomials 1 + at ∈ Z ≥ 0 [ t ] can be interpreted as h ∗ -polynomials of lattice simplices. Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Question B: Degree 2 Interpret h ∗ = 1 + a 1 t + a 2 t 2 ∈ Z ≥ 0 [ t ] as point in the positive orthant a = ( a 1 , a 2 ) ∈ R 2 ≥ 0 . P ( t )=1+ a 1 t + a 2 t 2 for lattice triangle P ⊆ R 2 } M := { a ∈ Z 2 ≥ 0 : h ∗ Proposition[H.,Nill,Oeberg] There is a family ( σ i ) i ∈ Z ≥ 0 of affine cones σ i ⊆ R 2 ≥ 0 such that M ∩ σ ◦ i = ∅ for all i . Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Question B: Degree 2 250 σ 0 200 150 σ 3 h ∗ 2 σ 2 100 50 σ 1 0 0 50 100 150 200 250 h ∗ 1 Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Simplices of given degree Question C What are the simplices of a given degree (any dimension)? Idea: Question C ⇒ Question B. Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Idea from Alg. Geom. Alg. Geometer are interested in M g,n = { smooth proj. curves C of genus g with n distinct marked points } / ∼ Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Idea from Alg. Geom. Alg. Geometer are interested in M g,n = { smooth proj. curves C of genus g with n distinct marked points } / ∼ Completeness is a desirable property. For a (possible) compactification, relax the smoothness condition M g,n = { proj. connected nodal curves C of genus g with n distinct, nonsing. marked points with a stability condition } / ∼ Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. “Moduli” of Lattice Simplices We are interested in M 0 ,s = { lattice simplices ∆ of deg. s with marked vertices } / ∼ Here “ ∼ ” means: “up to affine unimodular transformations”. Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. “Moduli” of Lattice Simplices We are interested in M 0 ,s = { lattice simplices ∆ of deg. s with marked vertices } / ∼ Here “ ∼ ” means: “up to affine unimodular transformations”. “genus 0 ” as a simplex is homeomorphic to a sphere. Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. “Moduli” of Lattice Simplices We are interested in M 0 ,s = { lattice simplices ∆ of deg. s with marked vertices } / ∼ Here “ ∼ ” means: “up to affine unimodular transformations”. “genus 0 ” as a simplex is homeomorphic to a sphere. Question What could be M 0 ,s ? Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Recall Definition Usually: ∆ = conv( v 1 , . . . , v d +1 ) ⊆ R d d -dimensional lattice simplex if v i ∈ Z d (aff. indep.) Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Recall Definition Usually: ∆ = conv( v 1 , . . . , v d +1 ) ⊆ R d d -dimensional lattice simplex if v i ∈ Z d (aff. indep.) Lattice stays the same: Z d . Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Recall Definition Usually: ∆ = conv( v 1 , . . . , v d +1 ) ⊆ R d d -dimensional lattice simplex if v i ∈ Z d (aff. indep.) Lattice stays the same: Z d . Vertices change: v 1 , . . . , v d +1 . Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Recall Definition Usually: ∆ = conv( v 1 , . . . , v d +1 ) ⊆ R d d -dimensional lattice simplex if v i ∈ Z d (aff. indep.) Lattice stays the same: Z d . Vertices change: v 1 , . . . , v d +1 . Idea Let’s do it vice versa. Lattice changes: Λ . Vertices stay the same: What is a good choice? Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Lattice of a Lattice Simplex From now on: e 1 , . . . , e d ∈ R d standard basis vectors. All vertices should be “equivalent” � bad choice: 0 , e 1 , . . . , e d . Better choice: e 1 , . . . , e d +1 ∈ R d +1 (Dimension increases by 1 ). Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Lattice of a Lattice Simplex ∆ = conv( v 1 , . . . , v d +1 ) ⊆ R d d -dimensional lattice simplex. Cone over ∆ C = cone( { 1 } × ∆) ⊆ R d +1 . Exists unique linear iso. ϕ : R d +1 → R d +1 with (1 , v i ) �→ e i . ϕ (∆) = conv( e 1 , . . . , e d +1 ) Λ ∆ := ϕ ( Z d +1 ) ⊆ R d lattice Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Example ∆ = conv( 0 , 2 e 1 , 2 e 2 ) ⊆ R 2 � 1 − 1 / 2 − 1 / 2 � � 1 1 1 � 1 0 0 � � = 0 1 / 2 0 0 2 0 0 1 0 0 0 2 0 0 1 0 0 1 / 2 � �� � ϕ � − 1 � − 1 � � 2 2 Λ ∆ = Z 3 + Z + Z 1 0 1 2 0 2 Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Example ∆ = conv( 0 , 2 e 1 , 2 e 2 ) ⊆ R 2 � 1 − 1 / 2 − 1 / 2 � � 1 1 1 � 1 0 0 � � = 0 1 / 2 0 0 2 0 0 1 0 0 0 2 0 0 1 0 0 1 / 2 � �� � ϕ � − 1 � − 1 � � � � 2 2 Λ ∆ = Z 3 + Z − 1 / 2 1 / 2 0 + Z � short: 1 0 − 1 / 2 0 1 / 2 1 2 0 2 Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Properties of Λ ∆ Proposition ∆ = conv( v 1 , . . . , v d +1 ) ⊆ R d d -dim. lattice simplex. ϕ : R d +1 �→ R d +1 ; ϕ (1 , v i ) = e i . Λ ∆ = ϕ � Z d +1 � . 1 Z d +1 ⊆ Λ ∆ � � x ∈ R d +1 : � d +1 2 Λ ∆ ⊆ i =1 x i ∈ Z . Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Properties of Λ ∆ Proposition ∆ = conv( v 1 , . . . , v d +1 ) ⊆ R d d -dim. lattice simplex. ϕ : R d +1 �→ R d +1 ; ϕ (1 , v i ) = e i . Λ ∆ = ϕ � Z d +1 � . 1 Z d +1 ⊆ Λ ∆ � � x ∈ R d +1 : � d +1 2 Λ ∆ ⊆ i =1 x i ∈ Z . Idea of Proof: 1 v i ∈ Z d ⇒ Z d +1 ⊆ Λ ∆ ϕ − → 2 ∆ ϕ (∆) Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Correspondence Definition A lattice Λ ∆ ⊆ R d +1 we call simplicial if 1 Z d +1 ⊆ Λ ∆ . � � x ∈ R d +1 : � d +1 2 Λ ∆ ⊆ i =1 x i ∈ Z . Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Correspondence Definition A lattice Λ ∆ ⊆ R d +1 we call simplicial if 1 Z d +1 ⊆ Λ ∆ . � � x ∈ R d +1 : � d +1 2 Λ ∆ ⊆ i =1 x i ∈ Z . Theorem The assignment ∆ �→ Λ ∆ induces a bijection � d -dim. lattice sim- � � simplicial lattices � / ∼ 1 ↔ / ∼ 2 plices ∆ ⊆ R d Λ ⊆ R d +1 1 ∼ 1 = up to affine unimodular equivalence 2 ∼ 2 = up to permutation of the coordinates Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Chabauty Topology C ℓ := { C ⊆ X closed subgroup } � � x ∈ R d +1 : � d +1 ⊆ R d +1 closed subgp. X := i =1 x i ∈ Z Johannes Hofscheier LSD
Motivation LLS LSD Deg. 2 Appl. Chabauty Topology C ℓ := { C ⊆ X closed subgroup } � � x ∈ R d +1 : � d +1 ⊆ R d +1 closed subgp. X := i =1 x i ∈ Z Chabauty topology Basis of neighborhoods of C ∈ C ℓ N K,U ( C ) where K ⊆ X compact and U ⊆ X open with 0 ∈ U . Johannes Hofscheier LSD
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