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Lattice polytopes Algebraic, geometric and combinatorial aspects

Winfried Bruns

FB Mathematik/Informatik Universität Osnabrück

wbruns@uos.de

Sedano, March 2007

Winfried Bruns Lattice polytopes

Sources

W. B. and J. Gubeladze, Polytopes, rings, and K -theory, Springer 2008 (?) Commutative algebra and combinatorics W.B. and J. Herzog, Cohen-Macaulay rings, Cambridge University Press 1998 (rev. ed.) E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer 2005 R. P. Stanley, Combinatorics and commutative algebra, Birkhäuser 1996 (2nd ed.) Toric varieties G. Ewald, Combinatorial convexity and algebraic geometry, Springer 1996 T. Oda, Convex bodies and algebraic geometry (An introduction to the theory of toric varieties), Springer 1988 W. Fulton, Introduction to toric varieties,Princeton University Press 1993

Winfried Bruns Lattice polytopes

Lecture 1

Basic notions

Winfried Bruns Lattice polytopes

Polyhedral geometry

Definition

A polyhedron is the intersection of finitely many closed affine halfspaces.

A polytope is a bounded polyhedron.

A cone is the intersection of finitely many linear halfspaces.

A closed affine halfspace is a set

HC̨ D fx 2 Rd W ˛.x/ � 0g where ˛ is an affine form, i. e. a polynomial function of degree 1. It is linear if ˛ is a linear form. Its bounding hyperplane is

H˛ D fx 2 Rd W ˛.x/ D 0g:

Winfried Bruns Lattice polytopes

0

A polyhedron, a polytope and a cone

The dimension of P is dim aff.P/.

Winfried Bruns Lattice polytopes

Definition

A face of P is the intersection P \ H where H is a support hyperplane, i. e. P � HC and P 6� H. A facet is a maximal face. A vertex is a face of dim 0.

P;; improper faces.

For full-dimensional cones the (essential) support hyperplanes Hi D fx W �i.x/ D 0g are unique:

Proposition

Let P � Rd , dim P D d. If the representation P D HC1 \ � � � \ HCs is irredundant, then the hyperplanes Hi are uniquely determined (up to enumeration). Equivalently, the affine forms ˛i are unique up to positive scalar factors.

Winfried Bruns Lattice polytopes

Finite generation of cones

Theorem (Minkowski-Weyl)

Let C ¤ ; be a subset of Rm. Then the following are equivalent: there exist finitely many elements y1; : : : ; yn 2 Rm such that C D RCy1 C � � � C RCyn; there exist finitely many linear forms �1; : : : ; �s such that C is the intersection of the half-spaces HCi D

˚ x W �i.x/ � 0

� .

A simplicial cone is generated by linearly independent vectors.

Corollary

P is a polytope” P D conv.x1; : : : ; xm/ One considers the cone over P (see below) and applies the theorem.

A simplex is the convex hull of a affinely independent set.

Winfried Bruns Lattice polytopes

Definition

An (embedded) polyhedral complex ˘ is a finite collection of polyhedra P � Rd such that

with P each face of P belongs to ˘ ,

P; Q 2 ˘ ) P \ Q is a face of P (and of Q)

˘ is a fan if it consists of cones.

A polytopal complex consists of polytopes.

A simplicial complex consists of simplices. A simplicial fan consists of simplicial cones.

Winfried Bruns Lattice polytopes

A subdivision of ˘ is a polyhedral complex ˘ 0 such that each face of ˘ is the union of faces of ˘ 0.

A triangulation of a polytopal complex is a subdivision into a simplicial complex.

A triangulation of a fan is a subdivision into a simplicial fan.

There are always enough triangulations:

Theorem

Let ˘ be a polytopal complex and X a finite subset of j˘ j DSP2˘ P such that vert.˘/ � X. Then there there exists a triangulation � of ˘ such that X D vert.�/.

An analogous theorem holds for fans.

Winfried Bruns Lattice polytopes

Rationality

Proposition

Let C be a cone. The generating elements y1; : : : ; yn can be chosen in Qm (or Zm) if and only if the ˛i can be chosen as linear forms with rational (or integral) coefficients.

Such cones are called rational. For them there is a unique choice of the ˛i satisfying

˛i.Zd/ � Z, 1 2 ˛i.Zd/.

With this standardization, we denote them by �i and call them support forms.

Winfried Bruns Lattice polytopes

Affine monoids and their algebras

Definition

An affine monoid M is (isomorphic to) a finitely generated submonoid of Zd for some d � 0, i. e.

M CM � M (M is a semigroup); 0 2 M (now M is a monoid); there exist x1; : : : ; xn 2 M such that M D ZCx1 C � � � CZCxn.

Often affine monoids are called affine semigroups.

gp.M/ D ZM is the group generated by M. gp.M/ Š Zr for some r D rank M D rank gp.M/.

Winfried Bruns Lattice polytopes

Let K be a field (can often be replaced by a commutative ring). Then we can form the monoid algebra

K ŒM� D M a2M

KX a; X aX b D X aCb

X a D the basis element representing a 2 M.

M � Zd (affine)) K ŒM� �K ŒZd � D K ŒX˙11 ; : : : ; X˙1d � is a (finitely generated) monomial subalgebra.

Sometimes a problem: additive notation in M versus multiplicative notation in K ŒM� (and exponential notation is often cumbersome).

Proposition

Let M be a monoid. 1 M is finitely generated” K ŒM� is a finitely generated

K -algebra. 2 M is an affine monoid” K ŒM� is an affine domain.

Winfried Bruns Lattice polytopes

Proposition

The Krull dimension of K ŒM� is given by

dim K ŒM� D rank M:

Proof. K ŒM� is an affine domain over K . Therefore

dim K ŒM� D trdeg QF.K ŒM�/ D trdeg QF.K Œgp.M/�/ D trdeg QF.K ŒZr �/ D r

where r D rank M.

Winfried Bruns Lattice polytopes

Sources for affine monoids (and their algebras) are

monoid theory,

ring theory,

invariant theory of torus actions,

enumerative theory of linear diophantine systems,

lattice polytopes and rational polyhedral cones,

coordinate rings of toric varieties,

initial algebras with respect to monomial orders.

Winfried Bruns Lattice polytopes

Lattice polytopes

Definition

The convex hull conv.x1; : : : ; xm/ of points xi 2 Zd is called a lattice polytope.

P

Winfried Bruns Lattice polytopes

Definition

The polytopal monoid associated with P is

M.P/ D ZC ˚ .x ; 1/ W x 2 P \Zd�:

P

C.P/

Definition

The cone over (an arbitrary polytope) P is

C.P/ D RC ˚ .x ; 1/ 2 RdC1 W x 2 P�:

Winfried Bruns Lattice polytopes

Definition

The polytopal algebra associated with P is the monoid algebra

K ŒP� D K ŒM.P/�:

K ŒP� has a natural ZC-grading in which the generators have degree 1.

Objects of algebraic geometry:

M affine monoid: Spec K ŒM� is an affine toric variety

P lattice polytope: Proj K ŒP� is a projective toric variety

Toric varieties are not necessarily normal (Oberwolfach convention January 2006).

Winfried Bruns Lattice polytopes

Affine charts for projective toric varieties

Proj K ŒP� is covered by the affine varieties K ŒMv � where Mv is a corner monoid of P: one has v 2 vert.P/ and

Mv D ZC ˚ x � v W x 2 P \Zd�:

Then

Proj K ŒP� smooth” K ŒMv � Š K ŒX1; : : : ; Xd �” Mv Š ZdC for all v 2 vert.P/.

Especially each vertex of P must be contained in exactly d edges.

Winfried Bruns Lattice polytopes

Toric ideals

Presentation of affine monoid algebras: Let R D K Œx1; : : : ; xn�. Then we have a presentation

� W K ŒX � D K ŒX1; : : : ; Xn�! K Œx1; : : : ; xn�; Xn 7! xn: Let I D Ker � and M D f�.X a/ W a 2 ZnCg

Theorem

The following are equivalent: 1 M is an affine monoid and R D K ŒM�; 2 I is prime, generated by binomials X a � X b, a; b 2 ZnC; 3 I D IK ŒX˙1� \ K ŒX �, I is generated by binomials X a � X b, and

U D fa � b W X a � X b 2 Ig is a direct summand of Zn. Definition

A prime ideal I as above is called a toric ideal.

Winfried Bruns Lattice polytopes

X a � X b 2 I ” a1x1 C � � � C anxn D b1x1 C � � � C bnxn (using additive notation in the monoid): The binomials in I represent the linear dependencies of the vectors generating M, and, in the polytopal case, the affine dependencies of the lattice points of P ( P

ai D P

bi in this case).

Algorithmic approach for the computation of the toric ideal I (M D ZCx1 C � � � CZCxn):

Compute the kernel Zc1 C � � � CZcn�r of � W Zn 7! gp.M/, ei 7! xi , r D rank M Saturate the ideal generated by X c

C i � X c�i , i D 1; : : : ; n � r

with respect to X1; : : : ; Xn.

ci D cCi � c�i , cCi ; c�i ZC.

Winfried Bruns Lattice polytopes

Examples

0 1 2

K ŒP� D K ŒY 0Z ; Y 1Z ; Y 2Z � Š K ŒX1; X2; X3�=.X1X3 � X 22 /

K ŒP� D K ŒY 01 Y 02 Z ; Y 01 Y 12 Z ; Y 11 Y 02 Z ; Y 11 Y 12 Z � Š K ŒX1; X2; X3; X4�=.X1X4 � X2X3/

K ŒP� D K