Polytopal Linear Groups

Journal of Algebra 218, 715᎐737 Ž1999. Article ID jabr.1998.7831, available online at http:rrwww.idealibrary.com on

Polytopal Linear Groups Winfried Bruns FB MathematikrInformatik, Uni¨ ersitat ¨ Osnabruck, ¨ 49069 Osnabruck, ¨ Germany E-mail: [email protected]

and Joseph Gubeladze A. Razmadze Mathematical Institute, Alexidze St. 1, 380093 Tbilisi, Georgia E-mail: [email protected] Communicated by Craig Huneke Received June 24, 1998

1. INTRODUCTION The main objects of this paper are the graded automorphisms of polytopal semigroup rings, i.e., semigroup rings k w SP x where k is a field and SP is the semigroup associated with a lattice polytope P ŽBruns, Gubeladze, and Trung wBGTx.. The generators of k w SP x correspond bijectively to the lattice points in P, and their relations are the binomials representing the affine dependencies of the lattice points. The simplest examples of such rings are the polynomial rings k w X 1 , . . . , X n x with the standard grading. They are associated to the unit simplices ⌬ ny 1. The graded automorphism group GL nŽ k . of k w X 1 , . . . , X n x is generated by diagonal and elementary automorphisms. Our main result is a generalization of this classical fact to the graded automorphism group ⌫k Ž P . of an arbitrary polytopal semigroup ring: It says that each automorphism ␥ g ⌫k Ž P . has a Žnon-unique. normal form as a composition of toric and elementary automorphisms and symmetries of the underlying polytope. In view of this analogy we call the groups ⌫k Ž P . polytopal linear groups. As in the case of GL nŽ k ., the elementary and toric automorphisms generate ⌫k Ž P . if Žand only if. this group is connected. The elementary automorphisms are defined in terms of so-called column structures on P. 715 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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The proof of the main result consists of two major steps. First we prove that an arbitrary automorphism that leaves the ‘‘interior’’ of k w SP x invariant preserves the monomial structure, and is therefore a composition of a toric automorphism and a symmetry. ŽIf k w SP x is normal, the interior is just the canonical module.. Second we show how to ‘‘correct’’ a graded automorphism ␥ by elementary automorphisms such that the composition preserves the interior of k w SP x. This correction is based on the action of ␥ on the divisorial ideals of the normalization of k w S P x. Our approach applies to arbitrary fields k, and it can even be generalized to graded automorphisms of an arbitrary normal affine semigroup ring Žsee Remark 3.3.. A further application is a generalization from a single polytope to Žlattice . polyhedral complexes wBGx. ŽRings defined by lattice polyhedral complexes generalize polytopal semigroup rings in the same way as Stanley᎐Reisner rings generalize polynomial rings. As an application outside the class of semigroup rings we determine the graded automorphism groups of the determinantal rings. The geometric objects associated to polytopal semigroup rings are projective toric varieties. The description of the automorphism group of a smooth complete toric ⺓-variety given by a fan F in terms of the roots of F is due to Demazure in his fundamental work wDex. The analogous description of the automorphism group of quasi-smooth complete toric varieties Žover ⺓. has recently been obtained by Cox wCox. As we learnt wBux generalized Cox’s results after this work had been completed, Buhler ¨ to arbitrary complete toric varieties. In the last section we derive a description of the automorphism group of an arbitrary projective toric variety from our theorem on polytopal linear groups. This method has the advantage of working in arbitrary characteristic, and furthermore it generalizes to arrangements of toric varieties Žsee wBGx.. Notation and Terminology. Let n be a natural number and let P be a finite, convex, n-dimensional lattice polytope in ⺢ n ; i.e., P has vertices in ⺪ n ; ⺢ n. We will assume that the lattice points of P affinely generate the whole group ⺪ n Žthis can always be achieved by a suitable change of the lattice of reference .. Let k be a field. The polytopal algebra Žor polytopal semigroup ring . associated with P and k is the semigroup algebra k w SP x, where SP is the additive sub-semigroup of ⺪ nq 1 generated by Ž x, 1. N x g P l ⺪ n4 wBGTx. The cone spanned by SP will be denoted by C Ž P .. The k-algebra k w SP x and, more generally, k w⺪ nq 1 x s k wgpŽ SP .x is naturally graded in such a way that the last component of Žthe exponent vector of. a monomial is its degree. The semigroup SP consists of homogeneous elements, and lattice points of P correspond to degree 1 elements of SP . In order to avoid cumbersome notation we will in the following identify the

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lattice points of P with the degree 1 elements in SP , and, more generally, elements of ⺪ nq 1 with Laurent monomials in k w⺪ nq 1 x. It follows immediately from the definitions that ⌫k Ž P . is an affine k-group: It can be identified with the closed subgroup of GL N Ž k ., N s 噛Ž P l ⺪ n ., given by the equations that reflect the relations between the degree 1 monomials in S P . Recall that a sub-semigroup S ; ⺪ m , m g ⺞, is called normal if x g gpŽ S . Žthe group of differences of S . and cx g S for some natural c imply x g S. This is equivalent to the normality of k w S x Žfor example, see Bruns and Herzog wBH, Chap. 6x.. A lattice polytope P is called normal if SP is a normal semigroup. Observe that P is normal if and only if S P s ⺪ nq 1 l C Ž P .. A convention: when a semigroup is considered as the semigroup of monomials in the corresponding ring, we use multiplicative notation for the semigroup operation; otherwise additive notation is used.

2. COLUMN STRUCTURES ON LATTICE POLYTOPES Let P be a lattice polytope as above. DEFINITION 2.1. An element ¨ g ⺪ n, ¨ / 0, is a column ¨ ector Žfor P . if there is a facet F ; P such that x q ¨ g P for every lattice point x g P _ F. For such P and ¨ the pair Ž P, ¨ . is called a column structure. The corresponding facet F is called its base facet and denoted by P¨ . One sees easily that for a column structure Ž P, ¨ . the set of lattice points in P is contained in the union of raysᎏcolumnsᎏparallel to the vector y¨ and with end-points in F. This is illustrated by Fig. 1. Moreover, the group ⺪ n is a direct sum of the two subgroups generated by ¨ and by the lattice points in P¨ , respectively. In particular, ¨ is a unimodular element of ⺪ n. In the following we will identify a column vector ¨ g ⺪ n with the element Ž ¨ , 0. g ⺪ nq 1. The proof of the next lemma is straightforward.

FIG. 1. A column structure.

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LEMMA 2.2. For a column structure Ž P, ¨ . and any element x g SP , such that x f C Ž P¨ ., one has x q ¨ g SP Ž here C Ž P¨ . denotes the facet of C Ž P . corresponding to P¨ .. One can easily control column structures in such formations as homothetic images and direct products of lattice polytopes. More precisely, let Pi be a lattice n i-polytope, i s 1, 2, and let c be a natural number. Then cP1 is the homothetic image of P1 centered at the origin with factor c and P1 = P2 is the direct product of the two polytopes, realized as a lattice polytope in ⺪ n1qn 2 in a natural way. Then one has the following observations: Ž). For any natural number c the two polytopes P1 and cP1 have the same column vectors. Ž)). The system of column vectors of P1 = P2 is the disjoint union of those of P1 and P2 Žembedded into ⺪ n1qn 2 .. Actually, Ž). is a special case of a more general observation on the polytopes defining the same normal fan. The normal fan N Ž P . of a Žlattice . polytope P ; ⺢ n is the family of cones in the dual space Ž⺢ n .* given by N Ž P . s Ž  ␸ g Ž ⺢ n . * N Max P Ž ␸ . s f 4 , f a face of P . ; here Max P Ž ␸ . is the set of those points in P at which ␸ attains its maximal value on P Žfor example, see Gelfand et al. wGKZx.. For each facet F of P there exist a unique unimodular ⺪-linear form ␸ F : ⺪ n ª ⺪ and a unique integer a F such that F s  x g P N ␸ F Ž x . s a F 4 and P s  x g ⺢ n N ␸ F Ž x . G a F for all facets F 4 , where we denote the natural extension of ␸ F to an ⺢-linear form on ⺢ n by ␸ F , too. That ¨ is a column vector for P with base facet F can now be described as follows: one has ␸ F Ž ¨ . s y1 and ␸G Ž ¨ . G 0 for all other facets G of P. The linear forms y␸ F generate the semigroups of lattice points in the rays Ži.e., one-dimensional cones. belonging to N Ž P . so that the system of column vectors of P is completely determined by N Ž P .: Ž))). Lattice n-polytopes P1 and P2 such that N Ž P1 . s N Ž P2 . have the same systems of column vectors. We have actually proved a slightly stronger result: if P and Q are lattice polytopes such that N Ž Q . ; N Ž P ., then ColŽ P . ; ColŽ Q ..

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We further illustrate the notion of column vector by Fig. 2: the polytope P1 has four column vectors, whereas the polytope P2 has no column vector. Let Ž P, ¨ . be a column structure. Then for each element x g SP we set ht ¨ Ž x . s m, where m is the largest non-negative integer for which x q m¨ g S P . Thus ht ¨ Ž x . is the ‘‘height’’ of x above the facet of the cone C Ž SP . corresponding to P¨ in the direction y¨ . More generally, for any facet F ; P we define the linear form ht F : X ⺢ nq 1 ª ⺢ by ht F Ž y . s ␸ F Ž y 1 , . . . , yn . y a F ynq1 , where ␸ F and a F are chosen as above. For x g SP and, more generally, for x g C Ž P . l ⺪ nq 1 the height ht F Ž x . of x is a non-negative integer. The kernel of ht F is just the hyperplane supporting C Ž P . in the facet corresponding to F, and C Ž P . is the cone defined by the support functions ht F . Clearly, for a column structure Ž P, ¨ . and a lattice point x g P we have ht ¨ Ž x . s ht P ¨Ž x ., as follows immediately from Lemma 2.2. 3. ELEMENTARY AUTOMORPHISMS AND THE MAIN RESULT Let Ž P, ¨ . be a column structure and let ␭ g k. We identify the vector ¨ , representing the difference of two lattice points in P, with the degree 0 element Ž ¨ , 0. g ⺪ nq 1 , and also with the corresponding monomial in k w⺪ nq 1 x. Then we define an injecti¨ e mapping from SP to QFŽ k w S P x., the quotient field of k w SP x, by the assignment x ¬ Ž 1 q ␭¨ .

ht ¨ Ž x .

x.

Since ht ¨ extends to a group homomorphism ⺪ nq 1 ª ⺪ our mapping is a homomorphism from SP to the multiplicative group of QFŽ k w SP x.. Now it is immediate from the definition of ht ¨ and Lemma 2.2 that the Žisomorphic. image of SP lies actually in k w SP x. Hence this mapping gives rise to a graded k-algebra endomorphism e¨␭ of k w SP x preserving the degree of an element. By Hilbert function arguments e¨␭ is an automorphism. Here is an alternative description of e¨␭. By a suitable integral change of coordinates we may assume that ¨ s Ž0, y1, 0, . . . , 0. and that P¨ lies in

FIG. 2. Two polytopes and their column structures.

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the subspace ⺢ ny 1 Žthus P is in the upper halfspace.. Now consider the standard unimodular n-simplex ⌬ n with vertices at the origin and standard coordinate vectors. It is clear that there is a sufficiently large natural number c, such that P is contained in a parallel translate of c⌬ n by a vector from ⺪ ny 1. Let ⌬ denote such a parallel translate. Then we have a graded k-algebra embedding k w SP x ; k w S⌬ x. Moreover, k w S⌬ x can be identified with the cth Veronese subring of the polynomial ring k w x 0 , . . . , x n x in such a way that ¨ s x 0rx 1. Now the automorphism of k w x 0 , . . . , x n x mapping x 1 to x 1 q ␭ x 0 and leaving all the other variables invariant induces an automorphism ␣ of the subalgebra k w S⌬ x, and ␣ in turn can be restricted to an automorphism of k w SP x, which is nothing else but e¨␭. It is clear from this description of e¨␭ that it becomes an elementary ␭ matrix Ž e01 in our notation. in the special case when P s ⌬ n , after the identification ⌫k Ž P . s GL nq1Ž k .. Therefore the automorphisms of type e¨␭ will be called elementary. LEMMA 3.1. Let ¨ 1 , . . . , ¨ s be pairwise different column ¨ ectors for P with the same base facet F s P¨ i , i s 1, . . . , s. Then the mapping

␸ : ⺑sk ª ⌫k Ž P . ,

Ž ␭1 , . . . , ␭ s . ¬ e¨␭11 ( ⭈⭈⭈ ( e¨␭ss ,

is an embedding of algebraic groups. In particular, e¨␭ii and e¨␭jj commute for ␭i any i, j g  1, . . . , s4 and the in¨ erse of e¨␭ii is ey ¨i . Ž⺑sk denotes the additive group of the s-dimensional affine space.. Proof. We define a new k-algebra automorphism ␽ of k w SP x by first setting

␽ Ž x . s Ž 1 q ␭1¨ 1 q ⭈⭈⭈ q␭ s¨ s .

ht F Ž x .

x

for x g S P and then extending ␽ linearly. Arguments very similar to those above show that ␽ is a graded k-algebra automorphism of k w SP x. The lemma is proved once we have verified that ␸ s ␽ . Choose a lattice point x g P such that ht F Ž x . s 1. ŽThe existence of such a point follows from the definition of a column vector: there is of course a lattice point x g P such that ht F Ž x . ) 0.. We know that gpŽ S P . s ⺪ nq 1 is generated by x and the lattice points in F. The lattice points in F are left unchanged by both ␽ and ␸ , and elementary computations show that ␸ Ž x . s Ž1 q ␭1¨ 1 q ⭈⭈⭈ q␭ s¨ s . x; hence ␸ Ž x . s ␽ Ž x .. The image of the embedding ␸ given by Lemma 3.1 is denoted by ⺑Ž F .. Of course, ⺑Ž F . may consist only of the identity map of k w SP x, namely if there is no column vector with base facet F. In the case in which P is the

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unit simplex and k w SP x is the polynomial ring, ⺑Ž F . is the subgroup of all matrices in GL nŽ k . that differ from the identity matrix only in the non-diagonal entries of a fixed column. For the statement of the main result we have to introduce some subgroups of ⌫k Ž P .. First, the Ž n q 1.-torus ⺤nq1 s Ž k*. nq 1 acts naturally on k w SP x by restriction of its action on k w⺪ nq 1 x given by

Ž ␰ 1 , . . . , ␰ nq1 . Ž e i . s ␰ i e i ,

i g w 1, n q 1 x ,

where e i is the ith standard basis vector of ⺪ nq 1. This gives rise to an algebraic embedding ⺤nq 1 ; ⌫k Ž P ., and we will identify ⺤nq1 with its image. It consists precisely of those automorphisms of k w S P x which multiply each monomial by a scalar from k*. Second, the automorphism group ⌺Ž P . of the semigroup SP is a finite subgroup of ⌫k Ž P . in a natural way. It is the group of integral affine transformations mapping P onto itself. Third, we have to consider a subgroup of ⌺Ž P . defined as follows. Assume ¨ and y¨ are both column vectors. Then for every point x g P l ⺪ n there is a unique y g P l ⺪ n such that ht ¨ Ž x . s hty¨ Ž y . and x y y is parallel to ¨ . The mapping x ¬ y gives rise to a semigroup automorphism of SP : it ‘‘inverts columns’’ that are parallel to ¨ . It is easy to see that these automorphisms generate a normal subgroup of ⌺Ž P ., which we denote by ⌺Ž P . inv . Finally, ColŽ P . is the set of column structures on P. Now the main result is THEOREM 3.2. Ža.

Let P be a con¨ ex lattice n-polytope and k a field.

E¨ ery element ␥ g ⌫k Ž P . has a Ž not uniquely determined. presenta-

tion

␥ s ␣ 1 ( ␣ 2 ( ⭈⭈⭈ ( ␣ r (␶ ( ␴ , where ␴ g ⌺Ž P ., ␶ g ⺤nq 1 , and ␣ i g ⺑Ž Fi . such that the facets Fi are pairwise different and 噛Ž Fi l ⺪ n . F 噛Ž Fiq1 l ⺪ n ., i g w1, r y 1x. Žb. For an infinite field k the connected component of unity ⌫k Ž P . 0 ; ⌫k Ž P . is generated by the subgroups ⺑Ž Fi . and ⺤nq1. It consists precisely of those graded automorphisms of k w SP x which induce the identity map on the di¨ isor class group of the normalization of k w SP x. Žc. dim ⌫k Ž P . s 噛ColŽ P . q n q 1. Žd. One has ⌫k Ž P . 0 l ⌺ Ž P . s ⌺ Ž P . inv and ⌫k Ž P .r⌫k Ž P . 0 f ⌺Ž P .r⌺Ž P . inv . Furthermore, if k is infinite, then ⺤nq1 is a maximal torus of ⌫k Ž P ..

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Remark 3.3. Ža. Our theorem is a ‘‘polytopal generalization’’ of the fact that any invertible matrix with entries from a field is a product of elementary matrices, permutation matrices, and diagonal matrices. The normal form in its part Ža. generalizes the fact that the elementary transformations e i␭j , j fixed, can be carried out consecutively. Žb. Observe that we do not claim the existence of a normal form as in Ža. for the elements from ⌫k Ž P . 0 if we exclude elements of ⌺Ž P . inv from the generating set. Žc. Let S ; ⺪ nq 1 be a normal affine semigroup such that 0 is the only invertible element in S. A priori S does not have a grading, but there always exists a grading of S such that the number of elements of a given degree is finite Žfor example, see wBH, Chap. 6x.. One can treat graded automorphisms of such semigroups as follows. It is well known that the cone C Ž S . spanned by S in ⺢ nq 1 is a finite rational strictly convex cone. An element ¨ g ⺪ nq 1 of degree 0 is called a column vector for S if there is a facet F of C Ž S . such that x q ¨ g S for every x g S _ F. The only disadvantage here is that the condition for column vectors involves an infinite number of lattice points, while for polytopal rings one only has to look at lattice points in a finite polytope Ždue to Lemma 2.2.. Then one introduces analogously the notion of an elementary automorphism e¨␭ Ž ␭ g k .. The proof of Theorem 3.2 we present below is applicable to this more general situation without any essential change, yielding a similar result for the group of graded k-automorphisms of k w S x. Žd. In tn attempt to generalize the theorem in a different direction, one could consider an arbitrary finite subset M of ⺪ n Žwith gpŽ M . s ⺪ n . and the semigroup SM generated by the elements Ž x, 1. g ⺪ nq 1 , x g M. However, examples show that there is no suitable notion of column vector in this generality: One can only construct the polytope P spanned by M, find the automorphism group of k w SP x, and try to determine ⌫k Ž M . as the subgroup of those elements of ⌫k Ž P . that can be restricted to k w SM x. ŽThis approach is possible because k w SP x is contained in the normalization of k w SM x.. Že. As a Žpossibly non-reduced. affine variety ⌫k Ž P . is already defined over the prime field k 0 of k since this is true for the affine variety Spec k w SP x. Let S be its coordinate ring over k 0 . Then the dimension of ⌫k Ž P . is just the Krull dimension of S or S m k, and part Žc. of the theorem must be understood accordingly. As an application to rings and varieties outside the class of semigroup rings and toric varieties we determine the groups of graded automorphisms of the determinantal rings. In plain terms, Corollary 3.4 answers the

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following question: let k be an infinite field, ␸ : k m n ª k m n a k-automorphism of the vector space k m n of m = n matrices over k, and r an integer, 1 F r - minŽ m, n.; when is rank ␸ Ž A. F r for all A g k m n with rank A F r? This holds obviously for transformations ␸ Ž A. s SATy1 with S g GL mŽ k . and T g GL nŽ k ., and for the transposition if m s n. Indeed, these are the only such transformations: COROLLARY 3.4. Let k be a field, X an m = n matrix of indeterminates, and R s K w X xrIrq1Ž X . the residue class ring of the polynomial ring K w X x in the entries of X modulo the ideal generated by the Ž r q 1.-minors of X, 1 F r - minŽ m, n.. Ža. The connected component G 0 of unity in G s gr.aut k Ž R . is the image of GL mŽ k . = GL nŽ k . in GL m nŽ k . under the map indicated abo¨ e, and is isomorphic to GL mŽ k . = GL nŽ k .rk* where k* is embedded diagonally. Žb. If m / n, the group G is connected, and if m s n, then G 0 has index 2 in G and G s G 0 j ␶ G 0 where ␶ is the transposition. Proof. The singular locus of Spec R is given by V Ž ᒍ . where ᒍ s Ir Ž X .rIrq1Ž X .; ᒍ is a prime ideal in R Žsee Bruns and Vetter wBV, Ž2.6., Ž6.3.x.. It follows that every automorphism of R must map ᒍ onto itself. Thus a linear substitution on k w X x for which Irq1Ž X . is stable also leaves Ir Ž X . invariant and therefore induces an automorphism of k w X xrIr Ž X .. This argument reduces the corollary to the case r s 1. For r s 1 one has the isomorphism R ª k w Yi Z j : i s 1, . . . , m, j s 1, . . . , n x ; k w Y1 , . . . , Ym , Z1 , . . . , Zn x induced by the assignment X i j ¬ Yi Z j . Thus R is just the Segre product of k w Y1 , . . . , Ym x and k w Z1 , . . . , Zn x, or, equivalently, R ( k w SP x where P is the direct product of the unit simplices ⌬ my 1 and ⌬ ny1. Part Ža. follows now from an analysis of the column structures of P Žsee observation Ž)). above. and the torus actions. For Žb. one observes that ClŽ R . ( ⺪; ideals representing the divisor classes 1 and y1 are given by Ž Y1 Z1 , . . . , Y1 Zn . and Ž Y1 Z1 , . . . , Ym Z1 . wBV, 8.4x. If m / n, these ideals have different numbers of generators; therefore every automorphism of R acts trivially on the divisor class group. In the case m s n, the transposition induces the map s ¬ ys on ClŽ R .. Now the rest follows again from the theorem above. ŽInstead of the divisoral arguments one could also discuss the symmetry group of ⌬ my 1 = ⌬ ny1..

4. PROOF OF THE MAIN RESULT Set SP s ⺪ nq 1 l C Ž P .. Then SP is the normalization of the semigroup SP and k w S P x is the normalization of the domain k w S P x. Let ⌫k Ž P . denote

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the group of graded k-algebra automorphisms of k w SP x. Since any automorphism of k w S P x extends to a unique automorphism of k w SP x we have a natural embedding ⌫k Ž P . ; ⌫k Ž P .. On the other hand, k w SP x and k w S P x have the same homogeneous components of degree 1. Hence ⌫k Ž P . s ⌫k Ž P .. Nevertheless we will use the notation ⌫k Ž P ., emphasizing the fact that we are dealing with automorphisms of k w SP x; ⌺Ž P . and ⌺Ž P . inv will refer to their images in ⌫k Ž P .. Furthermore, the extension of an elementary automorphism e¨␭ is also denoted by e¨␭ ; it satisfies the rule e¨␭ Ž x . s Ž1 q ␭¨ . ht ¨ Ž x . x for all x g SP . ŽThe equation ⌫k Ž P . s ⌫k Ž P . shows that the situation considered in Remark 3.3Žc. really generalizes Theorem 3.2; furthermore it explains the difference between polytopal algebras and arbitrary graded semigroup rings generated by their degree 1 elements.. In the following it is sometimes necessary to distinguish elements x g SP from products ␨ z with ␨ g k and z g SP . We will call x a monomial and ␨ z a term. Suppose ␥ g ⌫k Ž P . maps every monomial x to a term ␭ x y x , y x g SP . Then the assignment x ¬ y x is also a semigroup automorphism of SP . Denote it by ␴ . It obviously belongs to ⌺Ž P .. The mapping ␴y1 (␥ is of the type x ¬ ␰ x x, and clearly ␶ s ␴y1 (␥ g ⺤nq1. Therefore, ␥ s ␴ (␶ . Let intŽ C Ž P .. denote the interior of the cone C Ž P . and let ␻ s ŽintŽ C Ž P .. l ⺪ nq 1 . k w SP x be the corresponding monomial ideal. ŽIt is known that ␻ is a canonical module of k w SP x; see Danilov wDax, Stanley wStx, or wBH, Chap. 6x.. LEMMA 4.1. Ža. Suppose ␥ g ⌫k Ž P . preser¨ es the ideal ␻ . Then ␥ s ␴ (␶ with ␴ g ⌺Ž P . and ␶ g ⺤nq 1. Žb. One has ␴ (␶ ( ␴y1 g ⺤nq 1 for all ␴ g ⌺Ž P ., ␶ g ⺤nq1. Proof. Ža. By the argument above it is enough that ␥ maps monomials to terms. First consider a non-zero monomial x g SP l ␻ . We have ␥ Ž x . g ␻ . Since x is an ‘‘interior’’ monomial, k w SP x x s k w⺪ nq 1 x. On the other hand k w⺪ nq 1 x ; k w SP x␥ Ž x . . Indeed, since gpŽ SP . s ⺪ nq1 , it just suffices to observe that for any monomial z g SP there is a sufficiently large natural number c satisfying the following condition: The parallel translate of the Newton polytope N Ž␥ Ž x . c . by the vector yz g ⺢ nq 1 is inside the cone C Ž P . Žhere we use additive notation.. ŽObserve that N Ž␥ Ž x . c . is the homothetic image of N Ž␥ Ž x .., centered at the origin with factor c. Instead of Newton polytopes one could also use the minimal prime ideals of z, which we will introduce below; they all contain ␥ Ž x ... Hence all monomials become invertible in k w SP x␥ Ž x . .

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The crucial point is to compare the groups of units UŽ k w⺪ nq 1 x. s k* [ ⺪ and UŽ k w SP x␥ Ž x . .. The mapping ␥ induces an isomorphism nq 1

⺪ nq 1 f U k SP

ž

␥ Ž x.

/ rk*.

On the other hand we have seen that ⺪ nq 1 is embedded into UŽ k w SP x␥ Ž x . .rk* so that the elements from S P map to their classes in the quotient group. Assume that ␥ Ž x . is not a term. Then none of the powers of ␥ Ž x . is a term. In other words, none of the multiples of the class of ␥ Ž x . is in the image of ⺪ nq 1. This shows that rankŽUŽ k w SP x␥ Ž x . .rk*. ) n q 1, a contradiction. Now let y g SP be an arbitrary monomial and z a monomial in ␻ . Then yz g ␻ , and since ␥ Ž yz . is a term as shown above, ␥ Ž y . must be also a term. Žb. follows immediately from the fact that ␴ (␶ ( ␴y1 maps each monomial to a multiple of itself. In the light of Lemma 4.1Ža. we see that for Theorem 3.2Ža. it suffices to show the following claim: For every ␥ g ⌫k Ž P . there exist ␣ 1 g ⺑Ž F1 ., . . . , ␣ r g ⺑Ž Fr . such that

␣ r ( ␣ ry1 ( ⭈⭈⭈ ( ␣ 1 (␥ Ž ␻ . s ␻ and the Fi satisfy the side conditions of Theorem 3.2Ža.. For a facet F ; P we have constructed the group homomorphism ht F : ⺪ nq 1 ª ⺪. We now define the divisorial ideal of F by Div Ž F . s  x g SP N ht F Ž x . ) 0 4 k S P . It is clear that ␻ s F 1r DivŽ Fi ., where F1 , . . . , Fr are the facets of P. This shows the importance of the ideals DivŽ Fi . for our goals. We will use the following information about them. Ža. The ideals DivŽ Fi . are precisely the height 1 prime ideals generated by monomials. Moreover, the minimal prime overideals of any height 1 monomial ideal are among the ideals DivŽ Fi .. Note that ht F iŽ x . is precisely the discrete valuation of QFŽ k w SP x. that corresponds to DivŽ Fi . evaluated at x. Žb. Any divisorial fractional ideal of k w SP x is equivalent Ži.e., defines the same element in the divisor class group ClŽ k w SP x.. to a divisorial monomial ideal contained in k w SP x.

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Žc. Every divisorial monomial ideal I ; k w SP x has a unique presentation of the type r

Is

F DivŽ Fi . a

Ž i.

,

ai G 0

1

ŽDivŽ F .Ž j. is the jth symbolic power of Div Ž F ... For Ža. see wBH, Chap. 6x; Žb. is due to Chouinard wChx. ŽIt is in fact an immediate consequence of Nagata’s theorem on divisor class groups: if one inverts an ‘‘interior’’ monomial of S P , then the ring of quotients is the Laurent monomial ring k w⺪ nq 1 x; according to Ža., ClŽ k w SP x. is generated by the Div Ž Fi ... Statement Žc. is a consequence of Ža. since in a normal domain every divisorial ideal is an intersection of symbolic powers of the divisorial prime ideals in which it is contained. ŽFor all the facts about divisorial ideals that we will need in addition to Ža. above we refer the reader to Fossum wFox.. Before we prove the claim above Žreformulated as Lemma 4.4. we collect some auxiliary arguments. LEMMA 4.2. Let ¨ 1 , . . . , ¨ s be column ¨ ectors with the common base facet F s P¨ i and ␭1 , . . . , ␭ s g k. Then e¨␭11 ( ⭈⭈⭈ ( e¨␭ss Ž Div Ž F . . s Ž 1 q ␭1¨ 1 q ⭈⭈⭈ q␭ s¨ s . Div Ž F . and e¨␭11 ( ⭈⭈⭈ ( e¨␭ss Ž Div Ž G . . s Div Ž G . ,

G / F.

Proof. Using the automorphism ␽ from the proof of Lemma 3.1, we see immediately that e¨␭11 ( ⭈⭈⭈ ( e¨␭ss Ž Div Ž F . . ; Ž 1 q ␭1¨ 1 q ⭈⭈⭈ q␭ s¨ s . Div Ž F . . The left-hand side is a height 1 prime ideal Žbeing an automorphic image of such. and the right-hand side is a proper divisorial ideal inside k w SP x. Then, of course, the inclusion is an equality. For the second assertion it is enough to treat the case s s 1, ¨ s ¨ 1 , ␭ s ␭1. One has e¨␭ Ž x . s Ž 1 q ␭¨ .

ht F Ž x .

x,

and all the terms in the expansion of the right-hand side belong to Div Ž G . since ht G Ž ¨ . G 0. As above, the inclusion e¨␭ŽDiv Ž G .. ; Div Ž G . implies equality.

POLYTOPAL LINEAR GROUPS

LEMMA 4.3.

727

Let F ; P be a facet and let ␭1 , . . . , ␭ s g k _  04 and

¨ 1 , . . . ,¨ s g ⺪ nq 1 be pairwise different non-zero elements of degree 0. Suppose Ž1 q ␭1¨ 1 q ⭈⭈⭈ q␭ s¨ s .DivŽ F . ; k w SP x. Then ¨ 1 , . . . , ¨ s are column ¨ ectors

for P with the common base facet F. Proof. If x g P _ F is a lattice point, then x g Div Ž F .. Thus x¨ j is a degree 1 element of SP ; in additive notation this means x q ¨ j g P. The crucial step in the proof of our main result is the next lemma. LEMMA 4.4. Let ␥ g ⌫k Ž P ., and enumerate the facets F1 , . . . , Fr of P in such a way that 噛Ž Fi l ⺪ n . F 噛Ž Fiq1 l ⺪ n . for i g w1, r y 1x. Then there exists a permutation ␲ of w1, r x such that 噛Ž Fi l ⺪ n . s 噛Ž F␲ Ž i. l ⺪ n . for all i and

␣ r ( ⭈⭈⭈ ( ␣ 1 (␥ Ž Div Ž Fi . . s Div Ž F␲ Ž i. . , with suitable ␣ i g ⺑Ž F␲ Ž i. .. In fact, this lemma finishes the proof of Theorem 3.2Ža.: the resulting automorphism ␦ s ␣ r ( ⭈⭈⭈ ( ␣ 1 (␥ permutes the minimal prime ideals of ␻ and therefore preserves their intersection ␻ . By virtue of Lemma 4.1Ža. we then have ␦ s ␴ (␶ with ␴ g ⌺Ž P . and ␶ g ⺤nq 1. Finally, one just replaces each ␣ i by its inverse and each Fi by F␲ Ž i. . Proof of Lemma 4.4. As mentioned above, the divisorial ideal ␥ ŽDiv Ž F .. ; k w SP x is equivalent to some monomial divisorial ideal ⌬; i.e., there is an element ␬ g QFŽ k w SP x. such that

␥ Ž Div Ž F . . s ␬ ⌬ . The inclusion ␬ g Ž␥ ŽDiv Ž F .. : ⌬ . shows that ␬ is a k-linear combination of some Laurent monomials corresponding to lattice points in ⺪ nq 1. We factor our one of the terms of ␬ , say m, and rewrite the above equality as

␥ Ž Div Ž F . . s Ž my1␬ . Ž m⌬ . . Then my1␬ is of the form 1 q m1 q ⭈⭈⭈ qm s for some Laurent terms m1 , . . . , m s f k, while m⌬ is necessarily a divisorial monomial ideal of k w SP x Žsince 1 belongs to the supporting monomial set of my1␬ .. Now ␥ is a graded automorphism. Hence

Ž 1 q m1 q ⭈⭈⭈ qm s . Ž m⌬ . ; k SP is a graded ideal. This implies that the terms m1 , . . . , m s are of degree 0. Thus there is always a presentation

␥ Ž Div Ž F . . s Ž 1 q m1 q ⭈⭈⭈ qm s . ⌬ ,

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where m1 , . . . , m s are Laurent terms of degree 0 and ⌬ ; k w SP x is a monomial ideal Žwe do not exclude the case s s 0.. A representation of this type will be called admissible. In the following we will have to work with the number of degree 1 monomials in a given monomial ideal I. Therefore we let IP denote the set of such monomials; in other words, IP is the set of lattice points in P which are elements of I. Thus, we have r

ž

F DivŽ Fi . a

Ž i.

1

/

s  x g P l ⺪ n N ht F iŽ x . G a i , i g w 1, r x 4 P

for all a i G 0. ŽRecall that ht F i coincides on lattice points with the valuation of QFŽ k w SP x. defined by Div Ž Fi ... Furthermore we set c i s 噛 Ž IF i . . Then c i s 噛Ž P l ⺪ n . y 噛Ž Fi l ⺪ n ., and according to our enumeration of the facets, we have c1 G ⭈⭈⭈ G c r . For ␥ g ⌫k Ž P . consider an admissible representation

␥ Ž Div Ž F1 . . s Ž 1 q m1 q ⭈⭈⭈ qm s . ⌬ . Since ␥ is graded, 噛Ž ⌬ P . s c1; this is the dimension of the degree 1 homogeneous components of the ideals. As mentioned above, there are integers a i G 0 such that r

⌬s

F DivŽ Fi . a

Ž i.

.

1

It follows easily that if Ý1r a i G 2 and a i 0 / 0 for i 0 g w1, r x, then 噛Ž ⌬ P . c i 0 . This observation, along with the maximality of c1 , shows that exactly one of the a i is 1 and all the others are 0. In other words, ⌬ s Div Ž G1 . for some G1 g  F1 , . . . , Fr 4 containing precisely 噛Ž F1 l ⺪ n . lattice points. By Lemmas 4.2 and 4.3 there exists ␣ 1 g ⺑Ž G1 . such that

␣ 1 (␥ Ž Div Ž F1 . . s Div Ž G1 . . Now we proceed inductively. Let 1 F t - r. Assume there are facets G1 , . . . , Gt of P with 噛Ž Gi l ⺪ n . s 噛Ž Fi l ⺪ n . and ␣ 1 g ⺑Ž G1 ., . . . , ␣ t g ⺑Ž Gt . such that

␣ t ( ⭈⭈⭈ ( ␣ 1 (␥ Ž Div Ž Fi . . s Div Ž Gi . ,

i g w 1, t x .

ŽObserve that the Gi are automatically different.. In view of Lemma 4.2 we must show there is a facet Gtq1 ; P, different from G1 , . . . , Gt and

POLYTOPAL LINEAR GROUPS

729

containing exactly 噛Ž Ftq1 l ⺪ n . lattice points, and an element ␣ tq1 g ⺑Ž Gtq1 . such that

␣ tq1 ( ␣ t ( ⭈⭈⭈ ( ␣ 1 (␥ Ž Div Ž Ftq1 . . s Div Ž Gtq1 . . For simplicity of notation we put ␥ ⬘ s ␣ t ( ⭈⭈⭈ ( ␣ 1 (␥ . Again, consider an admissible representation

␥ ⬘ Ž Div Ž Ftq1 . . s Ž 1 q m1 q ⭈⭈⭈ qm s . ⌬ . Rewriting this equality in the form

␥ ⬘ Ž Div Ž Ftq1 . . s Ž my1 j Ž 1 q m1 q ⭈⭈⭈ qm s . . Ž m j ⌬ . , where j g  0, . . . , s4 and m 0 s 1, we get another admissible representation. Assume that by varying j we can obtain a monomial divisorial ideal m j ⌬, such that in the primary decomposition r

mj⌬ s

F DivŽ Fi . a

Ž i.

1

there appears a positive power of Div Ž G . for some facet G different from G1 , . . . , Gt . Then 噛ŽŽ m j ⌬ .P . F c tq1 Ždue to our enumeration. and the inequality is strict whenever Ý1r a i G 2. On the other hand 噛ŽDiv Ž Ftq1 .P . s c tq1. Thus we would have m j ⌬ s Div Ž G . and we could proceed as for the ideal Div Ž F1 .. Assume to the contrary that in the primary decompositions of all the monomial ideals m j ⌬ there only appear the prime ideals Div Ž G1 ., . . . , Div Ž Gt .. We have

Ž 1 q m1 q ⭈⭈⭈ qm s . ⌬ ; ⌬ q m1 ⌬ q ⭈⭈⭈ qm s ⌬ and

Ž 1 q m1 q ⭈⭈⭈ qm s . ⌬ s w ⌬ x s w m1 ⌬ x s ⭈⭈⭈ s w m s ⌬ x in ClŽ k w SP x.. Applying Ž␥ ⬘.y1 we arrive at the conclusion that Div Ž Ftq1 . is contained in a sum of monomial divisorial ideals ⌽ 0 , . . . , ⌽s , such that the primary decomposition of each of them only involves Div Ž F1 ., . . . , Div Ž Ft .. ŽThis follows from the fact that Ž␥ ⬘.y1 maps Div Ž Gi . to the monomial ideal Div Ž Fi . for i s 1, . . . , t; thus intersections of symbolic powers of Div Ž G1 ., . . . , Div Ž Gt . are mapped to intersections of symbolic powers of Div Ž F1 ., . . . , Div Ž Ft ., which are automatically monomial.. Furthermore, Div Ž Ftq1 . has the same divisor class as each of the ⌽i .

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Now choose a monomial M g ⺪ nq 1 l Div Ž Ftq1 . such that ht F1Ž M . q ⭈⭈⭈ qht F tŽ M . is minimal. Since the monomial ideal Div Ž Ftq1 . is contained in the sum of the monomial ideals ⌽ 0 , . . . , ⌽s , each monomial in it must belong to one of the ideals ⌽i ; so M g ⌽j for some j. There is a monomial d with Div Ž Ftq1 . s d⌽j , owing to the fact that Div Ž Ftq1 . and ⌽j belong to the same divisor class. It is clear that ht F iŽ d . F 0 for i g w1, t x and ht F iŽ d . - 0 for at least one i g w1, t x. In fact, ht F iŽ d . s ya i

for i s 1, . . . , t ,

where ⌽j s F 1t Div Ž Fi .Ž a i .. If we had a i s 0 for i s 1, . . . , t, then ⌽j s k w SP x, which is impossible since ⌽j has no non-zero elements in degree 0. By the choice of d the monomial N s dM belongs to Div Ž Ftq1 .. But ht F1Ž N . q ⭈⭈⭈ qht F tŽ N . - ht F1Ž M . q ⭈⭈⭈ qht F tŽ M . , a contradiction. Proof of Theorem 3.2Žb. ᎐ Žd.. Žb. Since ⺤nq 1 and the ⺑Ž Fi . are connected groups they generate a connected subgroup U of ⌫k Ž P . Žsee Borel wBo, Proposition 2.2x.. This subgroup acts trivially on ClŽ k w SP x. by Lemma 4.2 and the fact that the classes of the Div Ž Fi . generate the divisor class group. Furthermore U has finite index in ⌫k Ž P . bounded by 噛⌺Ž P .. Therefore U s ⌫k Ž P . 0 . Assume ␥ g ⌫k Ž P . acts trivially on ClŽ k w SP x.. We want to show that ␥ g U. Let E denote the connected subgroup of ⌫k Ž P ., generated by the elementary automorphisms. Since any automorphism that maps monomials to terms and preserves the divisorial ideals Div Ž Fi . is automatically a toric automorphism, by Lemma 4.1Ža. we only have to show that there is an element ␧ g E, such that

␧ (␥ Ž Div Ž Fi . . s Div Ž Fi . ,

i g w 1, r x .

Ž 1.

By Lemma 4.4 we know that there is ␧ 1 g E such that

␧ 1 (␥ Ž Div Ž Fj . . s Div Ž Fi j . ,

j g w 1, r x ,

Ž 2.

where  i1 , . . . , i r 4 s  1, . . . , r 4 . Since ␧ 1 and ␥ both act trivially on ClŽ k w SP x., we get Div Ž Fi j . s m i j Div Ž Fj . ,

j g w 1, r x ,

for some monomials m i j of degree 0. By Lemma 4.3 we conclude that if m i j / 1 Žin additive notation, m i j / 0., then both m i j and ym i j are column vectors with the base facets Fj and

POLYTOPAL LINEAR GROUPS

731

Fi j , respectively. Observe that the automorphism y1 ␧ i j s e1m i ( eym ( e1m i g E i j

j

j

interchanges the ideals Div Ž Fj . and Div Ž Fi j ., provided m i j / 1. Now we can complete the proof by successively ‘‘correcting’’ the equations Ž2.. Žc. We have to compute the dimension of ⌫k Ž P .. Without loss of generality we may assume that k is algebraically closed, passing to the algebraic closure of k if necessary Žsee Remark 3.3Že... For every permutation ␳ :  1, . . . , r 4 ª  1, . . . , r 4 we have the algebraic map ⺑ Ž F␳ Ž1. . = ⭈⭈⭈ = ⺑ Ž F␳ Ž r . . = ⺤nq1 = ⌺ Ž P . ª ⌫k Ž P . , induced by composition. The left-hand side has dimension 噛ColŽ P . q n q 1. By Theorem 3.2Ža. we are given a finite system of constructible sets, covering ⌫k Ž P .. Hence dim ⌫k Ž P . F 噛ColŽ P . q n q 1. To derive the opposite inequality, we can additionally assume that P contains an interior lattice point. Indeed, the observation Ž). in Section 2 and Theorem 3.2Ža. show that the natural group homomorphism ⌫k Ž P . ª ⌫k Ž cP ., induced by restriction to the cth Veronese subring, is surjective for every c g ⺞ Žthe surjection for the ‘‘toric part’’ follows from the fact that k is closed under taking roots.. So we can work with cP, which contains an interior point provided c is large. Let x g P be an interior lattice point and let ¨ 1 , . . . , ¨ s be different column vectors. Then the supporting monomial set of e¨␭ii Ž x ., ␭ g k*, is not contained in the union of those of e¨␭jj Ž x ., j / i Žjust look at the projections of x through ¨ i into the corresponding base facets.. This shows that we have 噛ColŽ P . linearly independent tangent vectors of ⌫k Ž P . at 1 g ⌫k Ž P .. Since the tangent vectors corresponding to the elements of ⺤nq 1 clearly belong to a complementary subspace and ⌫k Ž P . is a smooth variety, we are done. Žd. Assume ¨ and y¨ both are column vectors. Then the element 1 ␧ s e¨1 ( ey1 y¨ ( e ¨ g ⌫k Ž P .

0

Ž s ⌫k Ž P . 0 .

maps monomials to terms; more precisely, ␧ ‘‘inverts up to scalars’’ the columns parallel to ¨ so that any x g SP is sent either to the appropriate y g SP or to yy g k w S P x. Then it is clear that there is an element ␶ g ⺤nq 1 such that ␶ ( ␧ is a generator of ⌺Ž P . inv . Hence ⌺Ž P . inv ; ⌫k Ž P . 0 . Conversely, if ␴ g ⌺Ž P . l ⌫k Ž P . 0 then ␴ induces the identity map on ClŽ k wŽ SP x.. Therefore ␴ ŽDiv Ž Fi .. s m i j Div Ž Fi . for some monomials m i j , and the very same arguments we have used in the proof of Žb. show that ␴ g ⌺Ž P . inv . Thus ⌫k Ž P .r⌫k Ž P . 0 s ⌺Ž P .r⌺Ž P . inv .

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Finally, assume k is infinite and ⺤⬘ ; ⌫k Ž P . is a torus, strictly containing ⺤nq 1. Choose x g SP and ␥ g ⺤⬘. Then ␶y1 (␥ (␶ Ž x . s ␥ Ž x . for all ␶ g ⺤nq 1. Since k is infinite, one easily verifies that this is only possible if ␥ Ž x . is a term. In particular, ␥ maps monomials to terms. Then, as observed above Lemma 4.1, ␥ s ␴ (␶ with ⺤nq 1 , and therefore ␴ g ⌺ 0 s ⺤⬘ l ⌺. Lemma 4.1Žb. now implies that ⺤⬘ is the semidirect product ⺤nq 1 h ⌺ 0 . By the infinity of k we have ⌺ 0 s 1.

5. PROJECTIVE TORIC VARIETIES AND THEIR GROUPS Having determined the automorphism group of a polytopal semigroup ring, we show in this section that our main result gives the description of the automorphism group of a projective toric variety Žover an arbitrary algebraically closed field. via the existence of ‘‘fully symmetric’’ polytopes. ŽSee the Introduction for references to previous work on the automorphism groups of toric varieties.. We start with a brief review of some facts about projective toric varieties. Our terminology follows the standard references ŽDanilov wDax, Fulton wFux, and Oda wOdax.. To avoid technical complications, we assume from now on that the field k is algebraically closed. Let P ; ⺢ n be a polytope as above. Then ProjŽ k w SP x. is a projective toric variety Žthough k w SP x need not be generated by its degree 1 elements.. In fact, it is the toric variety defined by the normal fan N Ž P ., but it may be useful to describe it additionally in terms of an affine covering. For every vertex z g P we consider the finite rational polyhedral n-cone spanned by P at its corner z. The parallel translate of this cone by yz will be denoted by C Ž z .. Thus we obtain a system of cones C Ž z ., where z runs through the vertices of P. It is not difficult to check that N Ž P . is the fan in Ž⺢ n .* whose maximal cones are the dual cones C Ž z . * s  ␸ g Ž ⺢ n . * N ␸ Ž x . G 0 for all x g C Ž z . 4 . The affine open subschemes SpecŽ k w⺪ n l C Ž z .x. cover ProjŽ k w SP x.. The projectivity of ProjŽ k w SP x. follows from the observation that for all natural numbers c 4 0 the polytope cP is normal Žsee wOdax, or wBGTx for a sharp bound on c . and, hence, ProjŽ k w SP x. s ProjŽ k w Sc P x.. A lattice polytope P is called ¨ ery ample if for every vertex z g P the semigroup C Ž z . l ⺪ n is generated by  x y z N x g P l ⺪ n4 .

733

POLYTOPAL LINEAR GROUPS

It is clear from the discussion above the ProjŽ k w S P x. s ProjŽ k w SP x. if and only if P is very ample. In particular, normal polytopes are very ample, but not conversely: EXAMPLE 5.1. Let ⌸ be the simplicial complex associated with the minimal triangulation of the real projective plane. It has six vertices which we label by the numbers i g w1, 6x. Then the 10 facets of ⌸ have the following vertex sets Žwritten as ascending sequences .:

Ž 1, 2, 3 . , Ž 2, 3, 6 . ,

Ž 1, 2, 4 . , Ž 2, 4, 5 . ,

Ž 1, 3, 5 . , Ž 2, 5, 6 . ,

Ž 1, 4, 6 . , Ž 3, 4, 5 . ,

Ž 1, 5, 6 . , Ž 3, 4, 6 . .

Let P be the polytope spanned by the indicator vectors of the 10 facets Žthe indicator vector of Ž1, 2, 3. is Ž1, 1, 1, 0, 0, 0. etc... All the vertices lie in an affine hyperplane H ; ⺢ 6 , and P has indeed dimension 5. Using H as the ‘‘grading’’ hyperplane, one realizes R s k w SP x as the k-subalgebra of k w X 1 , . . . , X6 x generated by the 10 monomials ␮ 1 s X 1 X 2 X 3 , ␮ 2 s X1 X 2 X4 , . . . . Let R be the normalization of R. It can be checked by effective methods that R is generated as a k-algebra by the 10 generators of R and the monomial ␯ s X 1 X 2 X 3 X 4 X5 X6 ; in particular R is not normal. Then one can easily compute by hand that the products ␮ i ␯ and ␯ 2 all lie in R. It follows that RrR is a one-dimensional vector space; therefore ProjŽ R . s ProjŽ R . is normal, and P is very ample. For a very ample polytope P we have a projective embedding Proj Ž k w S P x . ; ⺠kN ,

N s 噛 Ž P l ⺪ n . y 1.

The corresponding very ample line bundle on ProjŽ k w SP x. will be denoted by L P . It is known that any projective toric variety and any very ample equivariant line bundle on it can be realized as ProjŽ k w SP x. and L P for some very ample polytope P. Moreover, any line bundle is isomorphic to an equivariant line bundle, and if LQ is a very ample equivariant line bundle on ProjŽ k w S P x. Žfor a very ample polytope Q . then N Ž P . s N Ž Q . Žsee wOda, Chap. 2x or wDax.. Therefore P and Q have the same column vectors Žsee observations Ž))). in Section 2.. Furthermore, LQ 1 and LQ 2 are isomorphic line bundles if and only if Q1 and Q2 differ only by a parallel translation Žbut they have different equivariant structures if Q1 / Q2 .. Let X be a projective toric variety and L P , LQ g PicŽ X . be two very ample equivariant line bundles. Then one has the elegant formula L P m LQ s L PqQ , where P q Q is the Minkowski sum of P, Q ; ⺢ n Žsee Teissier wTex.. ŽOf course, very ampleness is preserved by the tensor product, and therefore by Minkowski sums..

734

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In the dual space Ž⺢ n .* the column vectors ¨ correspond to the integral affine hyperplanes H interesting exactly one of the rays in N Ž P . Žthis is the condition ␸G Ž ¨ . G 0 for G / F . and such that there is no lattice point strictly between H and the parallel of H through 0 Žthis is the condition ␸ F Ž ¨ . s y1.. This shows that the column vectors correspond to Demazure’s roots wDex. In Fig. 3 the arrows represent the rays of the normal fans N Ž P1 . and N Ž P2 . and the lines indicate the hyperplanes corresponding to the column vectors Ž P1 and P2 are chosen as in Fig. 2.. LEMMA 5.2. If two lattice n-polytopes P1 and P2 ha¨ e the same normal fans, then the quotient groups ⌫k Ž P1 . 0rk* and ⌫k Ž P2 . 0rk* are naturally isomorphic. Proof. As in Section 4 we will work with ⌫k Ž Pi .. Put X s ProjŽ k w SP1 x s ProjŽ k w SP 2 x. and consider the canonical anti-homomorphisms ⌫k Ž Pi . 0 ª Aut k Ž X ., i s 1, 2. Let AŽ P1 . and AŽ P2 . denote the images. We choose a column vector ¨ Žfor both polytopes. and ␭ g k. We claim that the elementary automorphisms e¨␭Ž Pi . g ⌫k Ž Pi . 0 , i s 1, 2, have the same images in Aut k Ž X .. Denote the images by e1 and e2 . For i s 1, 2 we can find a vertex z i of the base facet Ž Pi . ¨ such that C Ž z1 . s C Ž z 2 .. Now it is easy to see that e1 and e2 restrict to the same automorphism of the affine subvariety Spec Ž k w C Ž z1 . l ⺪ n x. ; X, which is open in X. Therefore e1 s e2 , as claimed. It is also clear that for any ␶ g ⺤nq 1 the corresponding elements ␶ i g ⌫k Ž Pi ., i s 1, 2, have the same images in Aut k Ž X .. By Theorem 3.2Žb. we arrive at the equality AŽ P1 . s AŽ P2 .. It only remains to notice that k* s KerŽ ⌫k Ž Pi . 0 ª AŽ Pi .., i s 1, 2. EXAMPLE 5.3. Lemma 5.2 cannot be improved. For example, let P1 be the unit 1-simplex ⌬ 1 and P2 s 2 P1. Then ⺓w SP1 x s ⺓w X 1 , X 2 x, and ⺓w SP 2 x s ⺓w X 12 , X 1 X 2 , X 22 x is its second Veronese subring. Both polytopes have

FIG. 3. The normal fans of the polytopes P1 and P2 .

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the same symmetries and column vectors, and moreover the torus action on ⺓w SP 2 x is induced by that on ⺓w SP1 x. Therefore the natural map ⌫⺓ Ž P1 . ª ⌫⺓ Ž P2 . is surjective; in fact, ⌫⺓ Ž P1 . s GL 2 Ž⺓. and ⌫⺓ Ž P2 . s GL 2 Ž⺓.r "14 . If there were an isomorphism between these groups, then SL 2 Ž⺓. and SL 2 Ž⺓.r "14 would also be isomorphic. This can be easily excluded by inspecting the list of finite subgroups of SL 2 Ž⺓.. For a lattice polytope P we denote the group opposite to ⌫k Ž P . 0rk* by A k Ž P ., the projective toric variety ProjŽ k w SP x. by X Ž P .; the symmetry group of a fan F is denoted by ⌺Ž F .. Ž ⌺Ž F . is the subgroup of GL nŽ⺪. that leaves F invariant.. Furthermore we consider A k Ž P . as a subgroup of Aut k Ž X Ž P .. in a natural way. THEOREM 5.4. For a lattice n-polytope P the group Aut k Ž X Ž P .. is generated by A k Ž P . and ⌺Ž N Ž P ... The connected component of unity of Aut k Ž X Ž P .. is A k Ž P ., dimŽ A k Ž P .. s 噛ColŽ P . q n, and the embedded torus ⺤n s ⺤nq1rk* is a maximal torus of Aut k Ž X Ž P ... Proof. Assume for the moment that P is very ample and w L P x g PicŽ X Ž P .. is preserved by every element of Aut k Ž X Ž P ... Then we are able to apply the classical arguments for projective spaces as follows. We have k w SP x s [i G 0 H 0 Ž X, LiP .. Since w L P x is invariant under Aut k Ž X ., arguments similar to those in Hartshorne wHa, Example 7.1.1, p. 151x show that giving an automorphism of X is equivalent to giving an element of ⌫k Ž P .. In other words, the natural anti-homomorphism ⌫k Ž P . ª Aut k Ž X Ž P .. is surjective. Now Theorem 3.2 gives the desired result once we notice that ⌺Ž P . is mapped to ⌺Ž F .. Therefore, and in view of Lemma 5.2, the proof is completed once we show that there is a very ample polytope Q having the same normal fan as P and such that w LQ x is invariant under Aut k Ž X .. The existence of such a ‘‘fully’’ symmetric polytope is established as follows. First, we replace P by the normal polytope cP for some c 4 0 so that we may assume that P is normal. The k-vector space of global sections of a line bundle, which is an image of L P with respect to some element of Aut k Ž X Ž P .., has the same dimension as the space of global sections of L P , which is given by 噛Ž P l ⺪ n .. Easy inductive arguments ensure that the number of polytopes Q such that 噛Ž Q l ⺪ n . s 噛Ž P l ⺪ n . and, in addition, N Ž Q . s N Ž P . is finite. It follows that the set w LQ x, . . . , w LQ x4 of isomorphism classes of very ample equivariant line 1 t bundles to which w L P x is mapped by an automorphism of X Ž P . is finite. Since every line bundle is isomorphic to an equivariant one, any element ␣ g Aut k Ž X Ž P .. must permute the classes w LQ i x g Pic Ž X Ž P ... In

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FIGURE 4

particular, the element LQ 1 m ⭈⭈⭈ m LQ t g Pic Ž X . is invariant under Aut k Ž X Ž P ... But LQ 1 m ⭈⭈⭈ m LQ t s LQ 1q ⭈⭈⭈ qQ t and, hence, Q1 q ⭈⭈⭈ qQt is the desired polytope. EXAMPLE 5.5. In general the natural anti-homomorphism ⌫k Ž P . ª Aut k Ž X Ž P .. is not surjective. For example consider the polytopes P and Q in Fig. 4. Then ProjŽ k w SP x. s ProjŽ k w SQ x. s ⺠ 1 = ⺠ 1. However, the isomorphism corresponding to the exchange of the two factors ⺠ 1 cannot be realized in k w SQ x. ACKNOWLEDGMENTS This paper was written while the second author was an Alexander von Humboldt Research Fellow at Universitat The generous grant of the Alexander von Humboldt ¨ Osnabruck. ¨ Foundation is gratefully acknowledged. The second author was also supported by CRDF Grant GM1-115. We thank Matei Toma for stimulating discussions and Klaus Warneke for his technical help in the preparation of this paper.

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