Lattice polytopesAlgebraic, geometric and combinatorial aspects
Winfried Bruns
FB Mathematik/InformatikUniversitat Osnabruck
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
Winfried Bruns Lattice polytopes
Lecture 1
Basic notions
Winfried Bruns Lattice polytopes
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:
Winfried Bruns Lattice polytopes
0
A polyhedron, a polytope and a cone
The dimension of P is dim aff.P/.
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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.
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 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.
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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.
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 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.
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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.
Winfried Bruns Lattice polytopes
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/.
Winfried Bruns Lattice polytopes
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.
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 latticepolytope.
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 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).
Winfried Bruns Lattice polytopes
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.
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, 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.
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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.
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 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 /
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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:
Winfried Bruns Lattice polytopes
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:
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
(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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
Lecture 2
Unimodular covers and triangulations
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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//?
Winfried Bruns Lattice polytopes
Strategy
Triangulate polytope (or cone)
investigate simplices (or simplicial cones)
Winfried Bruns Lattice polytopes
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/
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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.
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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:
Winfried Bruns Lattice polytopes
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.
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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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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/.
Winfried Bruns Lattice polytopes
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).
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
d D 3: There exist empty simplices of arbitrary multiplicity!
Already clear from the existence of a nonnormal 3-polytope. Namely:
Winfried Bruns Lattice polytopes
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?
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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 ).
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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
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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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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/
Winfried Bruns Lattice polytopes
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!
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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/
Winfried Bruns Lattice polytopes
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 ??.
Winfried Bruns Lattice polytopes
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?
Winfried Bruns Lattice polytopes
Lecture 3
Hilbert functions and homological properties
Winfried Bruns Lattice polytopes
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?
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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 :
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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 .
Winfried Bruns Lattice polytopes
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:
Winfried Bruns Lattice polytopes
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/.
Winfried Bruns Lattice polytopes
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.)
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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/
�:
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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//
Winfried Bruns Lattice polytopes
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/:
Winfried Bruns Lattice polytopes
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:
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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)
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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 :
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
Lecture 4
Unimodality of h-vectors
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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).
Winfried Bruns Lattice polytopes
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).
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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.)
Winfried Bruns Lattice polytopes
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).
Winfried Bruns Lattice polytopes
P
y1
y2
�
P 0
Change of the triangulation and projection
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes
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
Winfried Bruns Lattice polytopes
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.
Winfried Bruns Lattice polytopes