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, . . . , }.
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