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Lattice polytopesAlgebraic, geometric and combinatorial aspects

Winfried Bruns

FB Mathematik/InformatikUniversitat Osnabruck

[email protected]

Sedano, March 2007

Winfried Bruns Lattice polytopes

Sources

W. B. and J. Gubeladze, Polytopes, rings, and K -theory, Springer2008 (?)Commutative algebra and combinatoricsW.B. and J. Herzog, Cohen-Macaulay rings, Cambridge UniversityPress 1998 (rev. ed.)E. Miller and B. Sturmfels, Combinatorial commutative algebra,Springer 2005R. P. Stanley, Combinatorics and commutative algebra, Birkhauser1996 (2nd ed.)Toric varietiesG. Ewald, Combinatorial convexity and algebraic geometry, Springer1996T. Oda, Convex bodies and algebraic geometry (An introduction tothe theory of toric varieties), Springer 1988W. Fulton, Introduction to toric varieties,Princeton University Press1993


Lecture 1

Basic notions


Polyhedral geometry

Definition

A polyhedron is the intersection of finitely many closed affinehalfspaces.

A polytope is a bounded polyhedron.

A cone is the intersection of finitely many linear halfspaces.

A closed affine halfspace is a set

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

H˛ D fx 2 Rd W ˛.x/ D 0g:


0

A polyhedron, a polytope and a cone

The dimension of P is dim aff.P/.


Definition

A face of P is the intersection P \ H where H is a supporthyperplane, 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 hyperplanesHi D fx W �i.x/ D 0g are unique:

Proposition

Let P � Rd , dim P D d. If the representation P D HC1 \ � � � \ HC

s isirredundant, then the hyperplanes Hi are uniquely determined (up toenumeration). Equivalently, the affine forms ˛i are unique up topositive scalar factors.


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 thatC D RCy1 C � � � C RCyn;

there exist finitely many linear forms �1; : : : ; �s such that C isthe intersection of the half-spaces HC

i 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.


Definition

An (embedded) polyhedral complex ˘ is a finite collection ofpolyhedra 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.


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 asimplicial 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 ofj˘ j DS

P2˘ P such that vert.˘/ � X. Then there there exists atriangulation � of ˘ such that X D vert.�/.

An analogous theorem holds for fans.


Rationality

Proposition

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

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

˛i.Zd/ � Z,

1 2 ˛i.Zd/.

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


Affine monoids and their algebras

Definition

An affine monoid M is (isomorphic to) a finitely generated submonoidof 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/.


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

K ŒM� DMa2M

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˙1

d � is a (finitelygenerated) monomial subalgebra.

Sometimes a problem: additive notation in M versus multiplicativenotation 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.


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.


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.


Lattice polytopes

Definition

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

P


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

�:


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 havedegree 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 conventionJanuary 2006).


Affine charts for projective toric varieties

Proj K ŒP� is covered by the affine varieties K ŒMv � where Mv is acorner 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.


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, andU 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.


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

ai DP

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 cCi � X c�

i , i D 1; : : : ; n � rwith respect to X1; : : : ; Xn.

ci D cCi � c�

i , cCi ; c�

i ZC.


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 0

2 Z ; Y 01 Y 1

2 Z ; Y 11 Y 0

2 Z ; Y 11 Y 1

2 Z �

Š K ŒX1; X2; X3; X4�=.X1X4 � X2X3/

K ŒP� D K ŒY �11 Y 0

2 Z ; Y 01 Y �1

2 Z ; Y 11 Y 1

2 Z ; ; Y 01 Y 0

2 Z �

Š K ŒX1; X2; X3; X4�=.X1X2X3 � X 34 /


An affine monoid M generates the cone

RCM D� X

aixi W xi 2 M; ai 2 RC�

Since M DPniD1 ZCxi is finitely generated, RCM is finitely

generated (and therefore a cone)

The structures of M and RCM are connected in many ways.


Gordan’s lemma and normality

As seen above, affine monoids define rational cones. The converseis also true.

Lemma (Gordan’s lemma)

Let L � Zd be a subgroup and C � Rd a rational cone. Then L\ Cis an affine monoid.

Proof. Let V D RL � Rd . Then V \Qd D QL and C \ V is arational cone in V) We may assume that L D Zd .

C is generated by elements y1; : : : ; yn 2 M D C \Zd .

x 2 C ) x D a1y1 C � � � C anyn ai 2 RC:

x D x 0 C x 00; x 0 D ba1cy1 C � � � C bancyn:


Clearly x 0 2 M. But

x 2 M ) x 00 2 gp.M/\ C ) x 00 2 M:

x 00 lies in a bounded set B)M generated by y1; : : : ; yn and the finite set M \ B.

More precisely we have shown:

Theorem

Suppose y1; : : : ; yn 2 M generate C. Then M is a finitely generatedmodule over N D ZCy1 C � � � CZCyn:

M D[

x2B\M

N C x :

The monoid M D L \ C has a special property: it is integrally closedin L:


Integral closure

Definition

A monoid M is integhrally closed in an overmonoid N ”x 2 N; kx 2 M for some k 2 Z; k > 0 ) x 2 M:

Two types of integral closedness are interesting for us:

M � Zn integrally closed in Zn,

M integrally closed in gp.M/. In this case M is called normal.

Being normal is necessary for integral closedness in any groupcontaining M.


Theorem

M � Zd integrally closed affine monoid” there exists arational cone C such that M D C \Zd ;

M � Zd affine monoid) the integral closure bM D RCM \Zd

is a finitely generated M-module and therefore an affine monoid.

This applies especially to the choice Zd Š gp.M/. In this caseNM D bM is the normalization.

Briefly: Normal affine monoids are discrete cones.

0


Normality of K ŒM�

An integral domain is normal if it is integrally closed in its field offractions Q: if x 2 Q satisfies an equation

xn C an�1xn�1 C � � � C a1x C a0; ai 2 R;

then x 2 R. Examples: all factorial domains.

Theorem

Let M be an affine monoid, K a field. Then K ŒM� is normal if and onlyif M is normal.


Positivity and Hilbert bases

Definition

A monoid M is positive if x ;�x 2 M ) x D 0.

Definition

A grading on M is a homomorphism deg W M ! Z. It is positive ifdeg x > 0 for x ¤ 0.

Proposition

For M affine the following are equivalent:1 M is positive;2 RCM is pointed (i. e. contains no full line);3 M is isomorphic to a submonoid of Zs for some s;4 M has a positive grading.

Proof. (c)) (d)) (a) trivial.


(a)) (b) Set C D RCM. One shows:fx 2 C W �x 2 Cg D Rfx 2 M W �x 2 Mg.Therefore: M positive) C pointed.

(b)) (c) Let C be positive. Recall: for each facet F of C there existsa unique linear form �F W Rd ! R with the following properties:

F D fx 2 C W �F .x/ D 0g, �F .x/ � 0 for all x 2 C;

� has integral coefficients, �.Zd/ D Z.

We have called them support forms of C.

Let s D # facets.C/ and define

� W Rd ! Rs; �.x/ D ��F .x/ W F facet

�:

Then �.M/ � �. NM/ � ZsC. Since C is positive, � is injective!We call � the standard embedding.


x 2 M is irreducible if x D y C z ) y D 0 or z D 0.

Proposition

The irreducible elements of a positive affine monoid form its uniqueminimal system of generators.

Proof. Every system of generators contains all the irreducibleelements. Enough to show: every element is the sum of irreducibles.Can assume: M � ZnC. Set deg x D s1 C � � � C xn. Now induction ondeg x. Either x is irreducible or it decomposes as x D y C z withdeg y ; deg z < deg x.

Definition

The set of irreducible elements is called the Hilbert basis of M.


Typical picture for normal monoids of rank 2:

0

Hilb.M/ is the set of lattice points in the bottom of M.

In higher dimension is situation is complicated.


Given a positive affine monoid M � Zd by a set of generators, onecan compute Hilbert bases of the normalization NM and and theintegral closure bM in Zd :

W.B., R. Koch, normaliz, available from

ftp.math.uos.de/pub/osm/kommalg/software/

normaliz computes various other data. It is accessible from Singularvia a library.


Lecture 2

Unimodular covers and triangulations


A nonnormal polytope

Let P be a lattice polytope. In general M.P/ is not normal.Example:

P D fx 2 R3 W xi � 0; 15x1 C 10x2 C 6x3 � 30g:One has

P D conv�

.0; 0; 0/; .5; 0; 0/; .0; 3; 0/; .0; 0; 2/�

Evidently, gp.M.P// D Z4, .1; 2; 4; 2/ 2 C.P/, but.1; 2; 4; 2/ … M.P/) M.P/ not integrally closed in Z4

BUT gp.M.P// D Z4 ) M.P/ is not normal.

In other words: M.P/ ¤ NM.P/, Hilb.M.P// ¤ Hilb. NM.P//.

Where can we find the remaining elements of Hilb.bM.P//?


Strategy

Triangulate polytope (or cone)

investigate simplices (or simplicial cones)


Simplicial cones

Recall: A cone is simplicial if it is generated by linearly independentvectors.

0v1

v2

par.v1; : : : ; vn/ D ˚q1v1 C � � � C qnvn W 0 � qi < 1; i D 1; : : : ; n

�E D Zn \ par.v1; : : : ; vn/


v1; : : : ; vn 2 Zn linearly independent. Set

M D ZCv1 C � � � CZCvn; U D gp.M/; C D RCM

Proposition

Then1 E is a system of generators of the M-module C \Zn;2 .x CM/ \ .y CM/ D ; for x ; y 2 E, x ¤ y;3 #E D ŒZn W U�;4 Hilb.C \Zn/ � fv1; : : : ; vng [ E.

Especially C \Zn is the disjoint union of the M-modules x C M,x 2 M.

Note: If deg vi D 1 for all i , then deg x < r for x 2 E.

Via triangulation this helps to bound the Hilbert basis is general.


Bounding the Hilbert basis

Theorem (Ewald-Wessels, B.-Gubeladze-Trung)

Let P be a lattice polytope, Then bM.P/ is generated by elements ofdegree � dim P � 1.More precisely, as a module over M.P/ its is generated by elementsof degree � dim P � 1.

Proof. Triangulate P such that each simplex contains only its verticesas lattice points:


Enough to consider a simplex P whose only lattice points are thevertices.

Proposition on simplicial cones implies: Can generate bM.P/ byelements with deg y �D dim P. (The simplex has dim P C 1vertices!)

Cute trick: assume x 2 E (notation as above) has degree d. Thenv1 C � � � C vdC1 � y has degree 1 and lies in P. But theny D v1 C � � � C vi�1 C viC1 C � � � C vdC1 for some i. Contradiction!

The bound in the theorem is the best possible in all dimensions.


In the corollary we apply attributes to P that, strictly speaking, aredefined for M.P/:

Corollary

Let Pbe a lattice polytope.

If dim P D 2, then P is integrally closed.

More generally, cP is integrally closed for c � dim P � 1.

Proof. If we cut C.P/ by a hyperplane at height c, we obtain cP.Therefore bM.cP/ D fx 2 bM.P/ W deg y � 0 .c/g:If deg y D mc, then y D x C z1 C � � � C zu with deg x � dim P � 1and deg zj D 1 (u D mc � deg x). Cut this sum into subsums ofdegree c.


This result follows the principle that a graded rings “improves” by thepassage to Veronese subrings, or – in projective algebraic geometry– a divisor “improves” it is replaced by a high multiple.


Quadratic generation of toric ideals

Let me mention another result that can be proved rather easily bycombinatorial methods:

Theorem (B.-Gubeladze-Trung)

Let c � dim P. Then

K ŒcP� has a presentation as a residue class ring of a polynomialring by an ideal with a quadratic Grobner basis.

The toric ideal defining K ŒcP� is generated in degree 2.

K ŒcP� is a Koszul algebra.


A recent result improves this in characteristic 0:

Theorem (Hering-Schenck-Smith)

CŒcP� has the Green-Lazarsfeld property Np forc � min.dim P C c � 1; c/.

In ring-theoretic terms: a graded ring R has

N0 if it is generated in degree 1;

N1 if it has N0 and the defining ideal is generated in degree 2;

Np, p � 2, if it has N1 and the syzygies up to step p are linear.

The theorem can be refined if one takes into account the degrees ofthe elements ion the interior of C.P/.


Open questions:

Suppose P lattice polytope such that Proj K ŒP� is smooth.

Is k ŒP� Koszul or at least defined by a toric ideal generated indegree 2?

Is K ŒP� normal?

The second question has a good chance for a positive answer sincethe corner monoids are free. However, normality of the cornermonoids does not imply normality of P (the other direction holds).


Can one explain normality of lattice polytopes?

– in the sense that it is equivalent to simpler, formally strongerproperties

I think the answer is no, but this is not the last word.


Recall: P D conv.x1; : : : ; xn/ � Rd , xi 2 Zd , is called a latticepolytope.

P

�1

�2

� D conv.v0; : : : ; vd/, v0; : : : ; vd affinely independent, is asimplex.


Set U� DPd

iD0 Z.vi � v0/.

�.�/ D ŒZd W U�� D multiplicity of �

� is unimodular if �.�/ D 1.

� is empty if vert.�/ D � \Zd .

Lemma

�.�/ D d Š vol.�/ D ˙ det

0B@

v1 � v0:::

vd � v0

1CA

When is P covered by its unimodular subsimplices?

For short: P has UC.


Multiplicity for simplicial cones:

Let v1; : : : ; vd 2 Zd be linearly independent elements, and supposethat each vi has coprime entries. Set C D RCv1 C � � � C RCvd. Then

�.C/ D j det.v1; : : : ; vd/j D �.conv.0; v1; : : : ; vd//:

C is unimodular if �.C/ D 1.


Low Dimensions

d D 1: 0 1 2 3 4�1 P has a uniqueunimodular triangulation.

d D 2:

Recall: lattice polytopes of dim 2are integrally closedTherefore every empty latticetriangle is unimodular) every2-polytope has a unimodulartriangulation.


d D 3: There exist empty simplices of arbitrary multiplicity!

Already clear from the existence of a nonnormal 3-polytope. Namely:


Proposition

P has UC) M.P/ D bM.P/ (P is integrally closed).

Recall: P is integrally closed”gp.M.P// D ZdC1 and

M.P/ is a normal monoid (M.P/ D CP \ gp.M.P//)

Does every integrally closed polytope have UC?


Generalization to rational cones

Let C be a rational cone. Then M D C \Zd is automatically anintegrally closed affine monoid.

Generalization of the condition UC is UHC: C the union of itsunimodular subcones that are generated by elements of Hilb.M/

Does every C have UHC?

Theorem (Sebo)

Cones of dimension 3 have even a unimdodular Hilbert triangulation.

This generalizes the theorem that polytopes of dimension 2 haveunimodular triangulations.


The integral Caratheodory property

The condition UHC can be weakened in an algebraic way. Recall

Theorem (Caratheodory)

Let C D RCx1 C � � � C RCxn be a cone of dimension d, and y 2 C.Then there exists i1; : : : ; id and aij 2 RC such thaty D ai1xi1 C � � � C aid xid .

This can be viewed as a consequence of the existence oftriangulations, but is more elementary.

Discrete version is the integral Caratheodory property ICP:

An positive affine monoid M has ICP if every y 2 M contained in asubmonoid generated by d elements of Hilb.M/, d D rank M.


Theorem (B.-Gubeladze)

If M has ICP, then

(a) M is normal

(b) every element of M is contained in a submonoid generated bylinearly independent elements of Hilb.M/.

Also UHC can be formulated for affine monoids:replave “linearly independent elements of Hilb.M/” in (b)by “basis of gp.M/ contained in Hilb.M/”.Thus UHC) ICP.

Since UHC and ICP imply normality, it is enough to consider monoidsof type C \Zd .

Does every rational cone C have ICP? (We assign properties to Cthat are defined for C \Zd ).


Counterexamples

(Bouvier and Gonzalez-Sprinberg, 1994) There exists a cone ofdimension 4 without a unimodular triangulation into subconesgenerated by elements of Hilb.M/, M D C \Z5.

(Kantor-Sarkaria, 2003) There exists a normal lattice polytopeof dimension 3 without a unimodular triangulation.

(B.-Gubeladze-Henk-Martin-Weismantel, 1998) There exists anormal lattice polytope of dimension 5 without ICP.

(2006) There exists a normal lattice polytope of dimension 5with ICP that violates UHC.


C6 \Z6 with Hilbert basis z1; : : : ; z10, is of form C.P5/, dim P5 D 5,P5 integrally closed, and violates UHC and ICP

z1 D .0; 1; 0; 0; 0; 0/; z6 D .1; 0; 2; 1; 1; 2/;

z2 D .0; 0; 1; 0; 0; 0/; z7 D .1; 2; 0; 2; 1; 1/;

z3 D .0; 0; 0; 1; 0; 0/; z8 D .1; 1; 2; 0; 2; 1/;

z4 D .0; 0; 0; 0; 1; 0/; z9 D .1; 1; 1; 2; 0; 2/;

z5 D .0; 0; 0; 0; 0; 1/; z10 D .1; 2; 1; 1; 2; 0/:

If we add

z011 D .0; �1; 2; �1; �1; 2/ z0

12 D .1; 0; 3; 0; 0; 3/

to the Hilbert basis, then we obtain a cone C06 of type C.P 0

5/ thatsatisfies ICP, but violates UHC.


Interesting fact: both monoids C6 \Z6 and C06 \Z6 have the

Frobenius group F20 as their automorphism group.

The results of very long searches for counterexamples in dimensions6 and 7 suggest the following:

C6 is the minimal counterexample to ICP and UHC.

All other counterexamples “contain” it.

Open problem: Do all cones of dimensions 4 and 5 have UHC?

See W.B., On the integral Caratheodory property, Exp. Math., toappear


Caratheodory rank

Definition

Let M be a positive affine monoid, y 2 M. Set

�.y/ D minfm W y D a1x1 C � � � C amxm; ai 2 ZC; xi 2 Hilb.M/g;and

CR.M/ D maxf�.x/ W x 2 Mg:Then CR.M/ is the Caratheodory rank of M.

Theorem (Cook-Vonlupt-Schrijver, Sebo)

Suppose rank M � 3. If M is normal, then CR.M/ � 2 rank.M/� 2.

Proof uses ideas very similar to the proof of the theorem ofEwald-Wessels.If M is not normal, then there is no bound for CR in terms of rank.


Clearly, M has ICP” CR.M/ D rank M.

C6 shows: in general CR.M/ > rank.M/, even if M is normal. But it isopen to what extent the bound in the theorem is sharp, or close tobeing sharp.

Using C6 and direct sums:

Proposition

For all n � 6 there exist normal affine monoids withCR.M/ � b7

6 rank Mc.


Strategy of the search for counterexamples to UHC andICP

M always a normal monoid, M D C \Zd .

Definition

Let x 2 Hilb.M/ and M 0 D ZC Hilb.M/ n fxg. We call x destructive, ifrank M 0 < rank M or Hilb.M/ n fxg is not the Hilbert basis ofRCM 0 \ gp.M/

Roughly speaking, Hilb.M/ n fxg is not the Hilbert basis of a cone offull dimension. A non-destructive element must span an extreme rayof RCM.

Definition

M (or RCM) is tight if every element of Hilb.M/ is destructive.


Suppose x generates an extreme ray of RCM. Then

MŒ�x � Š Z˚M 00; M 00 normal; rank M 00 D rank M � 1:

Lemma

Suppose x is non-destructive. Then

M 0:M 00 have UHC) M has UHC

CR.M/ � min.CR.M 0/; CR.M 00/C 1/.

Corollary

A minimal counterexample to UHC or ICP is tight.


Strategy for finding counterexamples:

generate a “random” normal monoid M;

shrink it to a tight normal M0;

check M0 for the property in question.

x

C0

C

Tightening a cone


Deciding UHC

Suppose D is a subcone of C. Then

D is contained in a unimodular subcone: discard D;

D does not properly intersect any unimodular subcone: UHCviolated;

otherwise split D along support hyperplane of a unimodularsubcone, and apply recursion.

D2

D1

U


unicover.D; n/

1 for i n to N2 do3 if D � Ui

4 then return5 if int.D/ \ int.Ui / ¤ ;6 then .D1; D2/ split.D; Ui /

7 unicover.D1; i/8 unicover.D2; i/9 return

10 output. D not u-covered /

11 return

main./

1 Create the list U1; : : : ; UN of u-subcones of C2 unicover.C; 1/


Unimodular triangulations of cones

If we omit the H in UHC for cones, then no condition remains:

Theorem

A rational cone C has a triangulation into unimodular subcones(spanned by integral vectors).

Proof. Start with arbitrary triangulation. Refine by iterated stellarsubdivision to reduce multiplicities.

Problem: bound the size of the vectors spanning the unimodularsubcones!


Multiples of polytopes again

Idea: find a unimodular cover of a lattice polytope by

triangulate each corner cone into unimodular subcones

extend the “basic simplices” to a tiling of Rd ;

hopefully the tiles contained in P cover P.

u vertex of P; then the corner cone at u is

RC.P � u/

First step:

v0

v2

v1

P


cP has the same corner cones as P, and if c � 0, then the tilesbecome small, enough tiles should lie in cP and cover it.

In the next step the “basic” tiles get into c0P

v0

v2

v1


Tile corner cones:

v0

v2

v1

Then we increase again so that tiles in c00P cover area betweenvertex and red line through barycenter (from all vertices)) c00P iscovered.


Consequence: for each P there exists c > 0 such that cP has UC.But much more is true: c can be bounded in terms of the dimension:

Theorem (B.-Gubeladze, v. Thaden)

Let P be a d-polytope. cP has UC for all c � cpold ,

cpold D O

�d16:5

��9

4

�.ld .d//2

;

.d/ D .d � 1/dpd � 1e:The step from corner cones to P is rather easy and ”polynomial”:

Lemma

Suppose the basic simplices of the corner cones lie in P. Then cPhas a unimodular cover for all c > d

pd.

So one must find a good bound for the unimodular triangulation ofrational cones! Enough to do simplicial cones.


Theorem

Let C be a rational simplicial d-cone and �C the simplex spanned byO and the extreme integral generators. Then

1 (M. v. Thaden) C has a triangulation into unimodular simplicialcones Di such that Hilb.Di/ � c�C for some

c � d2

4.�.C//7

�9

4

�.ld.�.C///2

:

2 C has a cover by unimodular simplicial cones Di such thatHilb.Di/ � c�C for some

c � d2

4.d C 1/

�.d/

�8�

9

4

�.ld..d///2

;

.d/ D ˙pd � 1

.d � 1/


von Thaden claims that he can improve the bounds to polynomialsize, at least for covers. (Should be the main result of his thesis.)

Best possible value: cpold D dim P � 1.

We know this for d D 2. It also holds for d D 3.

Results of Lagarias & Ziegler , Kantor & Sarkaria:

Proposition

dim P D 3) cP has UC for c � 2.

Theorem

4P has a unimodular triangulation for all P.

Wrong for 2P in general! 3P ??.


Knudsen-Mumford triangulations

Theorem (Knudsen-Mumford, 1973)

Let P be a lattice polytope. Then there exists c > 0 such that cP andc0CP for all c0 > 0 have unimodular triangulations.

Open problem: Does cP have a unimodular triangulation for allC � 0? Does there exist a uniform bound only in terms ofdimension?


Lecture 3

Hilbert functions and homological properties


The ADG conjectures

In 1966 H. Anand, V. C. Dumir, and H. Gupta investigated acombinatorial problem:

Suppose that n distinct objects, each available in k identical copies,are distributed among n persons in such a way that each personreceives exactly k objects.

What can be said about the number H.n; k/ of such distributions?


They formulated some conjectures:

(ADG-1) there exists a polynomial Pn.k/ of degree .n � 1/2 suchthat H.n; k/ D Pn.k/ for all k � 0;

(ADG-2) H.n; k/ D Pn.k/ for all k > �n; in particular Pn.�k/ D 0,k D 1; : : : ; n � 1;

(ADG-3) Pn.�k/ D .�1/.n�1/2Pn.k � n/ for all k 2 Z.

These conjectures were proved and extended by R. P. Stanley usingmethods of commutative algebra. Basic idea: interpret H.n; k/ (nfixed) as the Hilbert function of some graded ring.

The weaker version (ADG-1) of (ADG-2) has been included fordidactical purposes. A compact account has been given in

W.B., Commutative algebra arising from the Anand-Dumir-Guptaconjectures;http://www.math.uos.de/staff/phpages/brunsw/Allahabad.pdf


Reformulation:

aij D number of copies of object i that person j receives

) A D .aij/ 2 Zn�nC such that

nXlD1

ail DnX

mD1

amj D k ; i; j D 1; : : : ; n:

H.n; k/ is the number of such matrices A.

The system of equations is part of the definition of magic squares.Stanley calls the matrices A magic squares, though the usuallyrequires further properties for those.


Two famous magic squares:

8 1 6

3 5 7

4 9 2

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

The 3 3 square can be found in ancient Chinese sources and the4 4 appears in Albrecht Durer’s engraving Melancholia (1514). Ithas remarkable symmetries and shows the year of its creation.


A step towards algebra

Let Mn be the set of all matrices A D .aij/ 2 Zn�nC such that

nXlD1

ail DnX

mD1

amj ; i; j D 1; : : : ; n:

By Gordan’s lemma Mn is an affine, normal monoid, and A 7! magicsum k DPn

kD1 a1k is a positive grading on M .

rank Mn D .n � 1/2 C 1

Theorem (Birkhoff-von Neumann)

Mn is generated by the its degree 1 elements, namely thepermutation matrices.


Translation into commutative algebra

We choose a field K and form the algebra

R D K ŒMn�:

It is a normal affine monoid algebra, graded by the “magic sum”, andgenerated in degree 1. dim R D rank Mn D .n � 1/2 C 1.

) H.n; k/ D dimK Rk D H.R; k/ is the Hilbert function of R!

) (ADG-1): there exists a polynomial Pn of degree .n � 1/2 suchthat H.n:k/ D Pn.k/ for k � 0.

In fact, take Pn as the Hilbert polynomial of R.


A recap of Hilbert functions

Let K be a field, and R D ˚1kD0Rk a graded K -algebra generated by

homogeneous elements x1; : : : ; xn of degrees g1; : : : ; gn > 0.

Let M be a non-zero, finitely generated graded R-module.

Then H.M; k/ D dimK Mk <1 for all k 2 Z:

H.M; / W Z! Z is the Hilbert function of M.

We form the Hilbert (or Poincare) series

HM.t/ DXk2Z

H.M:k/tk :


Fundamental fact:

Theorem (Hilbert-Serre)

Let K be a field, and R D ˚1kD0Rk a graded K -algebra generated by

homogeneous elements x1; : : : ; xn of degrees g1; : : : ; gn > 0.

Let M be a non-zero, finitely generated graded R-module Then thereexists a Laurent polynomial Q 2 ZŒt ; t�1� such that

HM.t/ D Q.t/QniD1.1 � tgi /

:

More precisely: HM.t/is the Laurent expansion at 0 of the rationalfunction on the right hand side.


Refinement: M is finitely generated over a graded Noethernormalization K Œx1; : : : ; xd �:

S � R is a graded Noether normalization of M if

S D K Œx1; : : : ; xd � with algebraically independent elementsx1; : : : ; xd , d D dim M D dim R= Ann M;

M is a finitely generated S-module.

In other words, a Noether normalization allows us to study M as agraded module over a polynomial ring S such that AnnS M D 0. Wealso say that x1; : : : ; xd is a homogeneous system of parameters(hsop).

Special case: let us say that M is essentially standard graded (est) ifM is finitely generated over K ŒR1�. For this it is enough that R is est,and this certainly holds if R D K ŒR1�.

If M is est, then we can choose a Noether normalization in degree 1:deg x1 D � � � D deg xd D 1, at least after an extension of K .


Theorem

Suppose that M is est and let d D dim M. Then

HM.t/ D Q.t/

.1� t/d:

Moreover:There exists a polynomial PM 2 QŒX � such that

H.M; i/ D PM.i/; i > deg HM ;

H.M; i/ ¤ PM.i/; i D deg HM :

e.M/ D Q.1/ > 0, and if d � 1, then

PM D e.M/

.d � 1/ŠX d�1 C terms of lower degree:


Recall(ADG-1) there exists a polynomial Pn.k/ of degree .n � 1/2 such

that H.n; k/ D Pn.k/ for all k � 0;

With R D K ŒMn� this is equivalent to

HR.t/ D 1C h1t C : : : hutu

.1� t/d; d D rank Mn:

and so (ADG-1) has been proved. Furthermore

(ADG-2) H.n; k/ D Pr .n/ for all k > �n; in particular Pn.�k/ D 0,k D 1; : : : ; n � 1;

is equivalent to u � d D �n (hu ¤ 0). How can we translate

(ADG-3) Pn.�k/ D .�1/.n�1/2Pn.k � n/ for all k 2 Z

into a property of the rational function HR.t/? Note that the shift �nin (ADG-3) is the conjectured degree of HR.t/.


Lemma

Let R be an est graded K -algebra, dim R D d, with Hilbertpolynomial P. Suppose deg HR.t/ D g < 0. Then the following areequivalent:

P.�k/ D .�1/d�1P.k C g/ for all k 2 Z;

.�1/dHR.t�1/ D t�gHR.t/;

hi D hu�i for all i : the h-vector is palindromic.

The equivalence of the last two assertions is very easy, but theequivalence with the first is tricky. (Goes back to Polya, explicitlystated by Popoviciu.)


For (ADG-2) we have to compute the degree of the Hilbert series asa rational function.

For (ADG-3) it is best to work with the most compact form

.�1/dHR.t�1/ D t�gHR.t/:

We will see that the proofs of (ADG-2) and (ADG-3) can be based onhomological properties of normal affine monoid algebras.


Hochster’s theorem

Definition

Let M be a graded module over a positively graded K -algebra R.Then M is called Cohen-Macaulay if it is a free module over a gradedNoether normalization S of M.

An intrinsic definition shows that the choice of S is irrelevant. Notethat M always has a finite free resolution over S, and it isCohen-Macaulay if and only if the resolution has length 0.


Theorem (Hochster)

Let M be an affine normal monoid. Then K ŒM� is Cohen-Macaulayfor every field K .

There is no easy proof of this powerful theorem. For example, it canbe derived from the Hochster-Roberts theorem, using thatK ŒM� � K ŒX1; : : : ; Xs� can be chosen as a direct summand.

A special case is rather simple:

Proposition

Let M be a simplicial affine normal monoid. Then K ŒM� isCohen-Macaulay for every field K .

Let d D rank M. Choose elements x1; : : : ; xd 2 M generating RCM,and set

par.x1; : : : ; xd / D� dX

iD1

qixi W qi 2 Œ0; 1/

�:


With N D ZCx1 C � � � CZCxd : we know from the previous lecture:

M DSz2E z C N,

N is free,The union is disjoint.

In commutative algebra terms:K ŒM� is finite over K ŒN�.K ŒN� Š K ŒX1; : : : ; Xd �.K ŒM� is a free module over K ŒN�.

) K ŒM� is Cohen-Macaulay.

0x1

x2


A combinatorial consequence of the Cohen-Macaulayproperty

Theorem

Let M be a graded Cohen-Macaulay module over the positivelygraded K -algebra R and x1; : : : ; xd a hsop. for M, ei D deg xi . Let

HM.t/ D hata C � � � C hbtbQdiD1.1� tei /

; ha; hb ¤ 0:

Then hi � 0 for all i .

If M D R D K ŒR1�, then hi > 0 for all i D 0; : : : ; b.

Proof. hata C � � � C hbtb is the Hilbert series of

M=.x1M C � � � C xdM/:

So each hi is the dimension of a vector space.


For (ADG-3) we have to show .�1/.n�1/2C1HR.t�1/ D tnHR.t/ forR D K ŒMn�.

Strategy:

Find an R-module ! with H!.t/ D .�1/dHR.t�1/, d D dim R

Possible for R Cohen-Macaulay: ! is the canonical module of R.

Compute ! for R D K ŒMn� and show that ! Š R.g/,g D deg HR.t/.

R.g/ free module of rank 1 with generator in degree �g. ThusHR.g/.t/ D t�gHR.t/.

More generally:

Compute ! for R D K ŒM� with M affine normal.


The canonical module

In the following: R positively graded Cohen-Macaulay K -algebra,x1; : : : ; xd hsop, deg xi D gi

First R D S D K ŒX1; : : : ; Xd �:

.�1/dHS.t�1/ D .�1/dQdiD1.1 � t�gi /

D tg1C��CgdQdiD1.1 � t�gi /

D H!.t/

with ! D !S D S.�.g1 C � � � C gd//


Now R free over S D K Œx1; : : : ; xd �, with homogeneous basisy1; : : : ; ym:

R ŠmM

jD1

Syj ŠuM

iD1

S.�i/hi ; hi D #fj W deg yj D ig;

HR.t/D .h0 C h1t C � � � C hutu/HS.t/ D Q.t/HS.t/

Set !R D HomS.R; !S/. Then, with s D g1 C � � � C gd

!R ŠuM

iD1

HomS.S.�i/hi ; S.�s// ŠuM

iD1

S.�i C s/hi ;

H!R .t/D .h0ts C � � � C huts�u/HS.t/ D Q.t�1/tsHS.t/

D .�1/dQ.t�1/HS.t�1/ D .�1/dHR.t�1/:


Let us rewrite H!.t/:

H!.t/ D huts�u C � � � C h0tsQdiD1.1� t�gi /

Since s D g1 C � � � C gd , one has s � u D � deg HR.t/. Therefore

Corollary

deg HR.t/ D �minfk W !k ¤ 0g:


R-module structure of !R

Multiplication in the first component makes !R D HomS.R; S/ anR-module:

a � '. / D '.a � /:

But: Is !R independent of S ? Indeed

Theorem

!R depends only on R (up to isomorphism of graded modules).

The proof requires homological algebra, after reduction from thegraded to the local case.


The canonical module of K ŒM�

In the following M affine, normal, positive monoid. We want to findthe canonical module of R D K ŒM� (Cohen-Macaulay by Hochster’stheorem).

Theorem (Danilov, Stanley)

The ideal I generated by the monomials in the interior of RCM is thecanonical module of K ŒM� (with respect to every positive grading ofM or even multigraded).

Various proofs via

differentials (Danilov),

combinatorics (Stanley),

local cohomology (Stanley; see Cohen-Macaulay rings),

divisors

The case of a simplicial normal monoid is again elementary.Winfried Bruns Lattice polytopes

After all these preparations, (ADG-2) and (ADG-3) follow easily.Consider a magic square A D .aij/. Then

A 2 int.Mn/” aij > 0 for all i; j ” A � 1 2Mn

where 1 is the magic square with all entries 1: Thusint.Mn/ D 1CMn.

The magic sum of 1 is n. So !R D R.�n/. With d D .n � 1/2 C 1this implies

HR.t�1/ D .�1/d tnHR.t/ W (ADG-3)

anddeg HR.t/ D �n W (ADG-2)


Recall that we have written

HR.t/ D 1C h1t C � � � C hutu

.1 � t/d:

Hochster’s theorem)Theorem

If R is a normal affine monoid algebra, then

hi � 0 for all i

(ADG-2)” hi D hu�i for all i . What else can be said about theh-vector?

Stanley conjectured: the h-vector for R D K ŒMn� is unimodal:

1 D h0 � h1 � � � � � hbu=2c

This will be our next topic. But we should not forget to introduce aclass of rings to which K ŒMn� belongs.


Gorenstein rings

Definition

A positively graded Cohen-Macaulay K -algebra is Gorenstein if!R Š R.h/ for some h 2 Z.

Actually, there is no choice for h:

Theorem (Stanley)

Let R be Gorenstein of Krull dimension d. Then

!R Š R.g/, g D deg HR.t/;

h0 D hu�i for i D 0; : : : ; u: the h-vector is palindromic;

HR.t�1/ D .�1/d t�gHR.t/.

Conversely, if R is a Cohen-Macaulay integral domain such thatHR.t�1/ D .�1/d t�hHR.t/ for some h 2 Z, then R is Gorenstein.


Counting lattice points in polytopes

This area was pioneered by E. Ehrhart. As we will see, we havealready proved central theorems about lattice point counting.

Let P � Rd be a lattice polytope. We set

E.P; k/ D #.P \ 1

kZd/ D #.kP \Zd/:

This is the Ehrhart function of P.

the corresponding power series is the Ehrhart series

EP.t/ DXkD0

E.P; k/tk :


We knowkP \Zd $ fx 2 bM.P/ W deg x D kg:

Therefore, with R D K ŒbM.P/�,

E.P; k/ D H.R; k/

where H is again the Hilbert function.

In general R is not generated in degree 1 as an algebra. But: bM.P/ isa finitely generated M.P/-module, in other words:

P lattice polytope) R D K ŒbM.P/� is an est graded algebra.


Theorem

Let P be a lattice polytope of dimension d. Then

EP.t/ D HR.t/ D 1C h1t C : : : hutu

.1� t/dC1

with

deg HR.t/ D u � .d C 1/ D �minfk W int.P/ \Zd ¤ ;g < 0:

Thus, with the Hilbert polynomial Q of R,

E.P; k/ D Q.k/ for all k � 0:

One calls Q the Ehrhart polynomial of P. In the theorem we haveomitted the reciprocity of HR $ P and H! $ int.P/ which in thiscontext is called Ehrhart reciprocity.


Lecture 4

Unimodality of h-vectors


Questions on unimodality

Let R be an est (essentially standard) graded K -algebra. Recall thatthe Hilbert series can be written

HR.t/ D 1C h1t C : : : hutu

.1 � t/d; d D dim R; hu ¤ 0:

If R is Cohen-Macaulay, then hi � 0 for all i .

If R is Gorenstein, then hu�i D hi for all i , and if R is aCohen-Macaulay domain, then this symmetry of the h-vector impliesthat R is Gorenstein.

For R D K ŒMn�, the (Gorenstein) algebra over the monoid of magicsquares, Stanley conjectured that the h-vector is unimodal:

1 D h0 � h1 � � � � � hbu=2c


This conjecture was proved by Athanasiadis (2003), and thenextended to a larger class of Gorenstein monoid algebras by B. andRomer (2005). See

W.B., T. Romer, h-vectors of Gorenstein polytopes, J. Comb. Th. Ser.A, 114 (2007), 65–76.

As counterexamples show, the property est is not sufficient for theunimodality of the h-vector: one needs that R D K ŒR1� is standardgraded (as K ŒMn� is).

As far as I know, there is no example of a standard gradedGorenstein domain with a non-unimodal h-vector, and to prove ordisprove this property is the greatest challenge in combinatorialcommutative algebra.


In the characterization of Gorenstein rings by the symmetry conditionhu�i D hi the hypothesis domain cannot be omitted.

If R D K ŒX1; : : : ; Xn�=I is a Gorenstein ring, then one often findsmonomial orders on the polynomial ring such thatR0 D K ŒX1; : : : ; Xn�= in.I/ is Cohen-Macaulay, but not Gorenstein.Nevertheless, HR.t/ D HR0.t/.

Here in.I/ is the initial ideal of I, i. e. the ideal generated by theleading monomials of the elements of I.

Question: suppose R D K ŒX1; : : : ; Xn�=I is Gorenstein. Can one finda monomial order on K ŒX1; : : : ; Xn� such thatR0 D K ŒX1; : : : ; Xn�= in.I/ is Gorenstein as well?

Actually, the proof of our result on h-vectors results from the(successful) attempt to answer this question for a class of monoiddomains. The connection between unimodality of the h-vector andinitial ideals is given by certain triangulations.


The g-theorem

There is a famous instance where unimodality of the h-vectors andmore is known.

Let � be a simplicial complex of dimension d. (The dimension of � isthe maximal dimension of a face (D simplex) of �). It is useful to setd D dim �C 1.

The i-th component of the f -vector counts the number of faces ofdimension i:

f .�/ D .f0; : : : ; fd�1/; fi D #fı 2 � W dim ı D igThe f -vectors of simplicial complexes are characterized by theKruskal-Katona theorem.


Example:

1

2

3

4

5

�

dim � D 2

f .�/ D .5; 6; 2/

h.�/ D .1; 2;�1/

I.�/ D �X1X4; X1X5; X2X4; X2X5

�h.�/ and I.�/ will be explained below.


Next one defines the h-vector h.�/ through the polynomial identity

dXiD0

hisi.1C s/d�i D

dXiD0

fi�1si .f�1 D 1/:

Let v1; : : : ; vn be the vertices of �. A subset N D fvi1; : : : ; vimg is anonface of � is N is not the vertex set of a face of �.

In the polynomial ring K ŒX1; : : : ; Xn� we set

X N D Xi1 � � �Xim ;

and define the Stanley-Reisner ring or face ring of � by

K Œ�� D K ŒX1; : : : ; Xn�=I.�/; I.�/ D .X N W N nonface of �/:

It is enough to take the minimal nonfaces.


Proposition

Let � be a simplicial complex of dimension d, R D K Œ��, andh.�/ D .1 D h0; : : : ; hu/. Then

HR.t/ D 1C h1t C � � � C hdtu

.1� t/u:

Thus we can understand h.�/ as the h-vector of a graded ring.


The most interesting simplicial complexes are the boundarycomplexes of simplicial polytopes. (A polytope is simplicial if all itsfacets are simplices.) They satisfy the Dehn-Sommerville equations

hi D hd�i ; d D dim P:

a condition that we have encountered already. (Note that hd D 1 andthat dim @P D d � 1.)

A famous theorem of McMullen is the upper bound theorem(conjectured by Motzkin):

Theorem

The h-vector of (the boundary of) a simplicial polytope is boundedabove by the the h-vector of the cyclic polytope of the samedimension and the same number of vertices.


The upper bound theorem was later on generalized by Stanley tosimplicial spheres, i, e. simplicial complexes for which j�j Š Sd�1.The main argument:

Theorem (Stanley)

Let � be a simplicial sphere. Then K Œ�� is a Gorenstein ring.

Note that a simplicial sphere need not be the boundary of a simplicialpolytope. Boundaries of polytopes are shellable (Brugesser-Mani),and this was used by McMullen. However, there exist non-shellablesimplicial spheres.


For a simplicial complex satisfying the Dehn-Sommerville equationsone can define the g-vector

gi D hi � hi�1; i D 0; : : : ; bd=2c:McMullen conjectured and Billera and Lee (sufficiency) and Stanley(necessity) proved that the g-vectors of simplicial polytopes can becharacterized as follows:

Theorem (g-theorem)

The h-vectors h.@P/ D .1 D h0; : : : ; hd/ of the boundaries ofsimplicial polytopes P are characterized by the conditions

hi D hd�i , i D 0; : : : ; d;

the corresponding g-vector is the h-vector of a standard gradedK -algebra.

In particular, gi � 0 for all i , and so hi�1 � hi , i D 0; : : : ; bd=2c.


There is a theorem of Macaulay that characterizes the h-vectors ofstandard graded K -algebras in terms of explicit inequalities (involvingbinomial expansions). Therefore we say that the g-vector is aMacaulay sequence if it is the h-vector of a standard gradedK -algebra.

Open problem: does the g-theorem hold for simplicial spheres?

The advantage of simplicial polytopes is that one can assign a toricvariety to them, and Stanley used topological properties of thisvariety.

Strategy: In order to prove unimodality (or even the Macaulaycondition) for an h-vector, show it is the h-vector of the boundary of asimplicial polytope (or at least a simplicial sphere, and hope for thesolution of the open problem).


Unimodality for Gorenstein polytopes

Let us say that P is a Gorenstein polytope if

R D K ŒbM.P/� is standard graded (” M.P/ D bM.P/),

R is Gorenstein.

Stanley’s conjecture on the unimodality of the h-vector for magicsquares can now be generalized as follows:

Question: suppose P is a Gorenstein polytope. Is the h-vector h.P/

of R unimodal?

The answer is no if we do not require that R is standard graded.Counterexample by Mustata and Payne (connection with stringyHodge numbers).


Towards unimodality for Gorenstein polytopes

We have already observed that the canonical module of the algebraK ŒM� for an affine normal monoid M has the interiorint.M/ D M \ int.RCM/ of M as its monomial basis. This implies theequivalence of (1) and (2) in

Lemma

Let M be a normal affine monoid. Then the following are equivalent:1 K ŒM� is Gorenstein;2 int.M/ D x CM for some x 2 M;3 there exists x 2 M with �F .x/ D 1 for all facets F of RCM and

the corresponding support form �F .

the equivalence of (3) is an old observation, but crucial for thefollowing.


Let x be as in the lemma, and write x D y1 C � � � C yv with yi 2 M.Then the �F .yi/ are pairwise disjoint 0-1-vectors. Roughly speaking,this is a strong unimodularity condition.

Theorem

Let M be a normal affine monoid with R D K ŒM� Gorenstein andchoose x D y1 C � � � C yv as above. Then

the elements X y1 � X y2; : : : ; X yv�1 � X yv form a regularR-sequence;

the residue class ring R=(this sequence) is again a nor malaffine monoid Gorenstein algebra.

The proof is based on the construction of a unimodular triangulationthat can be projected along the vector subspace generated by thedifferences yi � yiC1.


Now suppose that M D M.P/ for a Gorenstein polytope. Then x canbe decomposed into a sum of degree 1 elements. Modulo a regularsequence of 1-forms the h-vector does not change, and the propertyof being standard graded is preserved:

Corollary

Suppose P is a Gorenstein polytope. Then there exists a Gorensteinpolytope P 0 with the following properties:

P 0 has exactly one interior lattice polytope (namely theprojection of x);

h.P 0/ D h.P/.

This reduces the problem of unimodality to the integrally closedreflexive polytopes. (P reflexive” P contains exactly one interiorlattice polytope and K ŒbM.P/� is Gorenstein.)


In order to get closer to the g-theorem we need a unimodulartriangulation () K ŒbM.P/� standard graded).

Theorem

Let P be a Gorenstein polytope with a unimodular triangulation. Thenthere exists a Gorenstein polytope P 0 with the following properties:

P 0 has exactly one interior lattice polytope;

@P has a triangulation � such that h.P/ D h.�/.

For the proof one has to start with the given unimodular triangulationand to modify it to one that can be projected.The g-theorem appears at the horizon: at least � is a simplicialsphere (and somewhat more than that).


P

y1

y2

�

P 0

Change of the triangulation and projection


In order to apply the g-theorem (as it is now), the hypothesis on thetriangulation must be strengthened:

Corollary

Let P be a Gorenstein polytope with a regular unimodulartriangulation. Then there exists a Gorenstein polytope P 0 with thefollowing properties:

P 0 has exactly one interior lattice polytope;

@P has a triangulation � such that h.P/ D h.�/;

� can be deformed to the boundary complex of a simplicialpolytope.

Therefore h.P/ is a Macaulay sequence and, in particular, isunimodal.


A regular subdivision arises as the decomposition of P into thedomains of linearity of a piecewise convex affine function.

A regular subdivision and a nonregular refinement


Regularity is also a keyword in the Grobner basis theory of toricideals (D deformation to monomial ideals). See B. Sturmfels,Grobner bases and convex polytopes, AMS 1996.

Corollary

Let P be a Gorenstein polytope with a regular unimodulartriangulation, and let IP be the toric ideal definingK ŒP� D K ŒX1; : : : ; Xn�=IP . Then there exists a monomial order onK ŒX1; : : : ; Xn� such that K ŒX1; : : : ; Xn�= in.IP/ is also Gorenstein.

Here regularity of the triangulation is essential.
