A combinatorial description of finite O-sequences and aCM genera

Contents lists available atScienceDirect

Journal

of

Symbolic

Computation

www.elsevier.com/locate/jsc

A

combinatorial

description

of

finite

O-sequences

and

aCM

genera

Francesca Cioffi

a

,

1

, Paolo Lella

b

,

2

,

Maria

Grazia Marinari

c a_{UniversitàdegliStudidiNapoli“FedericoII”,DipartimentodiMatematicaeApplicazioni,Complessouniv.} di MonteS.Angelo,ViaCintia,80126Napoli,Italy

b_{UniversitàdegliStudidiTrento,DipartimentodiMatematica,ViaSommarive14,38123Povo, Trento,Italy} c_{UniversitàdegliStudidiGenova,DipartimentodiMatematica,ViaDodecaneso35,16146Genova,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received28July2014 Accepted13March2015 Availableonline17March2015

MSC: 13C14 14N10 14H99 14Q05 05C20 Keywords: aCMgenus FiniteO-sequence Cohen–Macaulaycurve Directedgraph Partialorder

The goal of this paper is to explicitly detect all the arithmetic generaofarithmeticallyCohen–Macaulayprojectivecurveswitha givendegreed.Itiswell-knownthatthearithmeticgenus g ofa curve C canbeeasilydeduced fromtheh-vectorofthecurve;in the case where C isarithmetically Cohen–Macaulay ofdegree d, g mustbelongtotherangeofintegers0,. . . ,d−₂1.Wedevelop an algorithmic procedure that allows one to avoid constructing mostofthepossibleh-vectorsofC .Theessentialtoolsarea com-binatorial descriptionof the finiteO-sequences ofmultiplicity d,

andasortofcontinuityresultregardingthegenerationofthe gen-era. Theefficiency ofourmethodis supportedbycomputational evidence. Asaconsequence, we single outthe minimalpossible Castelnuovo–MumfordregularityofacurvewithCohen–Macaulay postulationandgivendegreeandgenus.

©2015ElsevierLtd.All rights reserved.

E-mailaddresses:francesca.cioffi@unina.it(F. Cioffi),paolo.lella@unitn.it(P. Lella),marinari@dima.unige.it(M.G. Marinari).

URLs:http://www.docenti.unina.it/francesca.cioffi(F. Cioffi),http://www.paololella.it(P. Lella),

http://www.dima.unige.it/~marinari/(M.G. Marinari).

1 _{PartiallysupportedbyPRIN2010-11Geometriadellevarietàalgebriche (2010S47ARA),cofinancedbyMIUR(Italy),andby}

GNSAGA.

2 _{PartiallysupportedbyPRIN2010-11Geometriadellevarietàalgebriche (2010S47ARA),cofinancedbyMIUR(Italy),byFIRB}

2012ModulispacesandApplications (RBFR12DZRV)andbyGNSAGA.

http://dx.doi.org/10.1016/j.jsc.2015.03.006

1966forageometricalpointofview,and

Elias

etal.,1996inthecontextoflocalalgebra).

Infact,notonly doestheh-vector encodea lotofinformationaboutthegeometryofthecurve; thearithmeticgenusofthecurveisalsoeasilydeducedfromit(Hartshorne,2010,Exercises8.11and 8.12),(Migliore,1998,Section1.4).ForanaCMprojectiveschemetheh-vectorisactuallytheHilbert function ofits Artinian reduction. Thisresult ismainly dueto the fundamental paperof Macaulay (1926)characterizingtheHilbertfunctionsofstandardgradedalgebras.

We stress the fact that thework ofMacaulay doesnot provide an algorithmic solution forthe problemofdeciding whetherornot anaCMcurve ofdegreed andgenus g exists.Thisremarkhas beenthestarting pointofourpaper.By investigatingthesetoffiniteO-sequencesofmultiplicity d andits properties we obtain our solution,both computational and theoretical,that relieson some closed formulaconsiderably reducing the amount of realcomputations. We have not beenable to findanalogousresultsinliterature.

Asafirststep,weprovideaverynaturalcombinatorialdescriptionoffiniteO-sequences,bymeans ofsuitableconnectedgraphs,andweobtainanefficientsearchalgorithmofthearithmeticgeneraof Cohen–Macaulaycurves(see

Algorithm 1

inSection2).

Then,forevery positiveintegerd,we denoteby Rd

=



0

,



d−₂1



∩ N

thesetofintegersto which thegenusofaCohen–Macaulaycurveofdegreed mustbelong,andwefocusourattentiononsmaller ranges Rs

d,consistingofthegenera ofCohen–Macaulaycurvesofdegreed andh-vector oflength s.

Byintroducingaconvenienttotalordering onthesetofO-sequencesofmultiplicityd andlength s, wecansingleouteachrangeRs_d (see

Corollary 2.10

,

Theorem 3.2

,

Proposition 3.4

).

Theintegersin Rd thatcannot berealizedasgenusofanaCMcurveofdegreed arecalledgaps.

ManyofthemarelocatedoutsideeveryrangeRs

d,someotherslienear themaximalgenusinRsd,for

valuesofs thatcanbeexactlydeterminedbysuitableclosedformulas(see

Propositions 4.3 and

4.9). Finally,we providean algorithm to computeall the generaof aCMcurvesfora givendegree d, avoiding to constructall the corresponding O-sequences(see Algorithm 2in Section 5). The strat-egysupportingthisalgorithmcombinestheprevious resultstogetherwithasortofcontinuity inthe generationofthegeneraofaCMcurvesdevelopedin

Lemma 5.1

andappliedin

Theorem 5.4

. Experi-mentalcomputationspointoutthatonlyasmallpercentageofintegersofRd needstobecheckedby

thesearchalgorithm(see

Tables 1 and

2).

InSection6,weapplyoursearchalgorithmtodetecttheminimalpossibleCastelnuovo–Mumford regularityofacurvewithCohen–Macaulaypostulationandgivendegreeandgenus(Proposition 6.1). Moreover, we answer toa question posed in

Cioffi

andDi Gennaro (2011) aboutthe Castelnuovo– MumfordregularityofevendimensionalprojectivesubschemeshavingthesameHilbertfunctionofa Cohen–Macaulayprojectivescheme(Example 6.3).

1. Generalities on O-sequences and aCM genera

Inthissection,westatesome notationandrecallsome basicresultsonO-sequences,referringto

BrunsandHerzog (1993)and

Valla (1998)

.

Giventwopositiveintegersa,t,thebinomialexpansionofa inbaset istheuniquewriting

a

=



k(_tt)



+



k(_tt₋−₁1)



+ · · · +



k(_jj)



(1.1)

where k

(

t

)

>

k

(

t

−

1

)

>

· · · >

k

(

j

)

≥

j

≥

1 with the convention that



_mn



=

0 whenever n

<

m and



n

0



at

:=



k(_tt₊)+₁1



+



k(t−_t1)+1



+ · · · +



k(_jj₊)+₁1



,

byaneasycomputation,onegets

(

a

+

1

)

t

>

at.Anumericalfunctionh

: N

→ N

isadmissible oran O-sequence ifh

(

0

)

=

1 andh

(

t

+

1

)

≤

h

(

t

)

tforeveryt

≥

1.

Ifhisan admissiblefunctionandh

(

t

)

=

0 forsomet,then h

(

t

+

i

)

=

0 foreveryi

>

0,andhis calledafiniteorArtinianO-sequence.ForafiniteO-sequence

(

h0

,

. . . ,

hs−1

)

weassume hs−1

=

0.The integers isthelength oftheO-sequenceandtheintegere

(

h

)

:=



s_i=−₀1hiisitsmultiplicity.

ItiswellknownthatthereisabijectivecorrespondencebetweenthesetoffiniteO-sequencesof multiplicityd andthesetofHilbertfunctionsofaCohen–Macaulaystandardgradedalgebraof multi-plicityd overafieldK (Valla,1998,Theorem1.5).Infact,alltheseHilbertfunctionscanbecomputed fromthefiniteO-sequences.Inparticular,ifthegradedalgebraistheringofregularfunctionsonan aCMcurve C (i.e. a closedsubscheme C

⊂ P

n_K ofdimension1), theHilbertfunction HC ofC isthe

2-thintegral ofafiniteO-sequencesh

= (

h0

,

h1

,

. . . ,

hs−1

)

,i.e. letting HC

(

0

)

:=

HZ

(

0

)

:=

h

(

0

)

=

1 and HZ

(

t

)

=

HZ

(

t

−

1

)

+

h

(

t

)

foreveryt

>

0,wehave

HC

(

t

)

=

HC

(

t

−

1)

+

HZ

(

t

),

for every t

>

0.

Hence, histheso-calledh-vector ofC andtheHilbertpolynomialofC is pC

(

z

)

=

dz

+

1

−

g where,

afteraneasycomputation,wefindthatthearithmeticgenus ofC is

g

=

1

+ (

s

−

2)d

−

p

(

s

−

2)

=

s−1

j=2

(

j

−

1)hj

≥

0. (1.2)

Inthissituation,wesaythat HC isanaCMfunction oraCohen–Macaulaypostulation, pC

(

z

)

isanaCM polynomial and g isanaCMgenus.

Remark 1.1. Thefollowingfactsareimmediateremarks:

(i) thearithmeticgenusofanaCMcurveisnon-negative;

(ii) every positive integer g is thegenus ofsome aCMcurve:itis enoughto take anyO-sequence

(

1

,

h1

,

g

)

,withh₁1

≥

g;

(iii) if g isthearithmeticgenusofsome aCMcurve Cd ofdegreed,thenthereisalsoanaCMcurve Cd+1 ofdegreed

+

1 withthesamearithmeticgenus g;indeed,ifh

= (

1

,

h1

,

h2

,

. . . ,

hs−1

)

isthe h-vectorofCd,thenthesequenceh

= (

1

,

h1

+

1

,

h2

,

. . . ,

hs−1

)

isalsoanO-sequenceandisthe h-vectorofacurveCd+1 withHilbertpolynomial

(

d

+

1

)

z

+

1

−

g.Indeed,themultiplicityofthe O-sequencehisd

+

1 andthenweapplyformula

(1.2)

,inwhichtheintegerh1 doesnotoccur. Fromageometricpointofview,thismeansthatCd+1 canbeobtainedastheunionofCd anda

linethroughapointofCd.

2. A combinatorial description of finite O-sequences

Inthissection,we consideranaturalstructureonthesetofallfiniteO-sequences.Thisstructure will entailbothoursearch algorithmofthearithmeticgeneraofCohen–Macaulay curves,andsome usefulinformationabouttheaCMgenera,suchastheexistence ofminimalgeneracorrespondingto O-sequenceswithgivenlength(andmultiplicity).

Weleteidenoteanysequence,ofanylength,consistingentirelyof0 except1 inthei-thposition.

Moreover,weintroducethefollowingcompactnotationforsomeparticularsequences:

(1

α0

_,

_hα1 i1

,

h α2 i2

, . . . ,

h αk ik

)

:= (

1, . . . ,



1 α0times

,

h

i1

, . . . ,



hi1

α1times

, . . . ,

h

ik

, . . . ,



hi

k αktimes

).

Definition 2.1. TheO-sequencesgraph isthedirectedgraph

G

suchthat:

•

thesetofverticesV

(G)

consistsofthefiniteO-sequences;

Fig. 1. TheO-sequencegraphG uptomultiplicity7.ThedashededgesareedgesofGthatdonotbelongtothespanning treeT.

•

thesetofedges E

(

G)

consistsofthepairs

(

h

,

h

)

∈

V

(

G)

2 s.t. h

−

h

=

ei forsome i (i.e.

(

h

,

h

)

∈

E

(G)

ifhcanbeobtainedfromhbyincreasingby1 itsi-thentry).

Anedge

(

h

,

h

)

∈

E

(

G)

fromhtohislabeledbyei ifh

−

h

=

ei.

Letusconsider the map g

:

G → N

that associates with eachO-sequencethe genus of an aCM curvehavingthisO-sequenceash-vector.

Proposition 2.2. TheO-sequencesgraph

G

isarootedconnectedgraphwithoutloops.Therootisthe O-sequenceofmultiplicity1.

Proof. For anyh

= (

1

,

h1

,

. . . ,

hs−1

)

,thesequenceh

=

h

−

es−1isadmissiblesothatthereisanedge going fromh to h. Repeatingthisprocedure, we get thelength one O-sequence

(

1

)

whichcannot betheheadofanyedge,provingthat

G

isconnected.Therearenoloopsaseachedgeincreasesthe multiplicityby1.

2

Remark 2.3. Denotedbyd_G

(

h

)

thedistanceofthenodehfromtheroot,wehaved_G

(

h

)

=

e

(

h

)

−

1. Wearegoingtodefineasubgraph

T ⊂ G

whichwillturnouttobeaspanningtree.Inthisway, wecandesignad hocalgorithmstovisit thetreeinordertoquicklyfindtheO-sequenceswiththe propertieswewilllookfor.Theideafordetermining

T

istheoneusedintheproofof

Proposition 2.2

. Foreach node of

G

, we consideronly theedge comingfromthe O-sequenceobtainedloweringby 1 thevalue withthe greatestindex. Indeed,notice thateach O-sequenceh (of anylength s)hasa successorin

T

,ash

+

esisalwaysafiniteO-sequence,whereasthesequenceh

+

es−1 mightnotbe admissible.

Definition 2.4. WecallO-sequencestree thesubgraph

T ⊂ G

suchthat:

•

V

(T )

=

V

(G)

;

•

E

(T )

=



(

h

,

h

)

∈

E

(G)



h

=

h

+

esor h

=

h

+

es−1

,

if hs−s−22

>

hs−1



.

Fig. 2. ThesubgraphsGs_{oftheO-sequencegraphwithgivenlengths.Alongthegreydottededgesthelengthincreases,so}

suchedgesofGdonotbelongtoanysubgraphGs_{.ThedashededgesareedgesofG}s_{thatdonotbelongtothecorresponding}

spanningtreeTs_.

In mostsituations,we willwork withO-sequenceswithgivenmultiplicity (i.e. with nodesof

G

at thesame distancefrom theroot) orwithgiven length (see Fig.2). We denoteby

G

d the set of

O-sequencesofmultiplicityd andby

G

sthesetofO-sequencesoflengths.

Remark 2.5. As in the spanning tree

T

each vertex is the tail of at most 2 edges, we have that

|G

d

|

<

2

|G

d−1

|

.Moreover,since

|G

2

|

=

1,byrecursion

|G

d

|

<

2d−2.

Proposition 2.6. Thesubgraph

G

s

⊂

G

isarootedconnectedgraphwithroot

(

1s

)

containingaspanningtree

T

s_{withthesameroot(seeFig.2).}

Proof. We need to show that, for any O-sequence h

= (

1s

₎

_{of length s, there exists another}

O-sequence of the same length with multiplicity e

(

h

)

−

1. If k

=

max

{

1

≤

i

≤

s

−

1

|

hi

>

1

}

, then

h

= (

1

,

h1

,

. . . ,

hk

,

1s−k−1

)

andh

= (

1

,

h1

,

. . . ,

hk

−

1

,

1s−k−1

)

isadmissible.

2

Remark 2.7. Denoted by ds

G

(

h

)

the distance of the node h from the rootof

G

s, we havedsG

(

h

)

=

d_G

(

h

)

− (

s

−

1

)

=

e

(

h

)

−

s.

G

disnotasubgraphof

G

,astherearenoedgesof

G

betweenO-sequenceswiththesame

multi-plicity.Buttheedgesof

G

inducethefollowingnaturalpartialorderon

G

d.

Definition 2.8. TwoO-sequencesh1andh2in

G

d aredirectlycomparable ifthereexistsh0

∈

G

d−1such that h1

=

h0

+

ei and h2

=

h0

+

ej,i.e. h1

−

h2

=

ei

−

ej. On directly comparableO-sequences we

considertheorder

h1

≺

h2

⇐⇒

i

<

j (2.1)

Fig. 3. The order relations among directly comparable elements ofGd, d=1, . . . ,7.

Thepartialorder

≺

givesanaturalstructureofdirectedgraphto

G

d.Theedgesareallthepossible

pairs

(

h

,

h

)

∈

V

(G

d

)

2 suchthath

=

h

+

ej

−

eiand j

>

i (see

Fig. 3

).Asbefore,wedefineaspanning

treeofthegraphstructureof

G

d whichallowsustoefficientlyexaminethesetofO-sequenceswith

given multiplicity. The same procedure is also extended to the set of O-sequences

G

s

d with given

multiplicityd andlengths (see

Fig.

4). Moreover,welet

hs

(

d

)

:= (

1,d

−

s

+

1,1s−2

)

and gs

(

d

)

:=

g

(h

s

(

d

))

=



s−₂1



.

(2.2) Proposition 2.9.

(i) Thegraph

G

dcontainsaspanningtree

T

dwithroottheO-sequence

(

1

,

d

−

1

)

.

(ii) Thesubgraph

G

_dscontainsaspanningtree

T

_dswithroottheO-sequencehs

(

d

)

.Thus,

G

_dsisalsoconnected.

Proof. (i) Foreachvertexh

∈

G

d

\ {(

1

,

d

−

1

)}

,thespanningtree

T

dcontainstheedgees−1

−

e1going fromh

=

h

−

es−1

+

e1 toh,wheres isthelengthofh.

(ii)Foreachvertexh

= (

1

,

h1

,

. . . ,

hi

,

1d− i

j=0hj

₎

_∈

_G

s

d

\{(

1

,

d

−

s

+

1

,

1

s−2

₎

_}

_{(i.e. i}

_>

_{1),thespanning} tree

T

_ds containstheedgeei

−

e1goingfromh

=

h

−

ei

+

e1 toh.

2

Fig. 4. Graph descriptions of O-sequences with given multiplicity and length.

Algorithm 1 The algorithmforsearchingaCMgenerawithgivenconstraintsonthemultiplicity and thelengthoftheO-sequences.Atrialversionofthisalgorithmisavailableat

http://www.paololella.it/

HSC/Finite_O-sequences_and_ACM_genus.html.

1: procedure genusSearch(g

,

T



)

Input: g,anon-negativeinteger.



T

,aspanningtreechosenamong

T

,

T

d,

T

sand

T

ds. Output: an O-sequencehsuchthat g

(

h

)

=

g (ifitexists).

2: stack

:= {

root

_(

T )}

; 3: while stack

= ∅

do

4: h

:=

removeFirst

₍

_stack

₎

_; 5: if g

(

h

)

=

g then return h; 6: else if g

(

h

)

<

g then

7: addFirst

₍

_stack

_,

children

₍

_h

_,

T ))



_;

8: end if

9: end while

10: end procedure

Corollary 2.10. Theorderinducedon

G

dbythetotalorderon

N

throughthemapg

:

G

d

→ N

isarefinementof thepartialorder

≺

.Inparticular,hs

(

d

)

=

min

(

G

_ds

)

withrespectto

≺

,gs

(

d

)

istheminimalgenuscorresponding toanO-sequenceoflengths andmultiplicityd anditdoesnotdependond.

Proof. If h1

−

h2

=

ei

−

ej,then g

(

h1

)

=

g

(

h2

)

+ (

i

−

1

)

− (

j

−

1

)

=

g

(

h2

)

+

i

−

j,by formula

(1.2)

. Hence,weobtain

h1

≺

h2

⇐⇒

i

<

j

⇒

g

(h

1

) <

g

(h

2

)

andtheassertionabouttheminimumfollowsby

Proposition 2.9

.

2

Astheminimalgenusgs

(

d

)

doesnotdependonthevalueofd,fromnowonwewillsimplydenote itbygs.

b. ifg

₍

h

)

isgreaterthanthegenuswearelookingfor,thenwecanavoidtovisitthetreeof descen-dantsofh,asthegenusincreasesalongtheedges(Proposition 2.2and

Corollary 2.10

);

c_{. ifg}

₍

_h

₎

_{issmallerthanthegenuswearelookingfor,thenweneedtovisitthetreeofdescendants} ofh,soweaddthechildrenofhinthetree

T



atthebeginningofthelist(resp. atthetopofthe stack)containingtheverticesstilltobevisited.

3. Combinatorial ranges

Fromnowon,weassumed

>

2,as

G

d hasonlyoneelementford

∈ {

1

,

2

}

.

Forconvenience,weletGd(resp. Gsd)bethesetofallthearithmeticgeneraoftheaCMcurvesof

degreed (resp. ofdegreed withh-vectoroflengths),i.e. Gd

:= {

g

(

h

)

|

h

∈

G

d

}

(resp. G_ds

:= {

g

(

h

)

|

h

∈

G

s d

}

).

Lookingatthegraph

G

d,weimmediatelycanobservethewellknownfactthatGd

⊆



0

,

. . . ,



d−₂1



(see

Hartshorne,

1994,Theorem3.1).Denotingby

[

a

,

b

]

thesetofintegers

{

n

∈ N

|

a

≤

n

≤

b

}

,welet Rd

:=



0

,



d−₂1



.In therange Rd wesingle out smallersuitable ranges,takingintoaccountalsothe

lengthoftheO-sequences.

Recallthat, by the partial order

≺

introduced in Definition 2.8 andby Corollary 2.10, we have min

(

Gs_d

)

=

g

(

min

(

G

_ds

))

=

gs

=



s−₂1



,thusgs

<

gs+1andgs+1

−

gs

=

s

−

1.Inordertoobtainan analo-gousresultaboutamaximum,weextendthepartialorder

≺

tothefollowingtotalorderon

G

_ds.

Definition 3.1. GiventwoO-sequencesh

= (

1

,

h1

,

. . . ,

hs−1

)

andh

= (

1

,

h1

,

. . . ,

hs−1

)

of

G

ds,wedenote

by

<

thetotalorderon

G

s

dsuchthath

<

hifh

<

h,where



:=

max

{

j

:

hj

=

hj

}

.

Althoughthe usual orderon

N

does notinduce on

G

_ds thetotal order

<

(see Example 3.3), we noticethat min_≺

(G

s

d

)

=

min

(G

ds

)

withrespectto

<

.Furthermore,wecanconsideralsomax

(G

ds

)

with

respectto

<

andobtainthefollowingnon-obvious result.

Theorem 3.2. Leth

= (

1

,

h1

,

. . . ,

hs−1

)

andk

= (

1

,

k1

,

. . . ,

ks−1

)

betwoO-sequencesof

G

_ds.Ifk

<

hand g

(

k

)

>

g

(

h

)

,then there is an O-sequenceh

¯

∈

G

_ds such thath

¯

>

h and g

(¯

h

)

>

g

(

k

)

.Thus, max

(

Gs_d

)

=

g

(

max

(G

s

d

))

.

Proof. We canassume s

−

1

=

max

{

j

:

hj

=

kj

}

,hencehs−1

>

ks−1 becauseh

>

k.Bythehypotheses,

wehave g

(h)

=

s−2

j

(

j

−

1)hj

+ (

s

−

2)hs−1

<

s−2

j

(

j

−

1)kj

+ (

s

−

2)ks−1

=

g

(k)

whichimpliesthereexiststheintegert

:=

max

{

j

∈ {

2

,

. . . ,

s

−

2

}

:

hj

<

kj

}

andso

(1,

h1

, . . . ,

ht

,

ht+1

, . . . ,

hs−2

,

hs−1

)

∧ 



∨

(1,

k1

, . . . ,

kt

,

kt+1

, . . . ,

ks−2

,

ks−1

)

(3.1)



_h t

<

kt

,

hi

≥

ki

,

t

+

1

≤

i

≤

s

−

2,

hs−1

>

ks−1

.

Notethatk_tt

≥

h_tt

≥

ht+1

≥

kt+1.Hence,wecanconsidertheO-sequenceh

:=

k

−

bet

+



sj−=1t+1cjej,

where b

=

min

⎧

⎨

⎩

kt

−

ht

,

s−1

j=t+1 hj

−

kj

⎫

⎬

⎭

and cj

=

min

⎧

⎨

⎩

hj

−

kj

,

b

−

j−1

i=t+1 ci

⎫

⎬

⎭

andh_j

≤

hjforevery j

>

t.

Thecorrespondinggenusofhis

g

(h

)

=

g

(k)

− (

t

−

1)b

+

s−1

j=t+1

(

j

−

1)cj

>

g

(k) >

g

(h).

Ifneeded, replacingthe O-sequencekby handrepeating thesameargumentasbefore,we obtain an O-sequence h withh_j

=

hj forevery j

>

t and g

(

h

)

>

g

(

h

)

. Ifh

<

h, wecan repeat the same

argumentasbeforeuntilweobtainanO-sequenceh

¯

withh

¯

j

=

hjforevery j

>

t andh

¯

t

≥

ht

+

1.

2

Example 3.3. (a)ConsiderthetwoO-sequencesh

= (

1

,

6

,

4

,

2

,

1

)

andk

= (

1

,

4

,

7

,

1

,

1

)

of

G

₁₄5 .Wehave h

>

kand11

=

g

(

h

)

<

g

(

k

)

=

12 asinthe hypothesesof

Theorem 3.2

.Inthiscase,we obtaint

=

2, b

=

min

{

3

,

1

}

=

1, c3

=

min

{

1

,

3

}

=

1 andc4

=

min

{

0

,

2

}

=

0,so that h

¯

=

k

−

e2

+

e3

= (

1

,

4

,

6

,

2

,

1

)

withgenusg

(¯

h

)

=

13

>

g

(

k

)

andh

¯

>

h.

(b)ConsiderthetwoO-sequencesh

= (

1

,

13

,

3

,

3

,

3

)

andk

= (

1

,

6

,

13

,

2

,

1

)

of

G

₂₃5.Wehaveh

>

k and 18

=

g

(

h

)

<

g

(

k

)

=

20.Applying Theorem 3.2,ast

=

2,b

=

min

{

10

,

3

}

=

3,c3

=

min

{

1

,

10

}

=

1 andc4

=

min

{

2

,

9

}

=

2,wedetermineh

¯

=

k

−

3e2

+

e3

+

2e4

= (

1

,

6

,

10

,

3

,

3

)

>

handg

(¯

h

)

=

18

+

2

+

3

=

21

>

g

(

k

)

.

Lookingagainatthegraph

G

s,wecanfindawaytodetectg

(

max

(

G

_ds

))

.Wefirstnotethat,ifd

<

s, then

G

_dsisemptyandifd

=

s,thenwehaveauniqueO-sequence

(

1s

)

correspondingtoaplanecurve ofdegrees,i.e. withgenus



s−₂1



.Ford

=

s

+

1 wehavetheuniqueO-sequence

(

1

,

2

,

1s−2

)

,obtained from

(

1s

)

byincreasingh1by1 andcorrespondingtoacurveofdegrees

+

1 andgenus



s−1 2



.Inthe othercases,wededucemax

(G

s

d

)

assumingtoknowtheO-sequenceh

=

max

(G

ds−1

)

andconsequently thegenusg

(

h

)

=



s_j−₌1₂hj

(

j

−

1

)

=

max

(

Gds−1

)

(Theorem 3.2).Nextresultshowshowtofindmax

(G

ds

)

andthen g

(

max

(

G

_ds

))

.

Proposition 3.4. Givenanyd

>

s

≥

3,leth

=

max

(G

s

d−1

)

.If

ı

isthehighestindexsuchthath

+

eıisan

O-sequencein

G

_ds,thenmax

(

G

_ds

)

=

h

+

eıandg

(

max

(

G

ds

))

=

g

(

max

(

G

s

d−1

))

+ ı −

1.

Proof. By theassumption,wehavehı

<

hıı−−11,sothathı

+

1

≤

hı−ı−11 andhı+r

=

hı+_ı+rr₋−₁1,forevery

1

≤

r

≤

s

−

1

− ı

,thatis: h

= (

1, . . . ,hı

,

hıı

,

hı+ 1 ı+1

, . . . ,

h s−2 s−2

)

and h

+

eı

= (

1, . . . ,hı

+

1,hıı

,

hı+ 1 ı+1

, . . . ,

h s−2 s−2

).

Foreveryh

∈

G

_ds₋₁

\ {

h

}

,considertheinteger



:=

max

{

j

:

hj

=

hj

}

.Then,wehaveh

<

handh+r

=

h+r,forevery1

≤

r

≤

s

−

1

− 

,becauseh

=

max

(

G

_ds₋₁

)

.Notethatwehave



< ı

,otherwiseh_

<

h

Fig. 5. TherangesR4

dford=4,. . . ,10.Inthepicture,theedgesontheleftarelabeledwiththecorrespondingincreaseofthe

genus.

h

= (

1, . . . ,h_

, . . . ,

hı

,

hıı

, . . . ,

h s−2

s−2

).

If there were an O-sequence h

∈

G

_ds₋₁ such that h

+

eλ

>

h

+

eı for some index

λ

such that

h

+

eλ

∈

G

_ds,then

ı < λ

.Wehaveseenthathandhcertainly haveequalentriesforindicesgreater

than or equal to

ı

and hı

+

1

>

hı

=

hı. But, for indices j

> ı

, the value hj

=

hj

=

h j−1

j−1 cannot beincreasedby thedefinitionofO-sequences.Thus, weobtain max

(

G

_ds

)

=

h

+

eı.The lastassertion

followsby

Theorem 3.2

andformula

(1.2)

.

2

Foreveryd

>

2 ands

∈ {

d₂



+

1

,

. . . ,

d

}

,welet

hs

(

d

)

:= (

1,2d−s

,

12s−d−1

)

and gs

(

d

)

:=

g

(h

s

(

d

))

=



_s−1 2



+



d−s 2



.

(3.2)

Then,wehave:max

(

G

_ds

)

=

hs

(

d

)

,gd

=



d−1 2



=

gd

(

d

)

andgd−1

=



d−2 2



=

gd−1

(

d

)

.

Remark 3.5. Anotherdescriptionofthemaximal genusofarange Rs_d could besetintermsof min-imalHilbertfunctionswitha constantHilbertpolynomialandagivenregularity(see

Roberts,

1982, Examples4.6and4.8and

Cioffi

etal.,inpress).Bytheway,thecombinatorialdescriptionweprovide herearisesinaverynaturalwayandgivesmoreinformation,atleastfromacomputationalpointof view.

Theprevious resultstogether withthose ofSections2suggest to considerthefollowing smaller rangesin Rd.

Definition 3.6. Foreveryd

≥

s

≥

2,thesetofintegersbetweengsandmax

(

Gs_d

)

iscalled

(

d

,

s

)

-range anddenotedby Rs

d (see

Fig.

5), i.e. Rsd

:=



α

∈ N |



s−₂1



=

gs

≤

α

≤

max

(

Gs d

)



.

Corollary 3.7. Foreveryd

≥

s

≥

2,thearithmeticgenusofanaCMcurveofdegreed havingh-vectoroflength s belongstotherangeRs_d.

4. Unattainable aCM genera in Rd

Recallthatwearedenotingby Rdtherange



0

,



d−₂1



andthat Gd

⊆

Rd. Definition 4.1. AnintegerinRd

\

Gd iscalledagapinRd.

Example 4.2. Theintegers in therange



d−₂2



+

1

,



d−₂1



−

1



aregaps in Rd.More generally,every

integerofRdnotcontainedinany

(

d

,

s

)

-rangeisagap.

Nextresultallowsustocharacterizetheconsecutive

(

d

,

s

)

-rangesthatareseparated,i.e. ranges Rs d

andR_ds+1 suchthatgs+1

−

max

(

Gs d

)

>

1. Proposition 4.3. Foranyd

>

2,wehave

max(G_ds

) <

gs+1

−

1

⇐⇒

2d

+

1

−

√

8d

−

15

2

<

s

≤

d

−

1.

Thus,theintegersin

[

max

(

Gs

d

)

+

1

,

gs+1

−

1

]

aregapsinRd,for2d+1− √

8d−15

2

<

s

≤

d

−

1.

Proof. For s

≥



d₂



+

1,by

(3.2)

wehave:

gs

(

d

) <

gs+1

−

1

⇐⇒



s−1 2



+



d−s 2



<



₂s



−

1. Hence gs

(

d

)

−

gs+1

+

1

=

s 2_−(2d+1)s+d2_−d+4 2

<

0

⇒

2d+1−√8d−15 2

<

s

<

2d+1+√8d−15 2

,

and thus gs

(

d

)

<

gs+1

−

1 if and only if 2d+1− √ 8d−15 2

<

s

≤

d

−

1, because 2d+1−√8d−15 2

>



d 2



, 2d+1+√8d−15 2

>

d

−

1 and 2d+1−√8d−15 2

>

d

−

1 impliesd

<

3.

To provethat there are no other pairsof separatedranges, wenotice that gs

(

d

)

≥

gs+1

−

1

im-plies gs−1

(

d

)

≥

gs

−

1, forevery s. Indeed,as gs

=

gs+1

− (

s

−

1

)

and gs

(

d

)

≤

gs−1

(

d

)

+ (

s

−

2

)

by

Proposition 3.4,wehave

gs−1

(

d

)

−

gs

+

1

≥

gs

(

d

)

− (

s

−

2)

−

gs+1

+ (

s

−

1)

+

1

>

gs

(

d

)

−

gs+1

+

1

≥

0.

2

Example 4.4. Foreveryd

≤

11,thegapsinRd areonlythosedescribedin

Proposition 4.3

.Ford

=

12,

inadditiontothegapsdescribedin

Proposition 4.3

,wefindbydirectcomputationauniquefurther gap

¯

g

=

26,belongingonlytotherange R8₁₂

= [

21

,

28

]

.

Example 4.5. Byadirect computationofthefiniteadmissibleO-sequences,we notethat ford

=

15 the integer g

¯

=

25 belongs to the ranges R6₁₅ and R5₁₅. Nevertheless, whereas foreach h

∈

R5₁₅ we have g

(

h

)

=

25,thereish

= (

1

,

3

,

3

,

4

,

2

,

2

)

∈

R₁₅6 suchthat g

(

h

)

=

25.

Example 4.5suggeststhefollowingdefinition.

Definition 4.6. An integerintherange Rs_d iscalledahole oftherange Rs_d ifit isnot thearithmetic genusofanaCMcurve C ofdegreed withh-vectoroflengths.

Remark 4.7. Noteveryhole isagap.Forinstance,

Example 4.5

tells usthat theinteger 25 is nota gapinR15,althoughitisaholeof R5₁₅.While

Example 4.4

atteststhatthehole26 of R8₁₂isactually agapin R12.

(

1

,

2d−s

_,

₁2s−d−1

₎

_.Inthegraph

_G

s

d,theonlyedgesinvolvingthisvertexareed−2

−

e1 anded−s

−

e2.

Hence,by

Corollary 2.10

,foreachh

∈

G

_ds

\ {

hs

(

d

)}

g

(h)

≤

max



g



hs

(

d

)

− (

ed−s

−

e1

)



,

g



hs

(

d

)

− (

ed−s

−

e2

)



=

max

{

gs

(

d

)

− (

d

−

s

−

1),gs

(

d

)

− (

d

−

s

−

2)

} =

gs

(

d

)

− (

d

−

s

−

2).

2

Alltheholesdescribedinthepreviouslemmaaresurely gapsifweconsider s

>

2d+1− √

8d−15

2 as

in

Proposition 4.3

.Indeed,issuchcasestheseholesdonotbelongtoanyotherrange.

Proposition 4.9. InthehypothesesofLemma 4.8,foreveryi

=

1

,

. . . ,

d

−

s

−

3,theholegs

(

d

)

−

i isagapif s

−

1

−



d−₂s



+

i

>

0.Moreprecisely,

(i) thehighestholegd−4

(

d

)

−

1

=

d(d−211)

+

20 isalwaysagap; (ii) everyholedescribedinLemma 4.8isagapifs

>

2d−1−

√ 8d−31

2 .

Proof. The holegs

(

d

)

−

i isagapifgs

(

d

)

−

i

<

gs+1,i.e.



s 2



−



s−1 2



−



d−s 2



+

i

=

s

−

1

−



d−₂s



+

i

>

0. Theproofof(i)and(ii)isadirectcomputation.

2

Example 4.10. By

Proposition 4.9

,wefindthefollowinggapsin R28:thegap258 belongingonlyto therange R24

d ,240 and239 belongingonlyto R23d ,224,223 and222 belongingonlyto R22d and207,

208 and209 belongingto R21

28.Anyway,by adirectcomputation we findalsothegap188,actually theminimaloneinR28.

5. Computation of the aCM genera for curves of degree d

Proposition 4.9givesacharacterizationofthegapsinRdbelongingtothelastpart ofa

(

d

,

s

)

-range.

Wedidnotfindanalogousconditionsforgapsbelongingtothefirstpart ofa

(

d

,

s

)

-range.Inparticular, itseems hardtogive a characterizationoftheminimal gap.Hence, wewill lookforan algorithmic methodtorecognizethegapsinRd,avoidingtoconstructallthefiniteO-sequencesofmultiplicityd

thankstoasortofcontinuity inthegenerationofthearithmeticgenera.DenotebyGd

+

a thesetof

allarithmeticgeneraoftheaCMcurvesofdegreed augmentedbyanon-negativeintegera.

Lemma 5.1. Gd

⊇

d



−1 j=1



Gj

+



d−j 2



.

Proof. Let

(

1

,

h1

,

. . . ,

hs−1

)

be an O-sequence of multiplicity j

<

d corresponding to an aCM genus g. Assuming h_ii

>

hi+1, for some i

∈ {

1

,

. . . ,

s

−

2

}

, we can consider the finite O-sequence

(

1

,

h1

,

. . . ,

hi+1

+

1

,

. . . ,

hs−1

)

ofmultiplicity j

+

1,corresponding to the genus g

+

i. Then, wecan take also the finite O-sequence

(

1

,

h1

,

. . . ,

hi+1

+

1

,

hi+2

+

1

,

. . . ,

hs−1

)

ofmultiplicity j

+

2, corre-spondingtothegenus g

+

i

+ (

i

+

1

)

,andsoon.Performingthisconstructionfromi

=

1 untild

−

j, wereachthedesiredconclusion.

2

Remark 5.2. Bytheproofof

Lemma 5.1

,wecanobservethatthearithmeticgeneradeterminedbythe O-sequences

(

1

,

h1

,

. . . ,

hs−1

)

withhi

≥

hi+1,forevery0

<

i

<

s

−

1,areincludedinthosedetectedby

Lemma 5.1.Forexample,wehave:

G1 =G2= {0}, G3=G2∪ (G1+1)= {0,1}, G4 =G3∪ (G2+1)∪ (G1+3)= {0,1,3},

G5 =G4∪ (G3+1)∪ (G2+3)∪ (G1+6)= {0,1,2,3,6},

G6 =G5∪ (G4+1)∪ (G3+3)∪ (G2+6)∪ (G1+10)= {0,1,2,3,4,6,10},

G7 ⊃G6∪ (G5+1)∪ (G4+3)∪ (G3+6)∪ (G2+10)∪ (G1+15)= {0,1,2,3,4,6,7,10,15}. Note that forthe multiplicity d

=

7, we losethe arithmetic genus g

=

5 whichcorresponds tothe finiteO-sequence

(

1

,

2

,

3

,

1

)

.

Now,weexploit

Lemma 5.1

obtaininglargesetsofaCMgenera.Tothisaim,wedefinean increas-ingsequence

{

md

}

d≥1 bythefollowingprocedure:

if d

=

1 then m1

:=

0; else M

:=

md−1; for k

=

2

,

. . . ,

d

−

1 do if



k₂



−

1

≤

M then M

=

max

{

M

,

md−k

+



k 2



}

; end if end for md

:=

M; end if

Example 5.3. Inthefollowing table,we listthe valuesofthesequence

{

md

}

d≥1 andcomparethem withthevaluesofgd2+2,for1

≤

d

≤

45:

d 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 md 0 0 1 1 3 4 4 7 11 13 18 19 19 25 32 gd2+2 1 1 3 3 6 6 10 10 15 15 21 21 28 28 36 d 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 md 40 43 52 62 73 85 89 102 116 118 133 149 166 184 203 gd2+2 _{36 45 45 55 55 66 66 78 78 91 91 105 105 120 120} d 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 md 208 228 229 229 250 272 295 319 344 370 376 403 431 460 490 gd2+2 136 136 153 153 171 171 190 190 210 210 231 231 253 253 271

Theorem 5.4 (Continuity).Foralld

≥

1,everyintegerin

{

0

,

. . . ,

md

}

isthearithmeticgenusofanaCMcurve ofdegreed,i.e.

{

0

,

. . . ,

md

}

⊆

Gd,andmd

≥

g

d

2+2_,foreveryd

≥

_18.

Proof. The firststatementholdsby

Lemma 5.1

andbythedefinitionofmd.Forthesecondaffirmation,

note that itisenough toconsider odddegreesd.For18

≤

d

≤

36,see thetablesof

Example 5.3

.If d

≥

37, let s

:= 

d₂



+

2. By construction andby induction, we knowthat md

≥

md−1

≥

g

d−1 2 +2

=



s−2

2



2 5: for s

=

2

,

. . . ,

d

−

3 do 6: g

:=

min

(

undecided

)

; 7: while g

≤

upperBound

(

Rs d

)

do 8: if g

<

lowerBound

₍

_Rs d

)

then 9: remove

₍

g

_,

undecided

)

; 10: gaps

=

gaps

∪ {

g

}

; 11: else 12: searching

:=

genusSearch

₍

g

_,

T

s d

)

; 13: if searching

= ∅

then 14: remove

₍

_g

_,

_undecided

₎

_; 15: genera

=

genera

∪ {

g

}

; 16: end if 17: end if 18: g

=

next

₍

_g

_,

_undecided

₎

_; 19: end while 20: end for 21: return genera; 22: end procedure md

≥

max



md−1

,

md−(s−2)

+



_s−2 2



.

Beingd odd, we haved

− (

s

−

2

)

=

d

− 

d₂



= 

d₂



−

1

=

s

−

3

≥

18. Thus, by inductionwe obtain md

≥



_s−3 2 +1 2



+



s−2 2



,becausemd−(s−2)

=

m_d 2−1

=

ms−3

≥

g s−3 2 +2_. Notethat



s−32 +1 2



+



s−2 2



≥



s−1 2



if



s−32 +1 2



≥

s

−

2,thatistrueforeverys

≥

10.

2

Theorem 5.4 givesa lower bound for the value assumed by md, forevery d

≥

18. Anyway, we

canobtainmoreinformationby afull applicationof

Lemma 5.1

which,together withthe algorithm genusSearch (see Algorithm 1), provides an algorithmto compute all the arithmetic generaof the aCMcurvesofdegreed,avoidingtoconstructallthefiniteO-sequences.Thestrategyconsistsofthe followingsteps:

Step 1 by Lemma 5.1,we determine recursively theset ofintegers



Gd

⊂

Rd that are certainly aCM

genera.Let



G1

= {

0

}

,wehave



Gd

=



i



Gi

+



d−i 2



;

Step 2 byresultsinSection4wedeterminealltheintegersofRd thatarecertainlygaps;

Step 3 usingalgorithm genusSearch (Algorithm 1)weinvestigatetheremainingintegers.

6. An application: Castelnuovo–Mumford regularity of curves with Cohen–Macaulay postulation

In this section, we show how the search algorithm of aCM genera (Algorithm 1) allows us to detect theminimal Castelnuovo–Mumford regularitymaCM_d,g of a curve withCohen–Macaulay postu-lation,given its degree d and genus g. Moreover, by Example 6.3 we give a negative answer to a question posedin CioffiandDi Gennaro (2011,Remark2.5).Acomplete solutiontotheproblemof

Table 1

Inthistable,wereportsomenumericalinformation abouttheintegersinGd uptodegree250.Thefirstcolumncontains

thenumberandthepercentageofvaluesinRd whichareaCMgenerabyanapplicationofLemma 5.1(withoutcomputing

the O-sequences);inthe secondcolumn,the numberandthe percentageofgapsdeterminedapplyingProposition 4.3and

Proposition 4.9;inthethirdcolumn,thenumberandthepercentageofvaluesofRdforwhichwehavetousetheprocedure

genusSearchtodecidewhethertheyareaCMgenera;inthelastcolumn,thecardinalityofGdanditspercentagewithrespect

to|Rd|.

d Certain genera Certain gaps Undecided values |Gd|

25 176(63.77%) 88(31.88%) 13(4.71%) 187(67.75%) 50 835(71.00%) 289(24.57%) 53(4.51%) 870(73.98%) 75 2033(75.27%) 558(20.66%) 111(4.11%) 2099(77.71%) 100 3798(78.29%) 879(18.12%) 175(3.61%) 3894(80.27%) 125 6129(80.37%) 1244(16.31%) 254(3.33%) 6261(82.10%) 150 9040(81.99%) 1653(14.99%) 334(3.02%) 9207(83.50%) 175 12 528(83.24%) 2094(13.91%) 430(2.86%) 12 734(84.61%) 200 16 610(84.31%) 2574(13.07%) 518(2.63%) 16 854(85.55%) 225 21 276(85.19%) 3084(12.35%) 617(2.47%) 21 560(86.32%) 250 26 530(85.92%) 3623(11.73%) 724(2.34%) 26 856(86.98%) Table 2

Inthistable,wereporttheresultsofatestofAlgorithm 2uptodegree250.Thefirstthreecolumnscontaintheelapsedtime (inmilliseconds)forStep1,Step2andStep3ofAlgorithm 2.Inthefourthcolumn,thereisthetotaltimefortheexecution (Step1+Step2+Step3).ThelastcolumncontainsthetimerequiredfordeterminingthesetGdbyperformingacomplete

visitofthetree Td (even ford=75,weobtainanOutOfMemory Error).ThealgorithmsareimplementedintheJava languageandhavebeenrunonaMacBookProwithanIntelCore2Duo2.4GHzprocessor.

d Step 1 Step 2 Step 3 Algorithm 2 VisitTd

25 37.336 ms 0.164 ms 38.594 ms 76.094 ms 210.769 ms 50 82.774 ms 0.208 ms 212.868 ms 295.850 ms 15 155.87 ms 75 21.734 ms 0.155 ms 458.117 ms 480.006 ms O.O.M. 100 47.529 ms 0.103 ms 1390.027 ms 1437.659 ms O.O.M. 125 104.683 ms 0.279 ms 4684.598 ms 4789.56 ms O.O.M. 150 207.936 ms 0.183 ms 12 610.461 ms 12 818.58 ms O.O.M. 175 546.818 ms 0.227 ms 37 518.036 ms 38 065.081 ms O.O.M. 200 665.378 ms 0.364 ms 73 552.564 ms 74 218.306 ms O.O.M. 225 922.599 ms 0.36 ms 169 042.878 ms 169 965.837 ms O.O.M. 250 1395.378 ms 0.179 ms 359 836.564 ms 361 232.121 ms O.O.M.

detecting theminimalCastelnuovo–MumfordregularityofaschemewithagivenHilbertpolynomial isdescribedin

Cioffi

etal. (inpress).

Denoting by

ρ

theregularity ofaHilbertfunction,i.e. theminimaldegreefromwhichtheHilbert functionandtheHilbertpolynomialcoincide,wecanstatethefollowing:

Proposition 6.1. maCM_d,g

=

min



ρ







ρ

is the regularity of an aCM postulation with Hilbert polynomial dt

+

1

−

g



+

2

Proof. Let f beanaCMpostulationwithHilbertpolynomialdt

+

1

−

g andregularity

ρ

.Then,the minimal possibleCastelnuovo–Mumfordregularityofacurve withHilbertfunction f is

ρ

+

2.Asa matteroffact,by

Cioffi

andDiGennaro (2011,Proposition2.4)thisregularityisstrictlygreaterthan

ρ

+

1 andifthecurveisaCM,itisexactly

ρ

+

2.

2

By

Proposition 6.1

,thevalueofm_daCM_,g isdetermined byapplying

Algorithm 1

inordertofindan O-sequencehofmultiplicityd andg

(

h

)

=

g withtheshortestpossiblelength.Noticethatifthelength ofhiss,thentheregularityof



2hiss

−

2.Thus,wecanrewritethestatementin

Proposition 6.1

as

Hence, theminimal Castelnuovo–MumfordregularitymaCM

d,g is8. Applying theresultsof Cioffietal. (in press) (see http://www.paololella.it/HSC/Minimal_Hilbert_Functions_and_CM_regularity.html), we noticethattheminimalCastelnuovo–MumfordregularityofanyprojectiveschemewithHilbert poly-nomial p

(

t

)

=

15t

−

31 is 7.

Moregenerally,inthecaseofan aCMfunction f withregularity

ρ

andHilbertpolynomialwith odd degree,wehavethattheminimalpossibleCastelnuovo–Mumfordregularityofascheme X with HX

=

f isstrictlygreaterthan

ρ

+

1 (see

Cioffi

andDiGennaro,2011,Proposition2.4).Ifthedegree

oftheHilbertpolynomialiseven,ananalogousresultdoesnothold,asthefollowingexampleshows.

Example 6.3. Thefollowingstrongly-stableideal

I

= (

x2₆

,

x5x6

,

x25

,

x4x5

,

x3x5

,

x2x5

,

x1x5

,

x42x6

,

x3x4x6

,

x2x4x6

,

x1x4x6

,

x23x6

,

x2x3x6

,

x1x3x6

,

x23x6

,

x1x22x6

,

x21x2x6

,

x44

,

x3x34

,

x2x34

,

x41x6

,

x33x24

,

x43x4

,

x53

)

⊂

K

[

x0

, . . . ,

x6

],

where x0

<

x1

<

· · · <

x6, defines a non-aCM surface X

⊂ P

6 with the aCM postulation HX

=

(

1

,

7

,

21

,

44

,

. . . ,

6t2

₋

_10t

₊

₂₁

_,

_{. . .)}

_ofregularity

_ρ

₌

_{4 andtheCastelnuovo–MumfordregularityofX} is5

=

ρ

+

1.

Acknowledgement

TheauthorswouldliketothankMargheritaRoggeroforusefuldiscussionsaboutapreviousversion ofthispaper.

References

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