The regularity of a toric variety

The Regula i y o a To ic Va ie y. E. B iales-Mo ales and P. Pis´on-Casa es Depa amen o de ´ Algeb a, Facul ad de Ma em´a icas, Apa ado 1160, 41080 Se illa (Spain). Fax: + 34 95 4556938. E-mail: [email p o ec ed], [email p o ec ed] and A. Vigne on-Teno io Depa amen o de Ma em´a icas, E.U.E. Emp esa iales, Po -Ve a 54, 11403 Je ez de la F on e a (C´adiz, Spain). Fax: + 34 956 345104. E-mail: [email p o ec ed] We gi e a me hod compu ing he deg ees o he minimal syzygies o a o ic a ie y by means o combina o ial echniques. Indeed, we comple e he explici desc ip ion o he minimal ee esolu ion o he semig oup algeb a associa ed, using he simplicial ep esen a ion o Koszul homology appea ed in [8]. As an applica ion, we ob ain an algo i hm compu ing he Cas elnuo o-Mum o d egula i y o a p ojec i e o ic a ie y. This egula i y is explici y bounded by means o he semig oup gene a o s which pa ame ize he a ie y. Key Wo ds: To ic a ie ies, syzygies, simplicial complexes, egula i y. INTRODUCTION Le S⊂Zhbe a ini ely gene a ed commu a i e semig oup wi h ze o elemen , such ha S∩(−S) = {0}. Le {n1, . . . , n } ⊂ Sbe a se o gene a o s o S. Le kbe a ield, le k[S] be he semig oup k−algeb a associa ed o S, and le R=k[X1, . . . , X ] be he polynomial ing in a iables. k[S] is ob iously an S−g aded ing, and Ris S−g aded assigning he deg ee {ni} o Xi.Le m he i ele an ideal o R. The k−algeb a 1 2E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO mo phism, ϕ:R−→ k[S], de ined by ϕ(Xi) = {ni},is an S−g aded mo phism o deg ee ze o. Thus, he ideal IS=ke (ϕ) is an S−homogeneous ideal ha i is called a Semi- g oup Ideal o a To ic Ideal because i de ines a o ic a ie y. The condi ion S∩(−S) = {0}gua an ees he S−g aded Nakayama’s lemma (P oposi ion 1.4 in [4]). Then, he e exis s a minimal S−g aded ee esolu ion o k[S]. Le Nibe he co esponding i-syzygy module (N0=I) and conside he k− ec o spaces Vi(m) := (Ni)m (mNi)m , m ∈S. By Nakayama’s lemma: •A minimal gene a ing se o Niconsis s exac ly o dimk(Vi(m)) ele- men s o deg ee m, o each m. •The elemen s o deg ee min a minimal gene a ing se o Ni, co espond wi h a basis o Vi(m). In pa icula , since Ris noe he ian, one has Vi(m) = 0 o all mbu ini e many o alues. I is well-known ha he e exis me hods using G ¨obne Bases (Sch eiye ’s Theo em) compu ing a minimal gene a ing se o Ni(see o example [11]). Howe e , we a e in e es ed in o unde s and combina o ially hese gen- e a ing se s, and he e o e he minimal S−g aded ee esolu ion o k[S]. We in oduced he ollowing no a ion: Le Λ := {1, . . . , }, and i F⊂Λ, nF=Pi∈Fni(n∅= 0). I m∈S, he se ∆m={F⊂Λ|m−nF∈S} is an abs ac simplicial complex. These simplicial complexes appea o he i s ime in he li e a u e in [8]. They a e a gene aliza ion o some g aphs de ined in [20]. Al hough, bo h wo ks a e inside he con ex o nume ical semig oups, ac ually ∆m appea s in a lo o good pape s inside mo e ex ensi e con ex s. Fo example, conside ˜ Hi(∆m) he k− ec o space o he i- educed homology. The e exis s an isomo phism (∗)˜ Hi(∆m)≃Vi(m). The exis ence o his isomo phism is p o ed in [8]. The gene aliza ion o a semig oup Swi h ou ini ial condi ions appea s in [10]. A such isomo phism is explici y cons uc ed in [5]. O he e y in e es ing pape s inside his line THE REGULARITY OF A TORIC VARIETY 3 a e [1], [7],[15]. The connec ion be ween ee esolu ions and simplicial complexes comes om [13] (see also [21] and [6]). Con inuing wi h ou p oblem. No ice ha he p e ious isomo phism p o ides he ollowing cons uc ion o ob ain a minimal gene a ing se o Ni. CONSTRUCTION: STEP 1: Find he se Ci:= {m∈S|˜ Hi(∆m)6= 0}o S−deg ees o he minimal i-syzygies. STEP 2: Fo any m∈Ci, ake he images o he elemen s in a basis o he i- educed homology space ˜ Hi(∆m) by he isomo phism. S ep 2 is sol ed wi h an algo i hmic me hod in [5] (Rema k 3.6). Thus, Cons uc ion will be an algo i hm when one knows an e ec i e way o compu ing he se Ci. In pa icula , since he se C0is a i hme ically explici ed in [10], his cons uc ion p o ides an algo i hm o i= 0 (To ic/Semig oup ideals) by means o In ege P og amming, [4]. Using a deg ee bound o he elemen s in C0, o he al e na i e algo i hm appea s in [12]. In [9] new simplicial complexes associa ed o a pa i ion o Λ allows he au ho s o deduce applica ions o conc e e si ua ions. Howe e , hey canno compu e he ini e se s Ciin he gene al case. Indeed, hey es ablish as he main p oblem in he compu a ion o he minimal esolu ion o ind ini e se s C0 icon aining Ci. A ini e se con aining C1is e ec i ely compu ed in [17]. The e o e, a new me hod compu ing he i s syzygy module N1is ob ained. Mo eo e , an explici bound o he deg ee o he minimal i s syzygies is gi en. This bound is desc ibed by means o he gene a o s o he semig oups S. In his pape , we gene alize he esul s in [17] o all o highe o de syzygies (sec ion 1). The e o e, we ob ain a new me hod compu ing he minimal gene a ing se o he syzygy modules because we sol e e ec i ely S ep 1 in Cons uc ion. The esul s in his pape allow us o say ha Cons uc ion is an algo i hm. In his way, we comple e he combina o ial desc ip ion o he minimal S−g aded ee esolu ion o k[S] ini ia ed in [8]. As an applica ion, we conside he pa icula case o a p ojec i e o ic a ie y (sec ion 4). We ob ain an algo i hm compu ing i s Cas elnuo o- Mum o d egula i y. Wi hou using his compu a ion, we desc ibe an ex- plici bound o he egula i y by means o he semig oup gene a o s which pa ame ize he a ie y. The key idea o ind ini e se s C0 icon aining Ciis he ollowing: Fo any F⊂Λ and o any τ iangula ion o Fin o i-dimensional aces, we conside a diophan ine linea sys em. The se C0 iis ob ained as a union 4E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO o subse s o he Hilbe bases associa ed o all o hese sys ems. C0 iis ini e because he Hilbe bases a e ini e (Dickson’s lemma). C0 ican be compu ed in an e ec i e way because he e exis algo i hms compu ing he Hilbe basis (see, o example, [16]). We can conclude ha : he noe he ian p ope y o Ras well as In e- ge P og amming gua an ee he se Ciis ini e, bu In ege P og amming p o ides also he way compu ing his ini e se . 1. FINDING THE SET C I We conside , as in he in oduc ion, S=< n1, . . . , n >⊂Zha ini ely gene a ed semig oup wi h ze o elemen such ha S∩(−S) = {0}, he se Λ = {1, . . . , }, and o any m∈S he simplicial complex ∆m={F⊂Λ|m−nF∈S}, whe e nF=Pi∈Fni.Gi en a ield k,˜ Hi(∆m) is he k− ec o space o he i- educed homology. We a e looking o a ini e se C0 icon aining he se Ci:= {m∈S|˜ Hi(∆m)6= 0}. Recall ha om he isomo phism (∗) in he in oduc ion, Ciconsis s ex- ac ly o he S−g aded minimal i-syzygies o k[S], he semig oup k−algeb a associa ed o S. Fix m∈S, choose an o ien a ion on each ace o ∆m, and conside he augmen ed chain complex wi h alues in he ield k. Le ˜ Ci(∆m) be he k− ec o space gene a ed eely by he i-dimensional aces o ∆m, whe e dimF=]F −1 (dim ∅=−1), and le δi:˜ Ci(∆m)→˜ Ci−1(∆m) he k−linea mapping gi en by δi(F) = X F0∈∆m,dimF0=i−1 F F 0F0, whe e F F 0= 0 i F06⊂ F, and F F 0=±1 i F0⊂F,F F 0= 1 i he o ien a ion induced by Fon F0is equal o he o ien a ion chosen on F0, and F F 0=−1 o he wise. We a e in e es ed in he link ˜ Ci+1(∆m)δi+1 →˜ Ci(∆m)δi →˜ Ci−1(∆m) wi h i≥2. Indeed, he ildes can be omi ed. Howe e we keep hem because ou esul s a e ue e en i i≥0.Le ˜ Zi(∆m) = ke (δi) and ˜ Bi(∆m) = Im(δi+1) be he spaces o cycles and bo de s. Then, we ha e THE REGULARITY OF A TORIC VARIETY 5 ha ˜ Hi(∆m) = ˜ Zi(∆m)/˜ Bi(∆m). Lemma 1.1. Suppose ha ˜ Hi(∆m)6= 0, and le c∈˜ Zi(∆m)−˜ Bi(∆m), c=P j=1 λjFj,λj∈k− {0} o any j= 1, . . . , ,Fj6=Fli j6=l. Then, i F=S j=1 Fjone has ha ∀p∈F∃q, 1≤q≤ |p /∈Fqand Fq∪ {p}/∈∆m. P oo . We begin p o ing ha ∀p∈F∃j, 1≤j≤ , such ha p /∈Fj. Suppose ha p∈Fand p∈Fj o any j, 1≤j≤ . Then, c0= P j=1 λjFjFj−{p}(Fj− {p})∈˜ Ci−1(∆m). No ice ha c0= 0 because δi(c) = 0, and he e o e λj= 0, ∀j= 1, . . . , . Bu i is no possible because c6= 0. Fix p∈F. We can suppose ha p /∈F1. I F1∪ {p}/∈∆mwe ha e inished. Suppose hen ha F1∪ {p} ∈ ∆m. Se l:= {j|p /∈Fj}and F0 1=F1∪ {p}. We conside c1:= c−λ1F0 1F1δi+1(F0 1). No ice ha c1∈˜ Zi(∆m)− ˜ Bi(∆m). Mo eo e , c1= X j=2 λjFj+X dim(F0)=i,F 06=F1 F1F0F0. Se c1=P (1) j=1 λ(1) jF(1) j, whe e λ(1) j∈k− {0} o any j= 1, . . . , (1), F(1) j6=F(1) li j6=l,l(1) := ]{j|p /∈F(1) j}, and F(1) := S (1) j=1 F(1) j. The ollowing p ope ies a e sa is ied (∗)1:]F(1) ≤F. (∗∗)1: I p /∈F(1) j, hen he e exis s q, 2 ≤q≤ such ha F(1) j=Fq. (∗ ∗ ∗)1:l(1) < l. Indeed, we eplace he ace F1in cwi h p /∈F1, by se e al aces F06=F1 wi h p∈F0 o cons uc c1. We p oceed by induc ion on ]F. I ]F is minimal, in (∗)1 he equali y holds. The e o e, p∈F(1) and l(1) 6= 0. Using he simila a gumen s wi h c1, he e exis s j, 1 ≤j≤ (1), such ha p /∈F(1) j. By (∗∗)1,F(1) j=Fq o some q, 2 ≤q≤ . I F(1) j∪ {p}/∈∆m, we ha e inished. Bu , i F(1) j∪ {p} ∈ ∆m, we can 6E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO ob ain c2∈˜ Zi(∆m)−˜ Bi(∆m) om c1, simila ly o he cons uc ion o c1 om c. Se c2=P (2) j=1 λ(2) jF(2) jwhe e λ(2) j∈k−{0} o any j= 1, . . . , (2), F(2) j6=F(2) li j6=l. Then, i l(2) := ]{j|p /∈F(2) j}, and F(2) := S (2) j=1 F(2) j we ob ain he p ope ies: (∗)2:]F(2) ≤F(1). (∗∗)2: I p /∈F(2) j, hen he e exis s q, 2 ≤q≤ (1) such ha F(2) j= F(1) q. (∗ ∗ ∗)2:l(2) < l(1). Again, in (∗)2 he equali y holds because ]F is minimal. The e o e p∈ F(2) and l(2) 6= 0. Now, ou esul ollows by ecu ence because he p ope ies (∗∗∗) gua an ees ha his p ocess mus inish in a ini e numbe o s eps. Suppose hen ou esul is ue o any c0∈˜ Zi(∆m)−˜ Bi(∆m), c0= P 0 j=1 λ0 jF0 jwhe e λ0 j∈k− {0} o any j= 1, . . . , 0,F0 j6=F0 li j6=l, wi h ]F0< ]F whe e F0=S 0 j=1 F0 j. Then: I in (∗)1 he equali y does no hold, i is enough o apply he induc ion hypo hesis o c1. We ob ain ha he e exis s F(1) jsuch ha p /∈F(1) jand F(1) j∪{p}/∈∆m. By (∗∗)1,F(1) j=Fqwi h 2 ≤q≤ , and we ha e inished. I in (∗)1 he equali y holds, we epea he same a gumen ha in he case ]F minimal. I we don’ inish, we cons uc c2as be o e. Again, we ob ain he esul by induc ion i in (∗)2 he equali y does no hold. O he wise, we begin he p ocess. We inish in a ini e numbe o s eps by he p ope ies (∗∗∗) which a e ob ained in he ecu ence. Now, ou esul is p o ed. Lemma 1.1 associa es o m∈Swi h ˜ Hi(∆m)6= 0 se s F⊂Λ and F1, . . . , F ∈∆msuch ha : 1. F=S j=1 Fj 2. dim(Fj) = i,∀j= 1, . . . , 3. F /∈∆m No ice ha τ={F1, . . . , F }is a iangula ion o Fin o i-dimensional aces o ∆m. This sugges s he ollowing de ini ion. De ini ion 1.1. Le F⊂Λ. We say ha τ={F1, . . . , F }is an i- iangula ion o Fi he ollowing p ope ies a e sa is ies: 1. dim(Fj) = i,∀j= 1, . . . , 2. F=S j=1 Fj THE REGULARITY OF A TORIC VARIETY 7 We say ha τis an i- iangula ion o Fin ∆m, wi h m∈S, i Fj∈∆m, ∀j= 1, . . . , , and F /∈∆m. Lemma 1.2. I ˜ Hi(∆m)6= 0, he e exis s F⊂Λand τi- iangula ion o Fin ∆m. P oo . I is enough o ake c∈˜ Zi(∆m)−˜ Bi(∆m) and F, F1, . . . , F as in Lemma 1.1. Rema k 1. 1. I is clea ha gi en m∈Swi h ˜ Hi(∆m)6= 0, Fis no unique in gene al. Mo eo e , o a ixed F, he i- iangula ion τis no unique in gene al. Suppose ha τ={F1, . . . , F }is an i- iangula ion o Fin ∆m. No ice ha he p ope y F1∈∆mis equi alen o m−nF1∈S. This means ha he e exis s αj∈Nsuch ha m=P j=1 αjnj, wi h αj≥1 o any j∈F1. Se A he ma ix whose column ec o s a e he gene a o s o S,A:= (n1|. . . |n )∈ Mh× (Z). Se eF1∈N he ec o wi h coo dina es equal o ze o, excep ing he j h one which is equal o one, o any j∈F1. Then, F1∈∆mi and only i he e exis s α(1) ∈N such ha Aα(1) =m, and α(1) eF1, whe e he symbol s ands o he na u al pa ial o de in N . Wi h simila no a ions we can ob ain an analogous condi ion o Fj∈ ∆m. This is: o any j, 1≤j≤ ,Fj∈∆mi and only i he e exis s α(j)∈N such ha Aα(j)=m, and α(j)eFj. Se A( ) :=        A −A 0 0 0 0 0 0A −A 0 0 ··· 0 0 0 0 A −A 0 0 0 ......... 0 0 0 0 0 A −A        ∈ Mh( −1)× (Z) and eτ:= (eF1, . . . , eF )∈N . Then, om τ, an i- iangula ion o Fin ∆m, we can ob ain a ec o α= (α(1), . . . , α( ))∈N which sa is ies: 1. A( )α= 0. 2. αeτ 8E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO 3. m=Aα(1) =. . . =Aα( ). Bu we canno o ge ha we a e looking o he m∈Ssuch ha ˜ Hi(∆m)6= 0. Then, we mus p oceed on he con a y. Le F⊂Λ and τ={F1, . . . , F }an i- iangula ion o F. Now, mis no ixed. We a e going o look o he elemen s m∈Ssuch ha τis an i- iangula ion o Fin ∆m. As be o e, we conside eτ:= (eF1, . . . , eF )∈N . Se Rτ:= {α= (α(1), . . . , α( ))∈N | A( )α= 0 , α eτ}. No ice ha i α∈Rτand i is w i en α= (α(1), . . . , α( )) wi h α(j)∈ N , o any j, 1≤j≤ , i is ob ained Aα(1) =. . . =Aα( )=m∈S. Se ΣRτ:= {m∈S|m=Aα(1), α = (α(1), α(2), . . . , α( ))∈Rτ}. A he momen i is also known ha i m∈Sis such ha ˜ Hi(∆m)6= 0, using Lemma 1.1 we ind Fand τas be o e such ha m∈ΣRτ. Then, we ob ain ha Ci=[ F[ τ ΣRτ, whe e he union is o e all o τi- iangula ion o F, and F⊂Λ wi h ]F ≥i+ 2. (No ice ha Fcomes om Lemma 1.1 and hen ]F ≥i+ 2). The se ΣRτis no ini e in gene al. The e o e, we ha e ye no ound ou se C0 i. Howe e , Rτ,HRτ:= {α∈Rτ|αis minimal o } is ini e. We a e going o p o e ha Ciis con ained in a union as be o e, bu o subse s o ΣHRτ:= {m∈S|m=Aα(1), α = (α(1), α(2), . . . , α( ))∈ HRτ}. By his way, we cons uc ou desi ed ini e se C0 i. We need o in oduce some new no a ion. On he elemen s o S, we de ine a pa ial o de ≥S m≥Sm0⇔m−m0∈S I H⊂S, we shall say ha m∈His S−minimal in Hi m≥Sm0wi h m0∈H, implies ha m=m0. Se Cτ:= {m∈S|mis S−minimal in ΣRτ}. THE REGULARITY OF A TORIC VARIETY 9 Lemma 1.3. In he condi ions as abo e, o any m∈ΣRτ, he e exis s m0∈Cτand m00 ∈Ssuch ha m=m0+m00. P oo . I m∈Cτ, i is enough o ake m0=mand m00 = 0. O he wise, he e exis s m0 1∈ΣRτsuch ha m≥Sm0 1. Then, m=m0 1+m00 1, wi h m00 1∈S. I m0 1∈Cτwe ha e inished. O he wise, we begin he easoning wi h m0 1. By ecu ence we cons uc a sequence o elemen s m0 j−1=m0 j+m00 j, whe e m0 j∈ΣRτand m00 j∈S. The condi ion S∩(−S) = {0}gua an ees ha he numbe o di e en exp essions o mas sum o non null elemen s in Sis ini e (P oposi ion 1.2 in [4]). Then, ou p ocess mus inish. Suppose ha i inishes in he j h s ep. Then, m0 j∈Cτ. Now m= m0 j+m00 1+...+m00 j, hence i is enough ake m0=m0 jand m00 =m00 1+. . .+ m00 j. Lemma 1.4. In he condi ions as abo e, Cτ⊂ΣHRτ. The e o e, he se Cτis ini e. P oo . Le m∈Cτ. Then, m=Aα(1) wi h α= (α(1), . . . , α( ))∈Rτ. I α∈ HRτ, we ha e inished. Suppose ha α /∈ HRτ. Then α=α0+α00, wi h α0∈ HRτand α00 ∈N . Mo eo e , α00 =α−α0sa is ies ha A( )α00 = 0. Then, i m0=Aα0(1), m0∈ΣHRτ. By m−m0=m00 =Aα00(1) ∈Sand m is S−minimal, we ob ain ha m=m0∈ΣHRτ. P oposi ion 1.1. I m∈Sand ˜ Hi(∆m)6= 0, hen he e exis s F⊂Λ wi h ]F ≥i+ 2, and he e exis s τi- iangula ion o Fin ∆msuch ha m∈Cτ. P oo . I ˜ Hi(∆m)6= 0, le c∈˜ Zi(∆m)−˜ Bi(∆m), c=P j=1 λjFj,λj∈k− {0} o any j= 1, . . . , ,Fj6=Fli j6=l. Then, i F=∪ j=1Fjand τ= {F1, . . . , F }, we ha e ha τis an i- iangula ion o Fin ∆mand m∈ΣRτ. By Lemma 1.3 m=m0+m00 wi h m0∈Cτand m00 ∈S. I m00 = 0, hen m=m0∈Cτand we ha e inished. Suppose ha m00 6= 0. Le m00 =P j=1 βjnjwi h βj∈N o any j, 1≤j≤ . We can suppose ha β16= 0. I 1 ∈F, we apply lemma 1.1 o p= 1. Then, he e exis s q, 1≤q≤ , such ha 1 /∈Fqand Fq∪ {1}/∈∆m. Howe e , m−nFq−n1∈Sbecause Fq∈∆m,β16= 0 and 1/∈Fq. This is a con adic ion wi h Fq∪ {1}/∈∆m. The e o e, 1 /∈F. 16 E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO No ice ha α∈ HRτi and only i α−eτ∈ H{β∈N | A( )(β+eτ)=0}. Se Aτ:= (A( )| − A( )eτ). We a e looking o he ela ion wi h Hilbe basis o he homogeneous sys ems Aτβ0= 0, β0∈N +1. I is easy o e i y ha H{β∈N | A( )β=−A( )eτ}={β∈N |(β, 1) ∈ H(Aτ)}. Lemma 3.1. Wi h he no a ion and condi ions as be o e, he ollowing inequali y is sa is ied ||(α−eτ,1)||1≤(1 + 4||A||1,∞)h(di−1), being di:=  i+ 1 . P oo . Using he abo e easoning we ob ain ha (α−eτ,1) ∈ H(Aτ). Then, by Rema k 3.1 ||(α−eτ,1)||1≤(1 + ||Aτ||1,∞)s, whe e s= ank(Aτ). Recall ha he ma ix Aτhas h( −1) columns, whe e ]τ = ,τ= {F1, . . . , F },Fj⊂F, and ]Fj=i+1, o any j. Then =]τ ≤]F i+ 1 ≤  i+ 1 =di, and ankAτ≤h( −1) ≤h(di−1). Finally, le see us ha ||Aτ||1,∞≤4||A||1,∞. I is enough o no ice ha ||A||1,∞=max1≤j≤ −1{||(A(2)| − A(2)(eFj, eFj+1 ))||1,∞} and ||(A(2)| − A(2)(eFj, eFj+1 ))||1,∞= THE REGULARITY OF A TORIC VARIETY 17 max1≤p≤h{ X q=1 |apq|+ X q=1 | − apq|+X q∈Fj |apq|+X q∈Fj+1 | − apq|} ≤ 4||A||1,∞ Theo em 3.2. I m∈Sis an S−deg ee o a minimal i-syzygy o k[S], hen m=Axwi h x∈N such ha ||x||1≤(1 + 4||A||1,∞)h(di−1) + (i+ 1)di−1, whe e di= i+ 1 . P oo . Wi h he no a ion in Lemma 3.1, m=Aα(1) wi h α= (α(1), . . . , α( )) sa is ying his lemma. Now, i is enough o no ice ha ||α(1)||1≤ ||α||1=||α−eτ+eτ||1≤ ||α−eτ||1+||eτ||1=||(α−eτ,1)||1−1+(i+1) . By Lemma 3.1, ||α(1)||1≤(1 + 4||A||1,∞)h(di−1) + (i+ 1)di−1. We a e going o see how i is possible o imp o e Theo em 3.2 in he pa icula cases i= 0,1. Suppose i= 0. I is well-known ha any minimal gene a ing se o he ideal I( he 0-syzygies) is con ained in he G a e basis o A,G A G A:= {Xα−Xβ|(α, β)∈ H(A(2))} (see [22] o de ails). Then, i m∈Sis a minimal deg ee o I, we ob ain ha m=Aα=Aβ, wi h (α, β)∈ H(A(2)). Using again Rema k 3.1 we ha e ha ||α||1≤(1 + 2||A||1,∞)h because ||A(2)||1,∞≤2||A||1,∞. This is clea ly an imp o emen o Theo em 3.2 o he case i= 0. Suppose i= 1. As we ha e seen be o e, we can educe o a special ype o 1- iangula ions o Fin ∆m, he F-ca i ies. All o hem ha e he same shape, a polygon wi h ]F e ices. Thus, ollowing he s eps in he p oo o Theo em 3.2, we ob ain ha 18 E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO ||x||1≤(1 + 4||A||1,∞)h( −1) + 2 −1 because ]F ≤ . The imp o emen now is ha d1= 2> because ≥3. In [17] a simila esul appea s. E en o ge ing he o sion, he e exis s a small di e ence. In [17] he numbe D:= max{||Aτ||1,∞|τis an F-ca i y, ]F ≥3} eplaces o 4||A||1,∞. I is clea ha D≤4||A||1,∞( he same a gumen s used in Theo em 3.2 a e igh ). The ad an age o ou e sion is ha he bound is s aigh o wa d ob ained om A, wi hou being necessa y o cons uc any ma ix Aτ.I is also clea ha in Theo em 3.2 he numbe Di:= max{||Aτ||1,∞|τis a i- iangula ion o F, ]F ≥i+ 2} can eplace o 4||A||1,∞, bu ou e sion is mo e use ul o he p oo o Theo em 4.1. The e is no esul simila o Theo em 3.2 in he li e a u e, o i≥2. In he ollowing ema k we conside he gene aliza ion o he case Swi h o sion non i ial. Rema k 3. 2. S⊂Zh⊕Z/a1Z⊕. . . ⊕Z/asZ wi h aj∈Z, 1 ≤j≤s. Now, A∈M(h+s)× (Z) and A( )∈ M(h+s)( −1)× (Z). To use he easoning in he wi hou o sion case, we need o emo e he cong uences in he sys ems o kind A( )β=−A( )eτ. Then, i is enough o conside he auxilia y ma ices (see [17] o de ails) THE REGULARITY OF A TORIC VARIETY 19 T=             0 0 0 0 · · · 0 0 0 . . .. . .. . . 0 0 0 0 · · · 0 0 0 a1−a10 0 0 0 0 0 0 a2−a20· · · 0 0 ......... 0 0 0 0 0 as−as             ∈ M(h+s)×2s(Z), and g A( )=(A( )|˜ T), whe e ˜ T=   T0. . . 0 0 ... 0 0 . . . 0T   ∈ M( −1)(h+s)×( −1)2s(Z). Using he sys em wi hou cong uences o kind g A( )β=−A( )eτ, he ollowing esul is ob ained. I m∈Sis an S-deg ee o a minimal i-syzygy o k[S], hen m=Ax wi h x∈N such ha ||x||1≤(1 + 2a+ 4||A||1,∞)(h+s)(di−1) + (i+ 1)di−1, whe e a=max1≤j≤s|aj|and di= i+ 1 . 4. REGULARITY OF PROJECTIVE TORIC VARIETIES In his sec ion we suppose ha Iis homogeneous o he na u al g adua- ion. This is equi alen o he e exis s a ec o w∈Qhsuch ha ni·w= 1, o any i= 1, . . . , . (Lemma 4.14 in [22]). Geome ically, Ide ines a p ojec- i e a ie y in P −1(k). No ice ha i ∈Iis S-homogeneous o S-deg ee m∈S, hen deg( ) = ||α||1, o any α∈N wi h Aα=m. Suppose ha { 1, . . . , β1}is a minimal gene a ing se o I,S-deg( i) = pi∈Sand deg( i) = ||αi||1, whe e pi=Aαi, 1 ≤i≤β1. I g= (g1, . . . , gβ1)∈N1is a 1-syzygy o S-deg ee m∈S, hen S-deg(gi) = m−pi and deg(gi) = ||βi||1whe e Aβi=m−pi. Thus, m=A(αi+βi) and deg(g) = ||αi+βi||1. 20 E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO Mo e gene ally, i h= (h1, . . . , hβi)∈Niis an i-syzygy o S-deg ee m∈S, hen deg(h) = ||α||1, o any α∈N such ha m=Aα. By his way, we ob ain an explici bound o he Cas elnuo o-Mum o d egula i y o I. In he ollowing heo em, he symbol bc s ands o in eg al pa . Theo em 4.1. Wi h assump ions and no a ions as abo e, eg(I)≤(1 + 4||A||1,∞)h(d−1) + ( + 1)(d−1) whe e d= b /2c. P oo . The egula i y o Iis eg(I) = max1≤i≤ { i−i}, whe e iis he maximum deg ee o he i-syzygies o I(see, o example, [2]). By Theo em 3.2, i≤(1 + 4||A||1,∞)h(di−1) + (i+ 1)di−1, wi h di= i+ 1 . I is clea ha di≤d, o any i. Then i−i≤(1 + 4||A||1,∞)h(d−1) + ( + 1)(d−1). I is enough o ake maximum. Finally, no ice ha he e ec i e compu a ion o he egula i y o Idoesn’ equi e he compu a ion o he minimal esolu ion. I is su icien o use he S ep 1 o Cons uc ion in he in oduc ion o any i. We desc ibe his algo i hm. Algo i hm 1. Compu ing he egula i y Inpu : Se o gene a o s {n1, . . . , n }o S. (Recall ha hey mus lie on a a ional hipe plane.) Ou pu : The egula i y o he ideal Io S. 1. Fo any i, 1 ≤i≤ •Compu e he se C0 i(Theo em 1.1). •Check he elemen m∈C0 isuch ha ˜ Hi(∆m)6= 0 and ob ain Ci. THE REGULARITY OF A TORIC VARIETY 21 •Compu e i={||α||1|m=Aα∈Ci}. 2. Ou pu eg(I) = max{ i−i|i= 1, . . . , }. REFERENCES 1. A. ARAMOVA, J. HERZOG, F ee esolu ion and Koszul homology. J. o Pu e and Applied Algeb a, 105 (1995), 1-16. 2. D. BAYER, M. STILLMAN, A c i e ion o de ec ing m- egula i y. In en iones ma hema icae, 87 (1987), 1-11. 3. F. DI BIASE, R. URBANKE, An Algo i hm o Calcula e he Ke nel o Ce ain Polynomial Ring Homomo phisms, Expe imen al Ma hema ics, Vol.4, No. 3 (1995), 227-234. 4. E. BRIALES, A. CAMPILLO, C. MARIJU´ AN, P. PIS ´ ON, Minimal Sys ems o Gene a o s o Ideals o Semig oups, J. o Pu e and Applied Algeb a, 124 (1998), 7-30. 5. E. BRIALES, A. CAMPILLO, C. MARIJU´ AN, P. PIS ´ ON, Combina o ics o syzygies o semig oup algeb a. Colle . Ma h. 49 (1998), 239-256. 6. W. BRUNS, J. HERZOG, Cohen-Macaulay ings. Camb igde Uni e si y P ess, (1993). 7. W. BRUNS, J. HERZOG, Semig oup ings and simplicial complexes. J. o Pu e and Applied Algeb a,122 (1997), 185-208. 8. A. CAMPILLO, C. MARIJU´ AN, Highe ela ions o a nume ical semig oup. S´em. Th´eo . Nomb es Bo deaux 3(1991), 249-260. 9. A. CAMPILLO, P. GIM´ ENEZ, Syzygies o a ine o ic a ie ies. Jou nal o Algeb a o appea . 10. CAMPILLO A.; PIS ´ ON P. L’id´eal d’un semi-g upe de ype ini. C. R. Acad. Sci. Pa is, S´e ie I,316 (1993), 1303-1306. 11. D. EISENBUD, In oduc ion o Commu a i e Algeb a wi h a View Towa ds Alge- b aic Geome y. G adua e Tex s in Ma hema ics 150 (1995), Sp inge , New Yo k. 12. P.A. GARC´ IA-S´ ANCHEZ Semig upos a ines. Ph. D. Thesis, Uni e sidad de G anada (1996). 13. M. HOCHSTER, Cohen-Macaulay ings, combina o ics, and simplicial complexes, In BR. McDonald, R.A. Mo is edi o s, Ring Theo y II, LNPAM 26, Ma cel-Dekke (1995), 171-223. 14. S. HOSTEN, B. STURMFELS, GRIN, An Implemen a ion o G ¨obne Bases o In ege P og amming, In E. Balas and J. Clausen edi o s, In ege P og amming and Combina o ial Op imiza ion, LNCS 920, Sp inge -Ve lag (1995), 267-276. 15. I. PEEVA, B. STURMFELS, Syzygies o codimension 2 la ice ideals, Ma h. Z. 229 (1998), 163-194. 16. P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO, N−solu ions o linea sys ems o e Z.P ep in o Uni e si y o Se illa 43 (1998). 17. P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO, Fi s Syzygies o To ic Va ie ies and Diophan ine Equa ions in Cong uence, P ep in o Uni e si y o Se illa 52 (1999). 22 E. BRIALES-MORALES, P. PIS ´ ON-CASARES, A. VIGNERON-TENORIO 18. L. POTTIER, Bo nes e algo i hmes de calcul des gene a eu s des solu ions de sys ´emes diophan ines lin´ea ies. C.R. Acad. Sci. Pa is, Se ie I 311(1990), 813-816. 19. L. POTTIER, Minimal solu ions o linea diophan ine sys ems: bounds and algo- i hms. P oceedings o he Fou h In e na ional Con e ence on Rew i ing Techniques and Applica ions, I aly 1991, 162-173. 20. J.C. ROSALES, Semig upos num´e icos . PhD Thesis, Uni e sidad de G anada (1991). 21. R. STANLEY, Combina o ics and commu a i e algeb a 2nd ed. P og ess in Ma h- ema ics Vol.41, Bos on Basel Be lin, Bi kh¨ause (1996). 22. B. STURMFELS, G ¨obne bases o To ic Va ie ies. Tˆohoku Ma h. J. 43 (1991), 249-261. 23. A. VIGNERON-TENORIO, Semig oup Ideals and Linea Diophan ine Equa ions. Linea Algeb a and i s Applica ions, 295 (1999), 133-144.