Lattice polytopes Algebraic, geometric and combinatorial aspects
Winfried Bruns
FB Mathematik/Informatik Universitat¨ Osnabr¨uck
Sedano, March 2007
Winfried Bruns Lattice polytopes Sources
W. B. and J. Gubeladze, Polytopes, rings, and K -theory, Springer 2008 (?) Commutative algebra and combinatorics W.B. and J. Herzog, Cohen-Macaulay rings, Cambridge University Press 1998 (rev. ed.) E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer 2005 R. P. Stanley, Combinatorics and commutative algebra,Birkhauser¨ 1996 (2nd ed.) Toric varieties G. Ewald, Combinatorial convexity and algebraic geometry, Springer 1996 T. Oda, Convex bodies and algebraic geometry (An introduction to the theory of toric varieties), Springer 1988 W. Fulton, Introduction to toric varieties,Princeton University Press 1993
Winfried Bruns Lattice polytopes Lecture 1
Basic notions
Winfried Bruns Lattice polytopes Polyhedral geometry
Definition A polyhedron is the intersection of finitely many closed affine halfspaces. A polytope is a bounded polyhedron. A cone is the intersection of finitely many linear halfspaces.
A closed affine halfspace is a set
C d H˛ Dfx 2 R W ˛.x/ 0g
where ˛ is an affine form, i. e. a polynomial function of degree 1. It is linear if ˛ is a linear form. Its bounding hyperplane is
d H˛ Dfx 2 R W ˛.x/ D 0g:
Winfried Bruns Lattice polytopes 0
A polyhedron, a polytope and a cone
The dimension of P is dim aff.P/.
Winfried Bruns Lattice polytopes Definition A face of P is the intersection P \ H where H is a support hyperplane,i.e.P HC and P 6 H. A facet is a maximal face. A vertex isafaceofdim0. P; ; improper faces.
For full-dimensional cones the (essential) support hyperplanes Hi Dfx W i .x/ D 0g are unique:
Proposition Rd C C Let P , dim P D d. If the representation P D H1 \ \Hs is irredundant, then the hyperplanes Hi are uniquely determined (up to enumeration). Equivalently, the affine forms ˛i are unique up to positive scalar factors.
Winfried Bruns Lattice polytopes Finite generation of cones
Theorem (Minkowski-Weyl)
Let C ¤;be a subset of Rm. Then the following are equivalent:
m there exist finitely many elements y1;:::;yn 2 R such that C D RCy1 C CRCyn; ;:::; there exist finitely many linear forms 1 ˚ s such that C is C . / the intersection of the half-spaces Hi D x W i x 0 .
A simplicial cone is generated by linearly independent vectors.
Corollary
P is a polytope ” P D conv.x1;:::;xm/
One considers the cone over P (see below) and applies the theorem.
A simplex is the convex hull of a affinely independent set.
Winfried Bruns Lattice polytopes Definition An (embedded) polyhedral complex ˘ is a finite collection of polyhedra P Rd such that with P each face of P belongs to ˘, P; Q 2 ˘ ) P \ Q is a face of P (and of Q)
˘ is a fan if it consists of cones.
A polytopal complex consists of polytopes.
A simplicial complex consists of simplices. A simplicial fan consists of simplicial cones.
Winfried Bruns Lattice polytopes A subdivision of ˘ is a polyhedral complex ˘ 0 such that each face of ˘ is the union of faces of ˘ 0.
A triangulation of a polytopal complex is a subdivision into a simplicial complex.
A triangulation of a fan is a subdivision into a simplicial fan.
There are always enough triangulations:
Theorem Let ˘ beS a polytopal complex and X a finite subset of ˘ .˘/ j jD P2˘ P such that vert X. Then there there exists a triangulation of ˘ such that X D vert. /.
An analogous theorem holds for fans.
Winfried Bruns Lattice polytopes Rationality
Proposition
Let C be a cone. The generating elements y1;:::;yn can be chosen m m in Q (or Z ) if and only if the ˛i can be chosen as linear forms with rational (or integral) coefficients.
Such cones are called rational. For them there is a unique choice of
the ˛i satisfying d ˛i .Z / Z, d 1 2 ˛i .Z /.
With this standardization, we denote them by i and call them support forms.
Winfried Bruns Lattice polytopes Affine monoids and their algebras
Definition An affine monoid M is (isomorphic to) a finitely generated submonoid of Zd for some d 0, i. e. M C M M (M is a semigroup); 0 2 M (now M is a monoid);
there exist x1;:::;xn 2 M such that M D ZCx1 C CZCxn.
Often affine monoids are called affine semigroups. gp.M/ D ZM is the group generated by M. gp.M/ Š Zr for some r D rank M D rank gp.M/.
Winfried Bruns Lattice polytopes Let K be a field (can often be replaced by a commutative ring). Then we can form the monoid algebra M C K ŒM D KX a; X aX b D X a b a2M
X a D the basis element representing a 2 M.
Zd Œ ŒZd Œ ˙1;:::; ˙1 M (affine) ) K M K D K X1 Xd is a (finitely generated) monomial subalgebra. Sometimes a problem: additive notation in M versus multiplicative notation in K ŒM (and exponential notation is often cumbersome).
Proposition Let M be a monoid. 1 M is finitely generated ” K ŒM is a finitely generated K -algebra. 2 M is an affine monoid ” K ŒM is an affine domain.
Winfried Bruns Lattice polytopes Proposition The Krull dimension of K ŒM is given by
dim K ŒM D rank M:
Proof. K ŒM is an affine domain over K . Therefore
dim K ŒM D trdeg QF.K ŒM / D trdeg QF.K Œgp.M/ / D trdeg QF.K ŒZr / D r where r D rank M.
Winfried Bruns Lattice polytopes Sources for affine monoids (and their algebras) are monoid theory, ring theory, invariant theory of torus actions, enumerative theory of linear diophantine systems, lattice polytopes and rational polyhedral cones, coordinate rings of toric varieties, initial algebras with respect to monomial orders.
Winfried Bruns Lattice polytopes Lattice polytopes
Definition d The convex hull conv.x1;:::;xm/ of points xi 2 Z is called a lattice polytope.
P
Winfried Bruns Lattice polytopes Definition The polytopal monoid associated with P is ˚ d M.P/ D ZC .x; 1/ W x 2 P \ Z :
C.P/
P
Definition The cone over (an arbitrary polytope) P is ˚ dC1 C.P/ D RC .x; 1/ 2 R W x 2 P :
Winfried Bruns Lattice polytopes Definition The polytopal algebra associated with P is the monoid algebra
K ŒP D K ŒM.P/ :
K ŒP has a natural ZC-grading in which the generators have degree 1.
Objects of algebraic geometry: M affine monoid: Spec K ŒM is an affine toric variety P lattice polytope: Proj K ŒP is a projective toric variety
Toric varieties are not necessarily normal (Oberwolfach convention January 2006).
Winfried Bruns Lattice polytopes Affine charts for projective toric varieties
Proj K ŒP is covered by the affine varieties K ŒMv where Mv is a corner monoid of P: one has v 2 vert.P/ and ˚ d Mv D ZC x v W x 2 P \ Z :
Then
d Proj K ŒP smooth ” K ŒMv Š K ŒX1;:::;Xd ” Mv Š ZC
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:
a n Let I D Ker and M Df .X / W a 2 ZCg
Theorem The following are equivalent: 1 M is an affine monoid and R D K ŒM ; a b n 2 I is prime, generated by binomials X X ,a; b 2 ZC; ˙ 3 I D IK ŒX 1 \ K ŒX , I is generated by binomials X a X b, and U Dfa b W X a X b 2 Ig is a direct summand of Zn.
Definition A prime ideal I as above is called a toric ideal.
Winfried Bruns Lattice polytopes a b X X 2 I ” a1x1 C Canxn D b1x1 C Cbnxn (using additive notation in the monoid): The binomials in I represent the linear dependencies of the vectors generating M, and, in the polytopalP P case, the affine dependencies of the lattice points of P ( ai D bi in this case).
Algorithmic approach for the computation of the toric ideal I
(M D ZCx1 C CZCxn):
n Compute the kernel Zc1 C CZcn r of W Z 7! gp.M/, ei 7! xi , r D rank M C c c Saturate the ideal generated by X i X i , i D 1;:::;n r with respect to X1;:::;Xn.
C C; ZC ci D ci ci , ci ci .
Winfried Bruns Lattice polytopes Examples
012
Œ Œ 0 ; 1 ; 2 Œ ; ; =. 2/ K P D K Y Z Y Z Y Z Š K X1 X2 X3 X1X3 X2
ΠΠ0 0 ; 0 1 ; 1 0 ; 1 1 K P D K Y1 Y2 Z Y1 Y2 Z Y1 Y2 Z Y1 Y2 Z
Š K ŒX1; X2; X3; X4 =.X1X4 X2X3/
Œ Œ 1 0 ; 0 1 ; 1 1 ;; 0 0 K P D K Y1 Y2 Z Y1 Y2 Z Y1 Y2 Z Y1 Y2 Z Œ ; ; ; =. 3/ Š K X1 X2 X3 X4 X1X2X3 X4
Winfried Bruns Lattice polytopes An affine monoid M generates the cone X RCM D ai xi W xi 2 M; ai 2 RC P n Z R Since M D iD1 Cxi is finitely generated, CM 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 converse is also true.
Lemma (Gordan’s lemma) Let L Zd be a subgroup and C Rd a rational cone. Then L \ C is an affine monoid. Proof. Let V D RL Rd . Then V \ Qd D QL and C \ V is a rational cone in V ) We may assume that L D Zd . d C is generated by elements y1;:::;yn 2 M D C \ Z .
x 2 C ) x D a1y1 C Canyn ai 2 RC:
0 00 0 x D x C x ; x Dba1cy1 C Cbancyn:
Winfried Bruns Lattice polytopes Clearly x0 2 M.But
00 00 x 2 M ) x 2 gp.M/ \ C ) x 2 M: x00 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 generated module over N D ZCy1 C CZCyn: [ M D N C x: x2B\M
The monoid M D L \ C has a special property: it is integrally closed in 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 group containing M.
Winfried Bruns Lattice polytopes Theorem M Zd integrally closed affine monoid ” there exists a rational cone C such that M D C \ Zd ; d b d M Z affine monoid ) the integral closure M D RCM \ Z is a finitely generated M-module and therefore an affine monoid.
This applies especially to the choice Zd Š gp.M/. In this case MN D Mb 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 of fractions Q:ifx 2 Q satisfies an equation
n n 1 x C an 1x C Ca1x 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 only if 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.Itispositive if deg x > 0forx ¤ 0.
Proposition For M affine the following are equivalent: 1 Mispositive; 2 RCMispointed (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 CgDRfx 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 exists d a unique linear form F W R ! R with the following properties:
F Dfx 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 d s W R ! R ; .x/ D F .x/ W F facet :
s Then .M/ .MN / ZC.SinceC 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 0orz D 0.
Proposition The irreducible elements of a positive affine monoid form its unique minimal system of generators.
Proof. Every system of generators contains all the irreducible elements. Enough to show: every element is the sum of irreducibles. n Can assume: M ZC. Set deg x D s1 C Cxn. Now induction on deg x. Either x is irreducible or it decomposes as x D y C z with deg 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, one can compute Hilbert bases of the normalization MN and and the integral closure Mb 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 Singular via a library.
Winfried Bruns Lattice polytopes Lecture 2
Unimodular covers and triangulations
Winfried Bruns Lattice polytopes A nonnormal polytope
Let P be a lattice polytope. In general M.P/ is not normal. Example:
3 P Dfx 2 R 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/ ¤ MN .P/, Hilb.M.P// ¤ Hilb.MN .P//.
Where can we find the remaining elements of Hilb.Mb.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 independent vectors.
v2
v1 0 ˚ par.v1;:::;vn/ D q1v1 C Cqnvn W 0 qi < 1; i D 1;:::;n n E D Z \ par.v1;:::;vn/
Winfried Bruns Lattice polytopes n v1;:::;vn 2 Z linearly independent. Set
M D ZCv1 C CZCvn; U D gp.M/; C D RCM
Proposition
Then 1 E is a system of generators of the M-module C \ Zn; 2 .x C M/ \ .y C M/ D; for x ; y 2 E, x ¤ y; 3 #E D ŒZn W U ; n 4 Hilb.C \ Z / 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.
Winfried Bruns Lattice polytopes Bounding the Hilbert basis
Theorem (Ewald-Wessels, B.-Gubeladze-Trung) Let P be a lattice polytope, Then Mb.P/ is generated by elements of degree dim P 1. More precisely, as a module over M.P/ its is generated by elements of degree dim P 1.
Proof. Triangulate P such that each simplex contains only its vertices as lattice points:
Winfried Bruns Lattice polytopes Enough to consider a simplex P whose only lattice points are the vertices.
Proposition on simplicial cones implies: Can generate Mb.P/ by elements with deg y D dim P. (The simplex has dim P C 1 vertices!)
Cute trick: assume x 2 E (notation as above) has degree d. Then v1 C CvdC1 y has degree 1 and lies in P. But then y D v1 C Cvi 1 C viC1 C CvdC1 for some i. Contradiction!
The bound in the theorem is the best possible in all dimensions.
Winfried Bruns Lattice polytopes In the corollary we apply attributes to P that, strictly speaking, are defined 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 Mb.cP/ Dfx 2 Mb.P/ W deg y 0 .c/g:
If deg y D mc, then y D x C z1 C Czu with deg x dim P 1 and deg zj D 1(u D mc deg x). Cut this sum into subsums of degree c.
Winfried Bruns Lattice polytopes This result follows the principle that a graded rings “improves” by the passage 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 by combinatorial methods:
Theorem (B.-Gubeladze-Trung) Let c dim P. Then K ŒcP has a presentation as a residue class ring of a polynomial ring 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 for c 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 of the 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 kŒP Koszul or at least defined by a toric ideal generated in degree 2? Is K ŒP normal?
The second question has a good chance for a positive answer since the corner monoids are free. However, normality of the corner monoids 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 stronger properties
I think the answer is no, but this is not the last word.
Winfried Bruns Lattice polytopes d d Recall: P D conv.x1;:::;xn/ R , xi 2 Z , is called a lattice polytope.
1
P 2
D conv.v0;:::;vd /, v0;:::;vd affinely independent, is a simplex.
Winfried Bruns Lattice polytopes P d Z. / Set U D iD0 vi v0 .
d . / D ŒZ W U D multiplicity of
is unimodular if . / D 1.
is empty if vert. / D \ Zd .
Lemma 0 1 v1 v0 B : C . / D dŠ vol. / D˙det @ : A
vd v0
When is P covered by its unimodular subsimplices?
For short: P has UC.
Winfried Bruns Lattice polytopes Multiplicity for simplicial cones:
d Let v1;:::;vd 2 Z be linearly independent elements, and suppose that each vi has coprime entries. Set C D RCv1 C CRCvd . Then
.C/ Djdet.v1;:::;vd /jD .conv.0; v1;:::;vd //:
C is unimodular if .C/ D 1.
Winfried Bruns Lattice polytopes Low Dimensions
1 01234 d D 1: P has a unique unimodular triangulation.
d D 2:
Recall: lattice polytopes of dim 2 are integrally closed Therefore every empty lattice triangle is unimodular ) every 2-polytope has a unimodular triangulation.
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 Mb.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 an integrally closed affine monoid.
Generalization of the condition UC is UHC: C the union of its unimodular 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 have unimodular 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 CRCxn be a cone of dimension d, and y 2 C. ;:::; R Then there exists i1 id and aij 2 C such that
y D ai1 xi1 C Caid xid . This can be viewed as a consequence of the existence of triangulations, 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 a submonoid 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 by linearly 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 monoids of type C \ Zd .
Does every rational cone C have ICP? (We assign properties to C that are defined for C \ Zd ).
Winfried Bruns Lattice polytopes Counterexamples
(Bouvier and Gonzalez-Sprinberg, 1994) There exists a cone of dimension 4 without a unimodular triangulation into subcones generated by elements of Hilb.M/, M D C \ Z5. (Kantor-Sarkaria, 2003) There exists a normal lattice polytope of dimension 3 without a unimodular triangulation. (B.-Gubeladze-Henk-Martin-Weismantel, 1998) There exists a normal lattice polytope of dimension 5 without ICP. (2006) There exists a normal lattice polytope of dimension 5 with ICP that violates UHC.
Winfried Bruns Lattice polytopes 6 C6 \ Z with Hilbert basis z1;:::;z10,isofformC.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 0 . ; ; ; ; ; / 0 . ; ; ; ; ; / z11 D 0 1 2 1 1 2 z12 D 1 0 3 0 0 3 0 . 0 / to the Hilbert basis, then we obtain a cone C6 of type C P5 that satisfies ICP, but violates UHC.
Winfried Bruns Lattice polytopes Z6 0 Z6 Interesting fact: both monoids C6 \ and C6 \ have the Frobenius group F20 as their automorphism group.
The results of very long searches for counterexamples in dimensions 6 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., to appear
Winfried Bruns Lattice polytopes Caratheodory´ rank
Definition Let M be a positive affine monoid, y 2 M.Set
.y/ D minfm W y D a1x1 C Camxm; 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 of Ewald-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/,evenifM is normal. But it is open to what extent the bound in the theorem is sharp, or close to being sharp.
Using C6 and direct sums: Proposition For all n 6 there exist normal affine monoids with . / 7 CR M b6 rank Mc.
Winfried Bruns Lattice polytopes Strategy of the search for counterexamples to UHC and ICP
M always a normal monoid, M D C \ Zd .
Definition 0 Let x 2 Hilb.M/ and M D ZC Hilb.M/ nfxg.Wecallx destructive,if rank M0 < rank M or Hilb.M/ nfxg is not the Hilbert basis of 0 RCM \ gp.M/
Roughly speaking, Hilb.M/ nfxg is not the Hilbert basis of a cone of full dimension. A non-destructive element must span an extreme ray of RCM.
Definition M (or RCM)istight if every element of Hilb.M/ is destructive.
Winfried Bruns Lattice polytopes Suppose x generates an extreme ray of RCM. Then
00 00 00 MŒ x Š Z ˚ M ; M normal; rank M D rank M 1:
Lemma Suppose x is non-destructive. Then M0:M00 have UHC ) M has UHC CR.M/ min.CR.M0/; CR.M00/ 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 C
C0
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: UHC violated; otherwise split D along support hyperplane of a unimodular subcone, and apply recursion.
U
D2
D1
Winfried Bruns Lattice polytopes unicover.D; n/ 1 for i n to N 2 do
3 if D Ui 4 then return
5 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 C 2 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 stellar subdivision to reduce multiplicities.
Problem: bound the size of the vectors spanning the unimodular subcones! 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:
v2
P
v0 v1 Winfried Bruns Lattice polytopes cP has the same corner cones as P, and if c 0, then the tiles become small, enough tiles should lie in cP and cover it.
In the next step the “basic” tiles get into c0P
v2
v0 v1
Winfried Bruns Lattice polytopes Tile corner cones:
v2
v0 v1 Then we increase again so that tiles in c00P cover area between vertex and red line through barycenter (from all vertices) ) c00P is covered.
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) cpol Let P be a d-polytope. cP has UC for all c d , .ld .d//2 : 9 cpol D O d 16 5 ; d 4 p .d/ D .d 1/d d 1e:
The step from corner cones to P is rather easy and ”polynomial”:
Lemma Suppose the basic simplices of the cornerp cones lie in P. Then cP has a unimodular cover for all c > d d.
So one must find a good bound for the unimodular triangulation of rational cones! Enough to do simplicial cones.
Winfried Bruns Lattice polytopes Theorem
Let C be a rational simplicial d-cone and C the simplex spanned by O and the extreme integral generators. Then 1 (M. v. Thaden) C has a triangulation into unimodular simplicial
cones Di such that Hilb.Di / c C for some
.ld. .C///2 d 2 9 c . .C//7 : 4 4
2 C has a cover by unimodular simplicial cones Di such that Hilb.Di / c C for some
2 2 .ld. .d/// d 8 9 c .d C 1/ .d/ ; 4 4 ˙p .d/ D d 1 .d 1/
Winfried Bruns Lattice polytopes von Thaden claims that he can improve the bounds to polynomial size, at least for covers. (Should be the main result of his thesis.)
cpol Best possible value: d 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 and c0CP for all c0 > 0 have unimodular triangulations.
Open problem: Does cP have a unimodular triangulation for all C 0? Does there exist a uniform bound only in terms of dimension?
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 a combinatorial problem:
Suppose that n distinct objects, each available in k identical copies, are distributed among n persons in such a way that each person receives exactly k objects.
What can be said about the number H.n; k/ of such distributions?
Winfried Bruns Lattice polytopes They formulated some conjectures:
2 (ADG-1) there exists a polynomial Pn.k/ of degree .n 1/ such that 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;
.n 1/2 (ADG-3) Pn. k/ D . 1/ Pn.k n/ for all k 2 Z.
These conjectures were proved and extended by R. P. Stanley using methods of commutative algebra. Basic idea: interpret H.n; k/ (n fixed) as the Hilbert function of some graded ring.
The weaker version (ADG-1) of (ADG-2) has been included for didactical purposes. A compact account has been given in W.B., Commutative algebra arising from the Anand-Dumir-Gupta conjectures; 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
n n ) A D .aij / 2 ZC such that
Xn Xn ail D amj D k; i; j D 1;:::;n: lD1 mD1
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 usually requires further properties for those.
Winfried Bruns Lattice polytopes Two famous magic squares:
16 3 2 13 8 1 6 5 10 11 8 3 5 7 9 6 7 12 4 9 2 4 15 14 1
The 3 3 square can be found in ancient Chinese sources and the 4 4 appears in Albrecht D¨urer’s engraving Melancholia (1514). It has remarkable symmetries and shows the year of its creation.
Winfried Bruns Lattice polytopes A step towards algebra
n n Let Mn be the set of all matrices A D .aij/ 2 ZC such that
Xn Xn ail D amj ; i; j D 1;:::;n: lD1 mD1 M By Gordan’sP lemma n is an affine, normal monoid, and A 7! magic n M sum k D kD1 a1k is a positive grading on . 2 rank Mn D .n 1/ C 1
Theorem (Birkhoff-von Neumann)
Mn is generated by the its degree 1 elements, namely the permutation 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”, and 2 generated in degree 1. dim R D rank Mn D .n 1/ C 1.
) H.n; k/ D dimK Rk D H.R; k/ is the Hilbert function of R!
2 ) (ADG-1): there exists a polynomial Pn of degree .n 1/ such that 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
1 Let K be a field, and R D˚kD0Rk 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 X k HM .t/ D H.M:k/t : k2Z
Winfried Bruns Lattice polytopes Fundamental fact:
Theorem (Hilbert-Serre) 1 Let K be a field, and R D˚kD0Rk 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 there exists a Laurent polynomial Q 2 ZŒt; t 1 such that . / . / Q Q t : HM t D n . gi / iD1 1 t
More precisely: HM .t/is the Laurent expansion at 0 of the rational function on the right hand side.
Winfried Bruns Lattice polytopes Refinement: M is finitely generated over a graded Noether normalization K Œx1;:::;xd :
S R is a graded Noether normalization of M if
S D K Œx1;:::;xd with algebraically independent elements x1;:::;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 a graded module over a polynomial ring S such that AnnS M D 0. We also say that x1;:::;xd is a homogeneous system of parameters (hsop).
Special case: let us say that M is essentially standard graded (est) if
M 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 Ddeg 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
Q.t/ HM .t/ D : .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 . / e M d 1 PM D X C terms of lower degree: .d 1/Š
Winfried Bruns Lattice polytopes Recall 2 (ADG-1) there exists a polynomial Pn.k/ of degree .n 1/ such that H.n; k/ D Pn.k/ for all k 0;
With R D K ŒMn this is equivalent to
u 1 C h1t C :::hut HR.t/ D ; d D rank Mn: .1 t/d 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 .n 1/2 (ADG-3) Pn. k/ D . 1/ Pn.k n/ for all k 2 Z into a property of the rational function HR.t/? Note that the shift n in (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 Hilbert polynomial P. Suppose deg HR.t/ D g < 0. Then the following are equivalent: P. k/ D . 1/d 1P.k C g/ for all k 2 Z; d 1 g . 1/ HR.t / D t HR.t/;
hi D hu i for all i: the h-vector is palindromic.
The equivalence of the last two assertions is very easy, but the equivalence with the first is tricky. (Goes back to Polya, explicitly stated by Popoviciu.)
Winfried Bruns Lattice polytopes For (ADG-2) we have to compute the degree of the Hilbert series as a rational function.
For (ADG-3) it is best to work with the most compact form
d 1 g . 1/ HR.t / D t HR.t/:
We will see that the proofs of (ADG-2) and (ADG-3) can be based on homological 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 graded Noether normalization S of M.
An intrinsic definition shows that the choice of S is irrelevant. Note that M always has a finite free resolution over S, and it is Cohen-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-Macaulay for every field K .
There is no easy proof of this powerful theorem. For example, it can be derived from the Hochster-Roberts theorem, using that
K 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 is Cohen-Macaulay for every field K .
Let d D rank M. Choose elements x1;:::;xd 2 M generating RCM, and set Xd par.x1;:::;xd / D qi xi W qi 2 Œ0; 1/ : iD1
Winfried Bruns Lattice polytopes With N D ZCx1 C CZCxd : we know from the previous lecture: S M D z2E z C N, N is free, The union is disjoint. x In commutative algebra terms: 2 K ŒM is finite over K ŒN . K ŒN Š K ŒX ;:::;X . 1 d x K ŒM is a free module over K ŒN . 1 0 ) K ŒM is Cohen-Macaulay.
Winfried Bruns Lattice polytopes A combinatorial consequence of the Cohen-Macaulay property
Theorem Let M be a graded Cohen-Macaulay module over the positively
graded K -algebra R and x1;:::;xd ahsop.forM,ei D deg xi . Let
a b hat C Chbt H .t/ Q ; h ; h 0: M D d a b ¤ . ei / iD1 1 t
Then hi 0 for all i.
If M D R D K ŒR1 , then hi > 0 for all i D 0;:::;b.
a b Proof. hat C Chbt is the Hilbert series of
M=.x1M C Cxd M/:
So each hi is the dimension of a vector space.
Winfried Bruns Lattice polytopes .n 1/2C1 1 n For (ADG-3) we have to show . 1/ HR.t / D t HR.t/ for R D K ŒMn . Strategy:
d 1 Find an R-module ! with H! .t/ D . 1/ HR.t /, 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. Thus . / g . / HR.g/ t D t HR 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 :
C C . /d g1 gd d 1 1 t . 1/ H .t / D Q D Q D H! .t/ S d d . gi / . gi / iD1 1 t iD1 1 t
with ! D !S D S. .g1 C Cgd //
Winfried Bruns Lattice polytopes Now R free over S D K Œx1;:::;xd , with homogeneous basis y1;:::;ym:
Mm Mu hi R Š Syj Š S. i/ ; hi D #fj W deg yj D ig; jD1 iD1 u HR.t/D .h0 C h1t C Chut /HS.t/ D Q.t/HS.t/
Set !R D HomS.R;!S /. Then, with s D g1 C Cgd
Mu Mu hi hi !R Š HomS.S. i/ ; S. s// Š S. i C s/ ; iD1 iD1 . / . s s u/ . / . 1/ s . / H!R t D h0t C Chut HS t D Q t t HS t d 1 1 d 1 D . 1/ Q.t /HS.t / D . 1/ HR.t /:
Winfried Bruns Lattice polytopes Let us rewrite H! .t/:
s u s hut C Ch0t ! . / Q H t D d . gi / iD1 1 t
Since s D g1 C Cgd , 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/ an R-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 the graded 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 find the canonical module of R D K ŒM (Cohen-Macaulay by Hochster’s theorem).
Theorem (Danilov, Stanley) The ideal I generated by the monomials in the interior of RCMisthe canonical module of K ŒM (with respect to every positive grading of M 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 2 Mn where 1 is the magic square with all entries 1: Thus int.Mn/ D 1 C Mn.
2 The magic sum of 1 is n.So!R D R. n/.Withd D .n 1/ C 1 this implies
1 d n HR.t / D . 1/ t HR.t/ W (ADG-3) and
deg HR.t/ D n W (ADG-2)
Winfried Bruns Lattice polytopes Recall that we have written u 1 C h1t C Chut HR.t/ D : .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 the h-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 a class 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/,gD deg HR.t/;
h0 D hu i for i D 0;:::;u: the h-vector is palindromic; 1 d g HR.t / D . 1/ t HR.t/.
Conversely, if R is a Cohen-Macaulay integral domain such that 1 d h HR.t / D . 1/ t HR.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 have already proved central theorems about lattice point counting.
Let P Rd be a lattice polytope. We set
1 E.P; k/ D #.P \ Zd / D #.kP \ Zd /: k This is the Ehrhart function of P.
the corresponding power series is the Ehrhart series X k EP .t/ D E.P; k/t : kD0
Winfried Bruns Lattice polytopes We know kP \ Zd $fx 2 Mb.P/ W deg x D kg:
Therefore, with R D K ŒMb.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: Mb.P/ is a finitely generated M.P/-module, in other words: P lattice polytope ) R D K ŒMb.P/ is an est graded algebra.
Winfried Bruns Lattice polytopes Theorem Let P be a lattice polytope of dimension d. Then
u 1 C h1t C :::hut EP .t/ D HR.t/ D .1 t/dC1 with
d deg HR.t/ D u .d C 1/ D minfk W int.P/ \ Z ¤;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 have omitted the reciprocity of HR $ P and H! $ int.P/ which in this context 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 that the Hilbert series can be written
u 1 C h1t C :::hut HR.t/ D ; d D dim R; hu ¤ 0: .1 t/d
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 a Cohen-Macaulay domain, then this symmetry of the h-vector implies that R is Gorenstein.
For R D K ŒMn , the (Gorenstein) algebra over the monoid of magic squares, 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 then extended to a larger class of Gorenstein monoid algebras by B. and Romer¨ (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 the unimodality of the h-vector: one needs that R D K ŒR1 is standard graded (as K ŒMn is). As far as I know, there is no example of a standard graded Gorenstein domain with a non-unimodal h-vector, and to prove or disprove this property is the greatest challenge in combinatorial commutative algebra.
Winfried Bruns Lattice polytopes In the characterization of Gorenstein rings by the symmetry condition hu i D hi the hypothesis domain cannot be omitted.
If R D K ŒX1;:::;Xn =I is a Gorenstein ring, then one often finds monomial orders on the polynomial ring such that 0 R 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 the leading monomials of the elements of I.
Question: suppose R D K ŒX1;:::;Xn =I is Gorenstein. Can one find a monomial order on K ŒX1;:::;Xn such that 0 R 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 monoid domains. The connection between unimodality of the h-vector and initial ideals is given by certain triangulations.
Winfried Bruns Lattice polytopes The g-theorem
There is a famous instance where unimodality of the h-vectors and more is known.
Let be a simplicial complex of dimension d. (The dimension of is the maximal dimension of a face (D simplex) of ). It is useful to set d D dim C 1.
The i-th component of the f -vector counts the number of faces of dimension i:
f . / D .f0;:::;fd 1/; fi D #fı 2 W dim ı D ig
The f -vectors of simplicial complexes are characterized by the Kruskal-Katona theorem.
Winfried Bruns Lattice polytopes Example:
2 5 dim D 2 3 f . / D .5; 6; 2/
1 4 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
Xd Xd i d i i hi s .1 C s/ D fi 1s .f 1 D 1/: iD0 iD0 ;:::; ;:::; Let v1 vn be the vertices of . A subset N Dfvi1 vim g is a nonface of is N is not the vertex set of a face of .
In the polynomial ring K ŒX1;:::;Xn we set
N ; X D Xi1 Xim and define the Stanley-Reisner ring or face ring of by
N K Œ D K ŒX1;:::;Xn =I. /; I. / D .X 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 Π, and h. / D .1 D h0;:::;hu/. Then
u 1 C h1t C Chd t HR.t/ D : .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 boundary complexes of simplicial polytopes. (A polytope is simplicial if all its facets 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 and that 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 bounded above by the the h-vector of the cyclic polytope of the same dimension and the same number of vertices.
Winfried Bruns Lattice polytopes The upper bound theorem was later on generalized by Stanley to simplicial spheres, i, e. simplicial complexes for which j jŠSd 1. The main argument:
Theorem (Stanley) Let be a simplicial sphere. Then K Πis a Gorenstein ring.
Note that a simplicial sphere need not be the boundary of a simplicial polytope. Boundaries of polytopes are shellable (Brugesser-Mani), and this was used by McMullen. However, there exist non-shellable simplicial spheres.
Winfried Bruns Lattice polytopes For a simplicial complex satisfying the Dehn-Sommerville equations one 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 be characterized as follows: Theorem (g-theorem)
The h-vectors h.@P/ D .1 D h0;:::;hd / of the boundaries of simplicial polytopes P are characterized by the conditions
hi D hd i ,iD 0;:::;d; the corresponding g-vector is the h-vector of a standard graded K -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 of standard graded K -algebras in terms of explicit inequalities (involving binomial expansions). Therefore we say that the g-vector is a Macaulay sequence if it is the h-vector of a standard graded K -algebra.
Open problem: does the g-theorem hold for simplicial spheres?
The advantage of simplicial polytopes is that one can assign a toric variety to them, and Stanley used topological properties of this variety.
Strategy: In order to prove unimodality (or even the Macaulay condition) for an h-vector, show it is the h-vector of the boundary of a simplicial polytope (or at least a simplicial sphere, and hope for the solution 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 ŒMb.P/ is standard graded ( ” M.P/ D Mb.P/), R is Gorenstein.
Stanley’s conjecture on the unimodality of the h-vector for magic squares 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 Mustat¸aˇ and Payne (connection with stringy Hodge numbers).
Winfried Bruns Lattice polytopes Towards unimodality for Gorenstein polytopes
We have already observed that the canonical module of the algebra K ŒM for an affine normal monoid M has the interior int.M/ D M \ int.RCM/ of M as its monomial basis. This implies the equivalence 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 C Mforsomex2 M;
3 there exists x 2 Mwith 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 the following.
Winfried Bruns Lattice polytopes Let x be as in the lemma, and write x D y1 C Cyv 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 and choose x D y1 C Cyv as above. Then the elements X y1 X y2 ;:::;X yv 1 X yv form a regular R-sequence; the residue class ring R=(this sequence) is again a nor mal affine monoid Gorenstein algebra.
The proof is based on the construction of a unimodular triangulation that can be projected along the vector subspace generated by the differences yi yiC1.
Winfried Bruns Lattice polytopes Now suppose that M D M.P/ for a Gorenstein polytope. Then x can be decomposed into a sum of degree 1 elements. Modulo a regular sequence of 1-forms the h-vector does not change, and the property of being standard graded is preserved:
Corollary Suppose P is a Gorenstein polytope. Then there exists a Gorenstein polytope P0 with the following properties: P0 has exactly one interior lattice polytope (namely the projection of x); h.P0/ D h.P/.
This reduces the problem of unimodality to the integrally closed reflexive polytopes. (P reflexive ” P contains exactly one interior lattice polytope and K ŒMb.P/ is Gorenstein.)
Winfried Bruns Lattice polytopes In order to get closer to the g-theorem we need a unimodular triangulation ( ) K ŒMb.P/ standard graded).
Theorem Let P be a Gorenstein polytope with a unimodular triangulation. Then there exists a Gorenstein polytope P0 with the following properties: P0 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 triangulation and to modify it to one that can be projected. The g-theorem appears at the horizon: at least is a simplicial sphere (and somewhat more than that).
Winfried Bruns Lattice polytopes y1
P P0
y2 Change of the triangulation and projection
Winfried Bruns Lattice polytopes In order to apply the g-theorem (as it is now), the hypothesis on the triangulation must be strengthened:
Corollary Let P be a Gorenstein polytope with a regular unimodular triangulation. Then there exists a Gorenstein polytope P0 with the following properties: P0 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 simplicial polytope. Therefore h.P/ is a Macaulay sequence and, in particular, is unimodal.
Winfried Bruns Lattice polytopes A regular subdivision arises as the decomposition of P into the domains 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 toric ideals (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 unimodular triangulation, and let IP be the toric ideal defining K ŒP D K ŒX1;:::;Xn =IP . Then there exists a monomial order on K ŒX1;:::;Xn such that K ŒX1;:::;Xn = in.IP / is also Gorenstein.
Here regularity of the triangulation is essential.
Winfried Bruns Lattice polytopes