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a finite vector configuration @xmath6 defines the _ toric ideal _ @xmath7 in the polynomial ring @xmath8 = { \mathbb k } [ { \bf x}]$ ] where @xmath9 is a field . let @xmath10 and @xmath11 denote the cone , lattice and semigroup spanned by the @xmath12 and @xmath13-linear combinations of @xmath0 where @xmath13 is the set of non - negative integers . let @xmath14 be the krull dimension of @xmath15 which equals the rank of @xmath16 . assume @xmath17 . the configuration @xmath0 is _ normal _ if @xmath18 and _ graded _ if @xmath0 spans an affine hyperplane in @xmath19 . a finite set @xmath20 such that @xmath21 is called a _ hilbert basis _ of the semigroup @xmath22 . if @xmath0 is normal , the zero set of @xmath3 is a normal toric variety in @xmath23 of dimension @xmath24 , and when @xmath0 is also graded , it is a projectively normal toric variety in @xmath25 of dimension @xmath26 . see @xcite for details on toric ideals . a survey of recent results and open questions on normal configurations can be found in @xcite . it is well known that initial ideals of a polynomial ideal inherit important invariants of the ideal such as dimension , degree and hilbert function . thus it is natural to ask if certain initial ideals inherit further properties of the ideal such as cohen - macaulayness , betti numbers or reducedness ( of the corresponding scheme ) . a result of hochster  @xcite shows that when @xmath0 is normal , @xmath27 is cohen - macaulay . if @xmath0 is also graded , then @xmath27 is generated by homogeneous binomials of degree at most @xmath24 ( * ? ? ? * thm .  13.14 ) . motivated by this , sturmfels asked and conjectured the following . [ cmconjecture ] if @xmath0 is normal ( more generally , if @xmath3 is cohen - macaulay ) , is there a monomial initial ideal @xmath28 of the toric ideal @xmath3 that is cohen - macaulay  ? [ bigconjecture ]  ( * ? ? ? * conjecture 2.8 ) if @xmath0 is a graded , normal configuration then @xmath27 has a grbner basis whose elements have degree at most @xmath29 .    in this paper , we show that question  [ cmconjecture ] has a positive answer and conjecture  [ bigconjecture ] is true when @xmath0 is @xmath5-_normal_. these configurations were defined by hoc sten and thomas in @xcite . we recall the definition . let @xmath5 be a pure @xmath30-dimensional simplicial complex with vertex set @xmath31 : = \{1 , \ldots , n\}$ ] . we denote the set of _ facets _ ( @xmath24-element faces ) of @xmath5 by @xmath32 . for a set @xmath33 $ ] , let @xmath34 . we say that @xmath5 is a _ triangulation _ of @xmath0 if @xmath35 and @xmath36 for all @xmath37 . stanley - reisner _ ideal of @xmath5 is the squarefree monomial ideal @xmath38 . a cornerstone in the combinatorics of toric initial ideals is the result by sturmfels that the radical of a monomial initial ideal @xmath39 of the toric ideal @xmath3 is the stanley - reisner ideal of a triangulation @xmath40 of @xmath0 ( * ? ? ? the ideal @xmath39 is said to be _ supported on _ @xmath40 . triangulations supporting initial ideals of @xmath3 are the _ regular triangulations _ of @xmath0 . a triangulation @xmath41 of @xmath0 is _ unimodular _ if for each @xmath42 , @xmath43 . [ def_dnormal ] a configuration @xmath0 is * @xmath5-normal * if it has a regular triangulation @xmath5 such that for each @xmath44 , @xmath45 is a hilbert basis of @xmath46 . note that @xmath16 is used in the semigroups of definition  [ def_dnormal ] . all @xmath5-normal configurations are normal . in sections  [ cmsection ] and [ dnormalboundsection ] , we prove our main results . * theorem  [ cm ] . * let @xmath0 be a @xmath5-normal configuration . then there exists a term order @xmath47 such that @xmath48 and @xmath49 is cohen - macaulay . * theorem  [ dnormalbound ] . * let @xmath0 be a graded @xmath5-normal configuration . then there exists a term order @xmath47 such that @xmath48 and the grbner basis of @xmath3 with respect to @xmath47 consists of binomials of degree at most @xmath29 . it was shown in  @xcite that if @xmath0 is @xmath5-normal then @xmath3 has a monomial initial ideal that is free of embedded primes . in section  [ setup ] we recall the main features of this initial ideal . theorems  [ cm ] and [ dnormalbound ] are proved by showing that this same initial ideal is cohen - macaulay and , when @xmath0 is graded , generated in degree at most @xmath24 . our proofs are combinatorial and rely heavily on the structure of this initial ideal . the set of @xmath5-normal configurations is a proper subset of the set of normal configurations . they occur naturally in two ways . if @xmath0 has a regular unimodular triangulation @xmath5 , then @xmath0 is @xmath5-normal . if @xmath50 is simplicial and we assume that its extreme rays are generated by @xmath51 , then @xmath0 is @xmath5-normal with respect to the coarsest regular triangulation @xmath5 consisting of the unique facet @xmath52 . these were the only examples known so far . specific instances of normal configurations that are not @xmath5-normal for any @xmath5 are also known @xcite . in section  [ families ] we construct non - trivial families of both @xmath5-normal and non-@xmath5-normal configurations .    * theorem  [ babysfirstdeltanormalfamily ] . * there are families of @xmath5-normal configurations @xmath53 where @xmath54 is non - simplicial and @xmath55 has no regular unimodular triangulations . * theorem  [ babysfirstnormalfamily ] . * there is a family of normal , graded configurations @xmath56 , that are not @xmath5-normal for any regular triangulation @xmath5 . we now recall from @xcite the initial ideal of @xmath3 without embedded primes when @xmath0 is @xmath5-normal . the construction uses the _ standard pair decomposition _ of a monomial ideal @xmath57 @xcite which carries detailed information about @xmath58 , the set of associated primes of @xmath57 . recall that every monomial prime ideal of @xmath59 is of the form @xmath60 for some @xmath33 $ ] . the monomials of @xmath59 that do not lie in @xmath57 are the _ standard monomials _ of @xmath57 . the _ support _ of a monomial @xmath61 is defined to be the support of its exponent vector @xmath62  i.e. , @xmath63 \ , : \ , v_i \neq 0\}$ ] . @xcite let @xmath64 be a monomial ideal . for a standard monomial @xmath65 of @xmath57 and a set @xmath33 $ ] , @xmath66 is a * pair * of @xmath57 if all monomials in @xmath67 $ ] are standard monomials of @xmath57 . we call @xmath68 a * standard pair * of @xmath57 if :    1 . @xmath68 is a pair of @xmath57 , 2 . @xmath69 , and 3 . the set of monomials in @xmath70 $ ] is not properly contained in the set of monomials in @xmath71 $ ] for any @xmath72 satisfying ( 1 ) and ( 2 ) . the set of standard pairs of @xmath57 is unique and is called the standard pair decomposition of @xmath57 since this set provides a decomposition of the standard monomials of @xmath57 . if @xmath61 is a standard monomial of @xmath57 then there is a standard pair @xmath73 of @xmath57 such that @xmath65 divides @xmath61 and @xmath74 . in this case we say that @xmath61 is _ covered _ by @xmath68 . we also use @xmath68 to denote the set of all standard monomials covered by it . [ ap56 ] ( a ) @xcite let @xmath57 be a monomial ideal in @xmath59 . then ,    1 .   @xmath75 if and only if @xmath57 has a standard pair of the form @xmath76 . 2 .   @xmath77 is a minimal prime of @xmath57 if and only if @xmath78 is a standard pair of @xmath57 . \(b ) @xcite let @xmath79 be a monomial initial ideal of the toric ideal @xmath3 . then ,    1 .   if @xmath75 then @xmath80 is a face of the regular triangulation @xmath40 of @xmath0 , 2 . @xmath81 is a minimal prime of @xmath57 if and only if @xmath82 , and 3 .   for @xmath83 , the number of standard pairs of @xmath57 of the form @xmath84 is @xmath85 , the normalized volume of @xmath86 in @xmath40 . we call @xmath65 and @xmath80 the _ root _ and _ face _ of the standard pair @xmath68 . let @xmath87 ( respectively , @xmath88 ) be the matrix whose set of columns is @xmath0 ( respectively , @xmath89 ) . the normalized volume of @xmath90 is the absolute value of the determinant of @xmath91 divided by the g.c.d . of the non - zero maximal minors of @xmath87 . [ dnormal ] ( * ? ? ? 4.7 ) let @xmath0 be a @xmath5-normal configuration . then there exists a term order @xmath47 such that @xmath48 and @xmath92 is free of embedded primes . the term order @xmath47 needed in theorem  [ dnormal ] is described in @xcite and is not directly used in this paper . the ideal @xmath92 is shown to be free of embedded primes via an explicit description of its standard pairs . this structure is crucial for this paper and hence we recall it now . assume without loss of generality that @xmath93 . for @xmath94 , let @xmath95 be the half open _ fundamental parallelopiped _ of @xmath96 . then @xmath97 has @xmath85 elements including the origin . for @xmath98 , let @xmath99 be the cheapest monomial with respect to @xmath47 among all @xmath100 with @xmath101 . it was shown in @xcite that @xmath102 . the standard pairs of the initial ideal @xmath103 in theorem  [ dnormal ] are precisely the pairs @xmath104 as @xmath105 varies in @xmath106 for each @xmath86 in @xmath107 . by theorem  [ ap56 ] , @xmath103 is thus free of embedded primes .    for the remainder of this paper we will denote the toric initial ideal @xmath49 of theorem  [ dnormal ] by @xmath108 and its set of standard pairs by @xmath109 . [ example1 ] let @xmath0 be the vector configuration consisting of the @xmath110 columns of @xmath111 then @xmath0 is a graded _ supernormal _ configuration @xcite which means that it is @xmath5-normal with respect to every regular triangulation . consider the regular triangulation @xmath112 the configuration @xmath0 and its regular triangulation @xmath5 are shown in figure  [ ex1 ] . the toric ideal @xmath3 lives in @xmath113 $ ] . in figure [ ex1 ] , we have labeled the points of @xmath0 by the variables @xmath114 corresponding to the columns of @xmath87 , instead of by @xmath115 . the term order @xmath47 in theorem  [ dnormal ] can be induced via the weight vector @xmath116 refined by the reverse lexicographic order with @xmath117 . the initial ideal ( computed using macaulay  2 @xcite ) is @xmath118 its standard pairs , grouped by the facets of @xmath5 are : + @xmath119 + for @xmath120 , @xmath121 consists of nine lattice points  @xmath122 for each exponent vector @xmath123 of the roots @xmath124 . the last of these is @xmath125 . the monomials @xmath65 of @xmath59 such that @xmath126 in increasing order with respect to @xmath47 are : @xmath127 . thus , @xmath128 . in this section we prove that the initial ideal @xmath108 of theorem  [ dnormal ] is cohen - macaulay . this is done by showing that @xmath108 has a particular _ stanley filtration _ @xcite which implies cohen - macaulayness @xcite . stanley filtrations are special _ stanley decompositions_.    let @xmath64 be a monomial ideal . a * stanley decomposition * of @xmath57 is a set of pairs of @xmath57 , @xmath129 , that _ partition _ the standard monomials of @xmath57 . every monomial ideal @xmath57 has the trivial stanley decomposition @xmath130 . there can be many stanley decompositions of a monomial ideal . the standard pair decomposition of @xmath57 is usually not a partition of the standard monomials of @xmath57 . we will show that the standard pair decomposition @xmath109 of @xmath108 can be modified first to a stanley decomposition and then to a stanley filtration of the needed form . for @xmath33 $ ] let @xmath131 $ ] be the projection map where @xmath132 if @xmath133 and @xmath134 if @xmath135 . the following is a well known fact . @xcite[artproj ] if @xmath136 for a regular triangulation @xmath40 of @xmath0 , then @xmath137 is an artinian monomial ideal in @xmath138 with @xmath85-many standard monomials which are precisely the roots of the standard pairs @xmath139 of @xmath28 . [ rootsdivisors ] if @xmath140 is a standard pair in @xmath109 , then every divisor of @xmath65 is also the root of a standard pair in @xmath141 . [ rootslemma ] let @xmath140 and @xmath142 be two standard pairs in @xmath109 . if @xmath143 then @xmath144 . suppose @xmath145 . then @xmath146 with @xmath147 , @xmath148 and @xmath149 outside the vertices of @xmath150 and thus in particular , outside @xmath151 . hence , @xmath152 . [ simpcase ] if @xmath0 is normal , @xmath50 is simplicial ( generated without loss of generality by @xmath51 ) , and @xmath108 is the special initial ideal of theorem  [ dnormal ] supported on @xmath153 , then @xmath109 is a stanley decomposition . here @xmath0 is @xmath150-normal . all standard pairs in @xmath109 have face @xmath154 $ ] and roots the standard monomials of @xmath155}(j)$ ] . thus no two standard pairs of @xmath109 have the same root and by lemma  [ rootslemma ] , no two standard pairs intersect . however , if @xmath156 , then it is precisely the standard pairs in @xmath109 with a common root that stop @xmath109 from being a partition . such pairs always exist when @xmath156  for instance , @xmath157 is in @xmath109 for all @xmath94 . we will use the combinatorial notion of _ shellings _ to create new pairs that partition the standard monomials covered by each set of standard pairs with a common root . in section  [ setup ] we defined @xmath158 . the following lemma identifies the faces in all standard pairs that share a root . [ sigmasets ] if @xmath65 is a root of a standard pair in @xmath109 , then @xmath159 . recall from section  [ setup ] that if @xmath160 , then @xmath161 . conversely , suppose @xmath68 is a standard pair in @xmath109 and @xmath162 for some @xmath163 in @xmath164 . then @xmath165 . then @xmath166 . since @xmath167 , it is also in @xmath168 . further , @xmath65 is the cheapest monomial with respect to @xmath47 among all monomials @xmath61 in @xmath59 with @xmath169 . this implies that @xmath140 is also a standard pair of @xmath108 . [ shelling ] ( * ? ? ? * chapter 3 , definition 2.1 ) let @xmath170 be a pure simplicial complex . a * shelling * of @xmath170 is a linear ordering @xmath171 of the facets of @xmath170 such that for each @xmath172 , @xmath173 , the subcomplex supported in @xmath174 has a unique minimal face . a simplicial complex @xmath170 is * shellable * if it has a shelling . let @xmath175 be a face of a simplicial complex @xmath170 . then @xmath176 is the simplicial complex generated by all @xmath177 containing @xmath175 . we sometimes write @xmath178 for @xmath179 when @xmath170 is obvious . the following is a mild generalization of lemma 8.7 in @xcite . [ starshell ] let @xmath170 be a pure shellable simplicial complex with shelling order @xmath171 . if @xmath175 is any face of @xmath170 then the restriction of the global shelling order to @xmath180 yields a shelling of @xmath180 .    for a root @xmath65 in @xmath109 , let @xmath181 .    in the following arguments we fix a root @xmath65 of a standard pair in @xmath109 . by lemma  [ sigmasets ] , @xmath182 and the set of facets @xmath183 if @xmath61 divides @xmath65 , then @xmath184 . the regular triangulation @xmath150 is shellable @xcite . in the rest of this section , we fix a shelling of @xmath150 . by lemma  [ starshell ] , this induces a shelling of @xmath185 . let us assume without loss of generality that @xmath186 is the induced shelling order of the facets of @xmath187 . for @xmath188 , let @xmath189 be the unique minimal face described in definition  [ shelling ] . it is known that @xmath190    * example  [ example1 ] continued . * consider the root @xmath191 of @xmath192 and the shelling order @xmath193 and @xmath194 on @xmath150 . then @xmath195 is the induced shelling order of @xmath196 . from this we obtain the sets @xmath197 and @xmath198 . consider the interval @xmath199 @xcite [ localgeompart ] the simplicial complex @xmath200 is the disjoint union of the intervals @xmath201 , @xmath202 . [ subshellingremark ] note that by construction , the partial union @xmath203 is a partition of the subcomplex with maximal faces @xmath204 and that @xmath189 is not contained in this partial union for any @xmath205 .    for a root @xmath65 in @xmath109 and a facet @xmath206 define the monomial @xmath207    recall that we have fixed a shelling of @xmath150 and thus , by lemma  [ starshell ] , a shelling of @xmath185 for each root @xmath208 . therefore , if @xmath209 , @xmath210 is uniquely defined . we return to the fixed root @xmath65 and the shelling @xmath211 of @xmath185 . [ partition ] the standard monomials of @xmath108 in @xmath212 are partitioned by the pairs @xmath213 , @xmath202 .    by lemma  [ localgeompart ] , @xmath214 is a partition of @xmath215 . hence , if @xmath216 then @xmath217 for a unique @xmath201 . by construction , @xmath218 and so @xmath219 . this implies that @xmath220 where the inclusion @xmath221 follows from the previous line and @xmath222 from the fact that @xmath223 for each @xmath202 . to see that @xmath224 is a partition , suppose @xmath225 where @xmath226 . then @xmath227 has support in @xmath228 which implies that @xmath229 . however , for @xmath230 , @xmath231 as @xmath232 which means that @xmath65 lies only in @xmath233 .    * example  [ example1 ] continued . * as before , the monomial @xmath191 is a root of @xmath192 with the shelling order induced on @xmath234 as above . we obtain the monomials @xmath235 and @xmath236 from @xmath237 and @xmath238 . then @xmath239 with the latter union being disjoint . [ stanley ] let @xmath240 be the fixed shelling of @xmath150 . then @xmath241 is a stanley decomposition of @xmath108 . lemma  [ partition ] showed how to make the union of the standard pairs of @xmath109 with a common root a disjoint union of pairs of @xmath108 . by lemma  [ rootslemma ] , ( [ standecomp ] ) , the union of these disjoint unions is a stanley decomposition of @xmath108 . the above stanley decomposition can be organized to have more structure . [ filtration ]  @xcite let @xmath64 be a monomial ideal . a * stanley filtration * of @xmath57 is a stanley decomposition of @xmath57 with an ordering of the pairs @xmath242 such that for all @xmath243 the set @xmath244 is a stanley decomposition of @xmath245 . equivalently , the ordered set @xmath246 is a stanley filtration provided the modules @xmath247 form a filtration @xmath248 with @xmath249 $ ] . ( from  @xcite ) let @xmath250 $ ] . then @xmath251 is a stanley decomposition of @xmath57 but no ordering of these pairs is a stanley filtration of @xmath57 . alternatively , the ordered pairs @xmath252 form a stanley filtration of @xmath57 . we now show that the pairs in ( [ standecomp ] ) can be ordered to yield a stanley filtration of @xmath108 . the significance of this for us comes from a result of simon  @xcite , interpreted as follows by maclagan and smith  @xcite . [ filtgivescm ] if @xmath64 is a monomial ideal with a stanley filtration such that for each face @xmath80 of a pair in the filtration , the prime ideal @xmath77 is a minimal prime of @xmath57 , then @xmath57 is cohen - macaulay . the faces of pairs in ( [ standecomp ] ) already index minimal primes of @xmath108 . thus to show that @xmath108 is cohen - macaulay all we need to do is to order the pairs in ( [ standecomp ] ) so that the ordered decomposition is a stanley filtration . we do this using the following algorithm . [ list ] * input : * the stanley decomposition ( [ standecomp ] ) of @xmath108 . + * output : * a stanley filtration of @xmath108 with the same faces as those in ( [ standecomp ] ) .    1 : :    ( local lists ) for each @xmath253 ,    @xmath254 , order the pairs in ( [ standecomp ] ) with    face @xmath253 in any way such that if    @xmath255    precedes    @xmath256 then    @xmath61 does not divide @xmath65 . call    this list @xmath257 . 2 : :    ( global list ) the global list @xmath258 is obtained    by appending @xmath257 to the end of @xmath259 for    @xmath260 . let @xmath261 for @xmath262 . then @xmath263 is the total number of pairs in ( [ standecomp ] ) . write @xmath258 as @xmath264 $ ] where @xmath265 when @xmath266 ( @xmath267 ) and @xmath268 is the root of the @xmath269-th pair in the local list @xmath257 constructed in step 1 of the algorithm . for @xmath243 define the partial list @xmath270 $ ] and the ideal @xmath271 . we need to prove that @xmath272 is a stanley decomposition of @xmath273 . since @xmath272 is already a partition , it suffices to show that the set of monomials in the pairs in @xmath272 is the set of standard monomials of @xmath273 . \(i ) _ the standard monomials of @xmath273 are contained in the pairs in @xmath272 _ : a standard monomial @xmath65 of @xmath273 is a standard monomial of @xmath108 and hence is covered by a unique pair @xmath274 in @xmath258 . also , @xmath275 which implies that @xmath276 for any @xmath277 . hence @xmath278 and @xmath279 . \(ii ) _ the monomials in the pairs in @xmath272 are standard monomials of @xmath273 _ : suppose @xmath65 lies in the ( unique ) pair @xmath280 . since @xmath281 , it suffices to show that @xmath282 . suppose @xmath283 . then there exists @xmath284 , @xmath285 such that @xmath286 where @xmath287 is a monomial with support in @xmath288 . since @xmath289 and @xmath290 are both in @xmath31 \backslash ( \tau_p \cup \tau_l)$ ] , it follows that @xmath291 . since @xmath292 , by step 1 of the algorithm , @xmath293 . recall that @xmath294 and @xmath295 are standard pairs of @xmath108 . since @xmath291 , by corollary  [ rootsdivisors ] , @xmath296 is also in @xmath109 . this implies that @xmath297 and @xmath288 are two distinct facets in @xmath298 . since @xmath299 , @xmath300 however , this is a contradiction since @xmath288 precedes @xmath297 in the shelling order on @xmath150 and hence @xmath301 can not be in @xmath288 . thus @xmath302 and @xmath303 and thus not in @xmath273 . * example  [ example1 ] continued . * as before , @xmath193 and @xmath194 is a shelling order on @xmath150 . the ( ordered ) local lists in the stanley filtration produced by algorithm  [ list ] are : + @xmath304 $ ] , + @xmath305 $ ] , + @xmath306 $ ] , + @xmath307 $ ] . [ cm ] let @xmath0 be a @xmath5-normal configuration . then there exists a term order @xmath47 such that @xmath308 and @xmath49 is cohen - macaulay . algorithm  [ list ] shows that the initial ideal @xmath108 of theorem  [ dnormal ] has a stanley filtration that satisfies the conditions of theorem  [ filtgivescm ] . this theorem guarantees that @xmath108 is cohen - macaulay . we remark that even when @xmath0 is @xmath5-normal it is not true that all initial ideals of @xmath3 without embedded primes are cohen - macaulay . take @xmath0 to be the columns of @xmath309 then @xmath0 admits a unimodular regular triangulation and is hence @xmath5-normal . the toric ideal @xmath310 $ ] has codimension four and has 46 initial ideals without embedded primes . among them , the following two have projective dimension five . 1 .   @xmath311 2 . @xmath312    the initial ideals of @xmath27 were computed using the software package cats @xcite and then checked for embedded primes and cohen - macaulayness using macaulay  2 . we remark that the first example of a monomial toric initial ideal without embedded primes that is not cohen - macaulay was found by laura matusevich  @xcite . in that example , @xmath3 is not cohen - macaulay and thus @xmath0 is not normal . [ dnormalbound ] if @xmath0 is a graded @xmath5-normal configuration , then there exists a term order @xmath47 such that @xmath48 and the grbner basis of @xmath3 with respect to @xmath47 consists of binomials of degree at most @xmath29 . theorem  [ dnormalbound ] settles conjecture  [ bigconjecture ] for the subset of normal configurations that are @xmath5-normal . since @xmath0 is graded , @xmath3 is homogeneous with respect to the usual grading of @xmath59 where @xmath313 for @xmath314 . hence it suffices to show that @xmath3 has an initial ideal of degree at most @xmath24 . we will show that the initial ideal @xmath108 from theorem  [ dnormal ] satisfies this degree bound when @xmath0 is graded . proposition  13.15 in @xcite shows that conjecture  [ bigconjecture ] is true whenever @xmath0 admits a regular unimodular triangulation . ( see also proposition  13.18 in @xcite . ) such configurations form a proper subset of the set of @xmath5-normal configurations . if we are allowed to replace a graded normal @xmath0 by all the lattice points in a `` big enough '' multiple of the convex hull of @xmath0 , then it is known that this new configuration admits regular unimodular triangulations and thus has a grbner basis of degree at most @xmath24 . see @xcite and @xcite for many such results . conjecture  [ bigconjecture ] requires that @xmath0 be both graded and normal . [ needgradedandnormal ] _ graded , but not normal : _ when @xmath315 with @xmath316 , @xmath317 and @xmath318 , then @xmath319 . its two initial ideals are therefore generated in degree @xmath320 . + _ normal , but not graded : _ the normal configuration @xmath321 where @xmath322 has the toric ideal @xmath323 . hence @xmath324 is the unique element in both its reduced grbner bases . [ sharp ]   ( @xcite , chapter 13 ) the bound in conjecture  [ bigconjecture ] is best possible . consider the graded @xmath5-normal configuration @xmath325 where @xmath326 . ( note that @xmath50 simplicial ) . then @xmath327 . consider the initial ideal @xmath108 from theorem  [ dnormal ] for a graded @xmath150-normal @xmath0 . since @xmath0 is graded , we may assume without loss of generality that @xmath328 for @xmath314 . we will show that @xmath108 is generated in degree at most @xmath24 .    for a @xmath329 , recall that @xmath330 . define @xmath331 . then @xmath332 is a partition of @xmath31 $ ] . let @xmath333 be the artinian ideal in @xmath334 $ ] from theorem  [ artproj ] . recall that the standard monomials of @xmath335 are the roots of standard pairs in @xmath109 with face @xmath86 . since the supports of these roots lie in @xmath336 , @xmath337 $ ] is a monomial ideal @xmath338 with @xmath339 , and @xmath340    [ lemma1ofdnormalbound ] each minimal generator @xmath341 of @xmath342 is a minimal generator of @xmath108 of degree at most  @xmath24 . a minimal generator @xmath341 of @xmath342 is the projection via @xmath343 of a minimal generator @xmath344 of @xmath108 where @xmath345 . suppose @xmath346 . then @xmath341 is a standard monomial of @xmath108 with @xmath339 . hence @xmath341 is covered by a standard pair @xmath347 of @xmath108 . this implies that all monomials of the form @xmath348 as @xmath349 varies are standard monomials of @xmath108 which contradicts that @xmath350 is in @xmath108 . thus @xmath351 which implies that @xmath352 is a minimal generator of  @xmath108 .    since @xmath328 for @xmath353 $ ] , each lattice point in the half open fundamental parallelopiped @xmath121 of @xmath96 lies on one of the @xmath24 hyperplanes @xmath354 in @xmath19 . therefore , if @xmath355 , then the @xmath356-norm of @xmath357 which equals the first co - ordinate of @xmath358 which equals @xmath359 is at most @xmath26 . this implies that @xmath360 . thus all standard monomials of the artinian ideal @xmath335 have degree at most @xmath26 which implies that the minimal generators of @xmath335 ( and @xmath342 ) have degree at most @xmath24 .    * example  [ example1 ] continued . * for @xmath120 , @xmath361 . note that all minimal generators of @xmath342 are minimal generators of @xmath108 of degree at most three . [ simplexbound ] if @xmath0 is a graded normal configuration with @xmath50 simplicial then @xmath27 has a grbner basis consisting of binomials of degree at most @xmath24 .    assuming that @xmath50 is generated by @xmath362 , @xmath0 is @xmath150-normal where @xmath150 is the regular triangulation of @xmath0 with the unique facet @xmath363 $ ] . here @xmath364 . we argue that all minimal generators of @xmath108 have support in @xmath365 . suppose @xmath366 is a minimal generator of @xmath108 with @xmath367 = f    \neq \emptyset$ ] . let @xmath368 $ ] . then @xmath369 since otherwise @xmath366 would lie on the standard pair @xmath370)$ ] of @xmath108 which is a contradiction . write @xmath371 where @xmath372 and @xmath373 . since @xmath374 , @xmath375 is a standard monomial of @xmath108 which implies that @xmath376 is also a standard monomial of @xmath108 as @xmath375 lies on some standard pair with face @xmath154 $ ] . this is a contradiction and so @xmath377 . the above argument shows that @xmath108 and @xmath342 have the same minimal generators . the degree bound then follows from the proof of lemma  [ lemma1ofdnormalbound ] . theorem  [ simplexbound ] proves theorem  [ dnormalbound ] in the case where @xmath50 is simplicial . when @xmath50 is not simplicial , @xmath108 may have minimal generators that are not pre - images under @xmath343 of the minimal generators of @xmath342 ( or even @xmath335 ) as @xmath86 varies in @xmath378 . our next step is to show that for a @xmath206 , the minimal generators of @xmath108 that project under @xmath343 to the minimal generators @xmath379 of @xmath335 have degree at most @xmath24 . we need a preliminary lemma . let @xmath380 be a @xmath30-polytope in @xmath381 and let @xmath382 be the cone over @xmath380 . then there exists a matrix @xmath383 such that @xmath384 where each row of @xmath385 is the normal to a facet of @xmath382 . hence @xmath386 . let @xmath387 be the system obtained by reversing all the inequalities in @xmath380 : @xmath388    [ halfspaces ] the polyhedron defined by @xmath387 is empty . we may assume that @xmath380 has been translated so that the unit vector @xmath389 lies in the relative interior of @xmath380 . if @xmath390 then by our assumption , @xmath391 which implies that @xmath392 . this implies that @xmath393 where @xmath394 is the dual cone to @xmath382 . ( recall @xmath395 . ) thus there exists some @xmath396 such that @xmath397 . therefore , if we choose @xmath398 such that @xmath399 then @xmath400 , @xmath401 and @xmath402 let @xmath403 be the right hand side vector in the description of @xmath387 . then @xmath404 and by farkas lemma ( * ? ? ? * prop . 1.7 . ) , @xmath405 . [ lemma2ofdnormalbound ] let @xmath86 be a facet of @xmath150 . then for a @xmath406 , the minimal generators of @xmath108 that are preimages of the minimal generator @xmath407 of @xmath335 under the map @xmath343 are squarefree monomials of degree at most  @xmath24 . let @xmath329 , @xmath406 and @xmath408 be a minimal generator of @xmath108 with @xmath409 . all minimal generators of @xmath108 that project to @xmath407 under @xmath343 look like @xmath410 . if @xmath411 , then @xmath407 is the only minimal generator of @xmath108 that projects to @xmath407 and we are done . therefore , we consider the case where @xmath412 . suppose @xmath410 is not squarefree . then there exists an @xmath413 such that @xmath414 where @xmath415 is the @xmath416-th co - ordinate of @xmath417 . since @xmath410 is a minimal generator of @xmath108 , @xmath418 is a standard monomial of @xmath108 with @xmath419 . hence there exists @xmath420 such that a standard pair with face @xmath80 covers @xmath418 . this implies that @xmath421 . since @xmath422 , @xmath423 and thus , @xmath424 . if @xmath425 , then @xmath410 is covered by the standard pair @xmath78 which contradicts that @xmath410 is in @xmath108 . if @xmath426 , then @xmath427 lies in @xmath428 . since @xmath0 lies on the hyperplane @xmath429 , @xmath427 is in fact in the minimal hilbert basis of both @xmath430 and @xmath50 and hence @xmath431 is the unique vector in @xmath432 that satisfies @xmath433 . consequently @xmath434 is a standard pair of @xmath108 . but this implies that @xmath410 lies on this standard pair which is again a contradiction . therefore , @xmath410 is squarefree .    to argue that @xmath435 , it therefore suffices to prove that @xmath436 . suppose @xmath437 $ ] , @xmath438 }    : = \prod_{i \in \sigma } x_i$ ] and @xmath439}$ ] . then for each @xmath440 $ ] , @xmath418 is a standard monomial of @xmath108 and is therefore covered by a standard pair @xmath441 of @xmath108 . the face @xmath442 does not contain @xmath416 since otherwise @xmath410 would be a standard monomial of @xmath108 . in particular , @xmath443 $ ] for any @xmath440 $ ] . also , @xmath444 for each @xmath445 $ ] . we now show that we may assume @xmath446 = [ d]\backslash \ { i    \}$ ] for all @xmath440 $ ] . clearly , @xmath446 \subseteq [ d ]    \backslash \ { i \}$ ] since @xmath447 . suppose a monomial in @xmath448 \backslash \ { i \}]$ ] lies in @xmath108 . then it is divisible by a minimal generator of @xmath108 that projects to @xmath407 under @xmath343 , all of which are squarefree . such a minimal generator would properly divide @xmath410 which contradicts that @xmath410 is a minimal generator of @xmath108 . hence @xmath449 \backslash \{i\})$ ] is a pair of @xmath108 and therefore , contained in a standard pair of @xmath108 . we may assume that @xmath442 is the face of this standard pair . thus @xmath154 \backslash \{i\ } \subset \tau^i$ ] and @xmath446 = [ d]\backslash \ { i \}$ ] as claimed .    since @xmath0 is graded , @xmath450 index @xmath30-simplices in a regular triangulation of @xmath451 , the convex hull of @xmath0 . the simplex indexed by @xmath154 $ ] is geometrically @xmath452 = \{{\bf x } \in { { \mathbb r}}^d \ , : \ , { \bf s}_i \cdot { \bf x } \geq 0 , \ ,    x_1 = 1 , \ , i = 1,2,\ldots , d \ } $ ] where @xmath453 for all @xmath454 \backslash \{i\}$ ] and @xmath455 . now @xmath456 for each @xmath440 $ ] implies that @xmath457_{rev}$ ] where @xmath458_{rev } = \ { { \bf x } \in { { \mathbb r}}^d \ , : \ , { \bf s}_i \cdot { \bf x } \leq    0 , \ , x_1 = 1 , \ , i = 1,2,\ldots , d \ } $ ] . but by lemma  [ halfspaces ] , @xmath452_{rev } = \emptyset$ ] which creates a contradiction . therefore , @xmath459}$ ] is not a minimal generator of @xmath108 and all preimages @xmath410 of @xmath407 have degree at most @xmath24 . * example  [ example1 ] continued . * for @xmath120 , @xmath460 which index the variables @xmath461 . the minimal generators of @xmath108 that map to these variables under @xmath343 are @xmath462 . finally we consider the minimal generators of @xmath108 that do not project under @xmath343 to minimal generators of @xmath335 for any @xmath329 . such generators may exist . * example  [ example1 ] continued . * consider the minimal generator @xmath463 of @xmath108 . then @xmath464 is not a minimal generator of @xmath335 for any of the four facets @xmath86 of @xmath150 . [ lemma3ofdnormalbound ] let @xmath465 be a minimal generator of @xmath108 whose image under @xmath343 is not a minimal generator of @xmath335 for any facet @xmath86 of @xmath150 . then @xmath465 is a quadratic squarefree monomial . let @xmath80 and @xmath466 be facets of @xmath150 and let @xmath467 , @xmath468 with @xmath469 . then @xmath470 is not covered by any standard pair of @xmath108 and hence lies in @xmath108 . since @xmath0 is graded , @xmath471 and @xmath472 are standard pairs of @xmath108 which implies that @xmath473 is a minimal generator of @xmath108 . further , @xmath474 is not a minimal generator of @xmath335 for any @xmath475 . we will prove that @xmath476 is precisely the set of minimal generators of @xmath108 that do not project under @xmath343 to a minimal generator of @xmath335 for a @xmath477 . this will prove the lemma . suppose @xmath465 is a minimal generator of @xmath108 such that @xmath478 is not a minimal generator of @xmath335 for any @xmath329 . let @xmath479 .    _ case ( i ) @xmath480 for some @xmath481 _ : then @xmath482 $ ] . since @xmath483 is not a minimal generator of @xmath484 ( and hence @xmath342 ) , some minimal generator of @xmath342 properly divides @xmath483 . by lemma  [ lemma1ofdnormalbound ] , every minimal generator of @xmath342 is a minimal generator of @xmath108 which contradicts that @xmath483 is a minimal generator of @xmath108 . _ case ( ii ) @xmath485 for some @xmath481 _ : then @xmath483 is covered by the standard pair @xmath486 which contradicts that @xmath487 .    _ case ( iii ) @xmath488 for some @xmath329 , with @xmath489 and @xmath490 _ : write @xmath491 where @xmath492 and @xmath493 . then @xmath494 all of whose minimal generators are minimal generators of @xmath108 . this implies that a divisor of @xmath495 is a minimal generator of @xmath108 . therefore , @xmath483 is not a minimal generator of @xmath108 , a contradiction . the above cases have shown that there is no single @xmath481 such that @xmath496 . therefore , there exists two distinct @xmath497 and two indices @xmath498 such that @xmath499 and @xmath500 . _ case ( a ) @xmath413 _ : since @xmath501 , @xmath502 is not covered by any standard pair of @xmath108 and so lies in @xmath108 . since @xmath503 divides @xmath483 and @xmath483 is a minimal generator of @xmath108 it must be that @xmath504 . but then @xmath505 , @xmath501 is a minimal generator of @xmath335 which contradicts our choice of @xmath483 . therefore this case can not arise . _ case ( b ) @xmath425 _ : by a symmetric argument to the previous , this can not happen . therefore , the only possibility is that @xmath506 and @xmath507 . since @xmath508 and @xmath501 , @xmath509 . by the argument in the beginning of the proof , @xmath503 is a minimal generator of @xmath108 and so @xmath504 . thus @xmath483 lies in the set @xmath510 as claimed . _ proof of theorem  [ dnormalbound ] . _ lemmas [ lemma1ofdnormalbound ] , [ lemma2ofdnormalbound ] and [ lemma3ofdnormalbound ] account for all minimal generators of the initial ideal @xmath108 and show that they all have degree at most @xmath24 . since @xmath0 is graded , the reduced grbner basis of @xmath3 with initial ideal @xmath108 consists of homogeneous binomials . hence these binomials have degree at most @xmath24 . in this last section we construct non - trivial families of both @xmath5-normal and non-@xmath5-normal configurations . recall that any configuration @xmath2 is always @xmath5-normal with respect to all its regular unimodular triangulations @xmath5 . also , a configuration @xmath512 for which @xmath513 is simplicial is @xmath5-normal with respect to its coarsest ( regular ) triangulation @xmath514 if we assume that @xmath515 . call @xmath2 _ simplicial _ if @xmath513 is simplicial . we construct families of @xmath5-normal configurations that are not simplicial and do not admit regular unimodular triangulations . by computer search , firla and ziegler @xcite found hundreds of normal simplicial configurations @xmath2 in @xmath516 and @xmath517 ( in the course of writing @xcite ) that admit no unimodular triangulations . our first result in this section is a construction that extends a firla - ziegler configuration to a family of _ non - simplicial _ @xmath5-normal configurations  one in @xmath518 for each @xmath519  without unimodular triangulations . in the second part of this section we construct a family of normal configurations ( an @xmath520 for each @xmath521 ) that are not @xmath5-normal for any regular triangulation @xmath5 . each firla - ziegler normal simplicial @xmath522 without unimodular triangulations is the hilbert basis of the cone generated by @xmath523 , the first three unit vectors in @xmath524 , and a vector @xmath525 of the form @xmath526 with @xmath527 . in this subsection we let @xmath2 denote such a firla - ziegler configuration and let @xmath528 be the extreme rays of @xmath513 . by definition , @xmath2 is the unique minimal generating set of the semigroup @xmath529 and @xmath530 . the vector @xmath531 is contained in @xmath532 .    since @xmath533 and @xmath534 , @xmath535 . for every @xmath536 with @xmath537 , @xmath538 for @xmath539 since the @xmath540-linear combination of elements in @xmath541 that expresses @xmath349 as an element of @xmath542 must involve a positive multiple of @xmath62 . on the other hand , the @xmath13-linear combination of elements in @xmath0 that expresses @xmath543 as an element of @xmath544 is the sum of distinct vectors in @xmath545 . at least one of these @xmath546 vectors  say @xmath547  has a positive last co - ordinate which implies that @xmath548 . therefore , @xmath549 is in @xmath0 , the minimal hilbert basis of @xmath550 . [ firstfz ] the first firla - ziegler @xmath2 in @xmath551 has @xmath552 and @xmath2 consists of the columns of the matrix @xmath553 the hilbert basis of any rational polyhedral cone can be computed using the software package normaliz  @xcite .    from a firla - ziegler @xmath2 we will now recursively construct configurations @xmath55 for each @xmath519 such that @xmath554 is @xmath5-normal , @xmath54 is not simplicial and @xmath55 has no unimodular triangulations . for each @xmath519 let @xmath555 , @xmath556 and @xmath557 . here @xmath558 is the @xmath24-th unit vector in @xmath19 . letting @xmath559 ( a firla - ziegler configuration in @xmath551 ) and @xmath560 , recursively define @xmath561 , @xmath562 and @xmath563 . we assume that @xmath564 and @xmath565 are always the first and second elements of @xmath566 and that @xmath86 is the index set of @xmath567 in @xmath55 . let @xmath568 and @xmath569 . consider the triangulation @xmath570 of @xmath571 consisting of the maximal subcones @xmath572 and @xmath573 . [ a5 ] the configuration @xmath574 has the following properties :    1 . @xmath575 , 2 . @xmath574 is non - simplicial , 3 . @xmath574 is @xmath576-normal , and 4 . @xmath574 admits no unimodular triangulations .    1 . since @xmath577 , @xmath578 and the first three unit vectors of @xmath579 belong to @xmath580 , it follows that all unit vectors of @xmath579 lie in @xmath581 which gives the result . similarly , @xmath582 . 2 .   since @xmath583 lies in the interior of @xmath584 , the vectors @xmath585 and @xmath586 do not lie on a common facet of the cone . hence @xmath584 is a bipyramidal cone over @xmath587 with six extreme rays and is hence non - simplicial . the triangulation @xmath576 is the regular triangulation of @xmath574 with respect to the weight vector @xmath588 . we first argue that @xmath589 is a minimal generating set of the semigroup @xmath590 . suppose @xmath591 . since @xmath585 is the unique generator of @xmath592 with a positive fifth co - ordinate , @xmath593 where @xmath594 is the unique expression of @xmath595 as an @xmath596-combination of @xmath585 and the other extreme rays of @xmath592 . since @xmath597 , in fact , @xmath598 where the equality @xmath599 follows from the normality of @xmath600 . this in turn implies that @xmath601 . ( note that @xmath602 . ) similarly , @xmath603 is its own hilbert basis . thus , @xmath574 is @xmath576-normal . 4 .   suppose @xmath41 is a unimodular triangulation of @xmath574 and @xmath80 is a facet of @xmath41 . then by ( 1 ) , @xmath604 and @xmath605 . if @xmath606 , then @xmath607 which shows that @xmath608 , a contradiction . hence each maximal simplex in @xmath41 contains exactly one of @xmath585 or @xmath586 and @xmath41 induces a triangulation @xmath609 of @xmath610 . since @xmath41 is unimodular , @xmath609 gives a unimodular triangulation of @xmath600 which is a contradiction as @xmath600 has no unimodular triangulations . therefore , we conclude that @xmath574 has no unimodular triangulations . [ babysfirstdeltanormalfamily ] for each @xmath519 , the configuration @xmath55 has the following properties :    1 . @xmath611 , 2 . @xmath55 is non - simplicial , 3 . @xmath55 is @xmath570-normal , and 4 . @xmath55 admits no unimodular triangulations . this theorem is proved by induction using lemma  [ a5 ] as the base step .    1 . suppose the result is true for @xmath612 . then it follows that @xmath613 contains the first @xmath26 unit vectors of @xmath614 which are therefore also in @xmath615 and @xmath616 . since @xmath617 ( and @xmath618 ) , we also get that @xmath619 ( and @xmath620 ) . hence @xmath621 . 2 .   assume by induction that @xmath622 is non - simplicial and that @xmath623 lies in the interior of @xmath624 . then , @xmath625 lies in the interior of @xmath54 and hence @xmath626 and @xmath565 do not lie on a common facet of @xmath54 . this implies that @xmath627 has exactly two more extreme rays than @xmath628 and hence is non - simplicial . 3 .   as in lemma [ a5 ] , @xmath570 can be induced as the regular triangulation of @xmath55 with respect to the weight vector @xmath629 for each @xmath519 . we assume by induction that @xmath622 is @xmath630-normal and hence normal . the arguments that @xmath631 and @xmath632 are minimal generating sets for @xmath633 and @xmath634 respectively follow from a straight generalization of the arguments in lemma  [ a5 ] . 4 .   again we assume by induction that @xmath622 admits no unimodular triangulations . the rest of the argument is also a straight generalization of the arguments in lemma  [ a5 ] ( 4 ) . we have thus produced non - simplicial @xmath5-normal configurations without unimodular triangulations in every dimension beyond four , starting with a firla - ziegler @xmath2 in @xmath551 . the construction applies to all such firla - ziegler configurations . consider the graph @xmath635 shown in figure  [ ho ] . in @xcite hibi and ohsugi showed that the graded normal configuration @xmath636 admits no regular unimodular triangulations , although it does have non - regular unimodular triangulations . further , the @xmath637 points in @xmath638 are all extreme points of the convex hull of @xmath638 , denoted as @xmath639 . the @xmath640-polytope @xmath641 is _ empty _ which means that it has no lattice points other than its vertices . thus @xmath642 which is a cone over @xmath639 has @xmath637 extreme rays and is therefore non - simplicial . [ emptynotdeltanormal ] let @xmath643 be a normal graded non - simplicial configuration in @xmath644 such that @xmath451 is empty . if @xmath0 does not have a regular unimodular triangulation then @xmath0 is not @xmath5-normal for any regular triangulation @xmath5 .    without loss of generality , we can assume that @xmath645 . by the hypothesis , every regular triangulation @xmath5 of @xmath0 has a maximal face @xmath86 such that @xmath646 . thus the hilbert basis of @xmath96 contains at least one vector @xmath647 not in @xmath89 . since all vectors in @xmath0 are extreme rays of @xmath50 , none of them lie in @xmath96 unless they are in @xmath89 . this implies that @xmath648 is not normal and hence @xmath0 is not @xmath5-normal . the hibi - ohsugi configuration @xmath638 is not @xmath5-normal for any regular triangulation @xmath5 . from @xmath638 we now recursively construct configurations @xmath55 for each @xmath521 such that @xmath55 is normal and graded but not @xmath5-normal for any regular triangulation @xmath5 . for each @xmath521 let @xmath649 . letting @xmath650 , recursively define @xmath651    [ babysfirstnormalfamily ] for each @xmath521 , the configuration @xmath55 is normal and graded but not @xmath5-normal for any regular triangulation @xmath5 . it suffices to show that @xmath55 satisfies the conditions of lemma  [ emptynotdeltanormal ] for each @xmath24 . for a given @xmath24 , @xmath55 is graded since it lies in @xmath652 and @xmath653 is a @xmath640-polytope and hence empty . further , @xmath54 is non - simplicial as @xmath624 is non - simplicial for all @xmath521 . the configuration @xmath654 is normal . to see this let @xmath655 . since the only extreme ray of @xmath656 with non - zero eleventh co - ordinate is @xmath657 , @xmath658 . further , @xmath659 where @xmath660 is the unique expression of @xmath595 as an @xmath596-combination of the extreme rays of @xmath656 . the integral vector @xmath661 lies in @xmath662 since @xmath663 is normal and hence it lies in @xmath664 . thus @xmath665 . by induction , it follows that @xmath55 is normal for all @xmath521 . suppose @xmath654 had a regular unimodular triangulation . then @xmath666 would be a vertex in every maximal face of this regular unimodular triangulation of @xmath451 . this in turn induces a regular unimodular triangulation in @xmath667 and hence in @xmath663 , a contradiction . again , a straightforward inductive argument shows that @xmath55 has no regular unimodular triangulation for all @xmath521 . we thank diane maclagan for alerting us to the interpretation of simon s condition for a monomial ideal to be cohen - macaulay in her paper @xcite with greg smith . we also thank winfried bruns , joseph gubeladze , serkan hoten , francisco santos and gnter ziegler for discussions on constructing @xmath5-normal configurations . , computing the integral closure of an affine semigroup , effective methods in algebraic and analytic geometry , 2000 ( krakow ) . _ acta math . _ * 39 * , ( 2001 ) , 5970 . + software : _ normaliz _ , ftp.mathematik.uni-osnabrueck.de/pub/osm/kommalg/software .
a normal ( respectively , graded normal ) vector configuration @xmath0 defines the toric ideal @xmath1 of a normal ( respectively , projectively normal ) toric variety . 
 these ideals are cohen - macaulay , and when @xmath2 is normal and graded , @xmath1 is generated in degree at most the dimension of @xmath3 . based on this , sturmfels asked if these properties extend to initial ideals  when @xmath2 is normal , is there an initial ideal of @xmath1 that is cohen - macaulay , and when @xmath2 is normal and graded , does @xmath1 have a grbner basis generated in degree at most @xmath4  ? 
 in this paper , we answer both questions positively for @xmath5-normal configurations . 
 these are normal configurations that admit a regular triangulation @xmath5 with the property that the subconfiguration in each cell of the triangulation is again normal . 
 such configurations properly contain among them all vector configurations that admit a regular unimodular triangulation . 
 we construct non - trivial families of both @xmath5-normal and non-@xmath5-normal configurations .