CRITERIA FOR σ-AMPLENESS 1. Introduction In the past ten years a study of

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AMERICANMATHEMATICALSOCIETY
Volume13,Number3,Pages517{532
S0894-0347(00)00334-9
ArticleelectronicallypublishedonMarch29,2000
CRITERIA FOR (cid:27)-AMPLENESS
DENNIS S. KEELER
1. Introduction
Inthepasttenyearsastudyof\noncommutativeprojectivegeometry"has(cid:13)our-
ished. By using and generalizing techniques of commutative projective geometry,
onecanstudycertainnoncommutativeringsandobtainresultsforwhichnopurely
algebraic proof is known.
The most basic building block of the theory is the twisted homogeneous coordi-
nate ring. Let X be a projective scheme overan algebraicallyclosed (cid:12)eld k with (cid:27)
aschemeautomorphism,andletLbe aninvertiblesheafonX. In[ATV]atwisted
versionofthehomLogeneouscoordinateringB =B(X;(cid:27);L)ofX wasinventedwith
the grading B = B for
m
B =H0(X;L(cid:10)L(cid:27)(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)L(cid:27)m(cid:0)1)
m
where L(cid:27) = (cid:27)(cid:3)L is the pullback of L. Multiplication on sections is de(cid:12)ned by
a(cid:1)b=a(cid:10)b(cid:27)m when a2B and b2B .
m n
Soonaftertheirseminalpaper,ArtinandVandenBerghformalizedmuchofthe
theoryof these twistedhomogeneouscoordinaterings in [AV]. Inthe commutative
case, the most useful homogeneous coordinate rings are associated with an ample
invertiblesheaf. Ageneralizationofamplenesswasthereforeneededandde(cid:12)nedas
follows.
An invertible sheaf L is called right (cid:27)-ample if for any coherent sheaf F,
Hq(X;F (cid:10)L(cid:10)L(cid:27)(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)L(cid:27)m(cid:0)1)=0
for q > 0 and m (cid:29) 0. Similarly, L is called left (cid:27)-ample if for any coherent sheaf
F,
Hq(X;L(cid:10)L(cid:27)(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)L(cid:27)m(cid:0)1 (cid:10)F(cid:27)m)=0
for q > 0 and m (cid:29) 0. A divisor D is called right (resp. left) (cid:27)-ample if O (D) is
X
right(resp. left) (cid:27)-ample. If(cid:27) is the identity automorphism,then these conditions
are the same as saying L is ample. Artin and Van den Bergh proved that if L is
right(resp. left)(cid:27)-ample,thenBisa(cid:12)nitelygeneratedright(resp. left)noetherian
k-algebra [AV].
Twisted homogeneous coordinate rings have been instrumental in the classi(cid:12)ca-
tion of rings, such as the 3-dimensional Artin-Schelter regular algebras [ATV, St1,
ReceivedbytheeditorsDecember13,1999.
2000 Mathematics Subject Classi(cid:12)cation. Primary 14A22, 14F17, 14J50, 16P90, 16S38,
16W50.
Keywords andphrases. Noetheriangradedrings,noncommutativeprojectivegeometry,auto-
morphisms,vanishingtheorems.
Theauthor waspartiallysupportedbyNSFgrantDMS-9801148.
(cid:13)c2000 American Mathematical Society
517
518 DENNIS S. KEELER
St2] and the 4-dimensionalSklyaninalgebras [SS]. Artin and Sta(cid:11)ord showedthat
any connected (i.e. B =k) graded domain of GK-dimension 2 generatedby B is
0 1
the twisted homogeneous coordinate ring (up to a (cid:12)nite dimensional vector space)
of some projective curve X, with automorphism (cid:27) and (left and right) (cid:27)-ample L
[AS]. Therefore any such ring is automatically noetherian!
While the concept of noncommutative schemes has grown to encompass more
than just twisted homogeneous coordinaterings (cf. [AZ]), they remain a guide for
how such a scheme ought to behave. However, fundamental open questions about
these coordinate rings and (cid:27)-ample divisors have persisted for the past decade. In
[AV], the authors derived a simple criterion for a divisor to be (cid:27)-ample in the case
X is a curve, a smooth surface, or certain other special cases. With this criterion,
they showed that B must have (cid:12)nite GK-dimension. In other words, they showed
that B has polynomial growth. They ask
Questions 1.1 ([AV, Question 5.19]). (1) What is the extension of our simple
criterion to higher dimensions?
(2) Doestheexistenceofa(cid:27)-ampledivisorimplythatB haspolynomialgrowth?
The second question was asked again after [AS, Theorem 4.1].
One would also like to know if the conditions of right and left (cid:27)-ampleness are
related and if B could be right noetherian, but not left noetherian. One might ask
forwhich(commutative)schemesandautomorphismsa(cid:27)-ampledivisorevenexists
and if one can be easily found.
In this paper, all these questions will be settled very satisfactorily. We obtain
Theorem 1.2. The following are true for any projective scheme X over an alge-
braically closed (cid:12)eld.
(1) Right and left (cid:27)-ampleness are equivalent. Thus every associated B is (right
and left) noetherian.
(2) A projective scheme X has a (cid:27)-ample divisor if and only if the action of
(cid:27) on numerical equivalence classes of divisors is quasi-unipotent (cf. x3 for
de(cid:12)nitions). In this case, every ample divisor is (cid:27)-ample.
(3) GKdimB is an integer if B =B(X;(cid:27);L) and L is (cid:27)-ample. Here GKdimB
is the Gel’fand-Kirillov dimension of B in the sense of [KL].
The (cid:12)rst two results are handled in x5, while the third is covered in x6.
These facts are all consequences of
Theorem 1.3 (see Remark 5.2). Let X beaprojective schemewith automorphism
(cid:27). Let D be a Cartier divisor. D is (right) (cid:27)-ample if and only if (cid:27) is quasi-
unipotent and
D+(cid:27)D+(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1D
is ample for some m>0.
This is the \simple criterion"which wasalreadyknownif X is a smoothsurface
[AV, Theorem 1.7]. We obtain the result mainly by use of Kleiman’s numerical
theory of ampleness [K].
Besidestheresultsabove,wederiveothercorollariesinx5and(cid:12)ndboundsforthe
GK-dimension in x6 via Riemann-Roch theorems. We also examine what happens
in the non-quasi-unipotent case and obtain the following theorem.
CRITERIA FOR (cid:27)-AMPLENESS 519
Theorem 1.4 (see Remark 6.2). Let X beaprojective schemewith automorphism
(cid:27). Then the following are equivalent:
(1) The automorphism (cid:27) is quasi-unipotent.
(2) For all ample divisors D, B(X;(cid:27);O (D)) has (cid:12)nite GK-dimension.
X
(3) For all ample divisors D, B(X;(cid:27);O (D)) is noetherian.
X
This paper is part of the author’s Ph.D. thesis at the University of Michigan,
under the direction of J.T. Sta(cid:11)ord.
2. Reductions
Throughout this paper, we will work in the case of a projective scheme X over
an algebraically closed base (cid:12)eld of arbitrary characteristic. A variety will mean
a reduced, irreducible scheme. All divisors will be Cartier divisors unless other-
wise stated. For a projective scheme, the group of Cartier divisors, modulo linear
equivalence,isnaturallyisomorphictothePicardgroupofinvertiblesheaves. Since
muchofourworkwillentailintersectiontheory,wewillworkfromthedivisorpoint
of view. Several times we use the facts that the ample divisors form a cone, that
ampleness depends only on the numerical equivalence class of a divisor, and that
ampleness is preserved under an automorphism. Hence the cone of ample divisors
and its closure, the cone of numerically e(cid:11)ective divisors, are invariant under an
automorphism. As a reference for these and related facts we suggest [K].
Remark 2.1. The main results of this paper will be proved in terms of divisors
rather than line bundles. However, the reader should note that, unravelling the
de(cid:12)nitions,onehasO ((cid:27)D)(cid:24)=O (D)(cid:27)(cid:0)1. Itisthereforenotationallymoreconve-
X X
nienttoworkwitharight(cid:27)(cid:0)1-amplelinebundleL=O (D),sincethenD isright
X
(cid:27)(cid:0)1-ample if andonly if Hq(X;F(cid:10)O (D+(cid:27)D+(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1D))=0 for allq >0
X
and m(cid:29)0. Obviously, this will have no e(cid:11)ect on the (cid:12)nal theorems. Throughout
this paper, we will use the notation (cid:1) =D+(cid:27)D+(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1D.
m
Before deriving our main criterion for (cid:27)-ampleness, we must (cid:12)rst prove other
equivalent criteria. We will need
Lemma 2.2 ([Fj, p. 520, Theorem 1]). Let F be a coherent sheaf on a projective
scheme X and let H be an ample divisor on X. Then there exists an integer c
0
such that for all c(cid:21)c ,
0
Hq(X;F (cid:10)O (cH +N))=0
X
for q >0 and any ample divisor N.
Proposition 2.3. Let X be a projective scheme with (cid:27) an automorphism. Let D
be a divisor on X and (cid:1) = D + (cid:27)D +(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1D. Then the following are
m
equivalent:
(1) For any coherent sheaf F, there exists an m such that Hq(X;F(cid:10)O ((cid:1) ))
0 X m
=0 for q >0 and m(cid:21)m .
0
(2) For any coherent sheaf F, there exists an m such that F (cid:10) O ((cid:1) ) is
0 X m
generated by global sections for m(cid:21)m .
0
(3) For any divisor H, there exists an m such that (cid:1) (cid:0)H is very ample for
0 m
m(cid:21)m .
0
(4) ForanydivisorH,thereexistsanm suchthat(cid:1) (cid:0)H is ampleform(cid:21)m .
0 m 0
The (cid:12)rst condition is the original de(cid:12)nition of right (cid:27)(cid:0)1-ample.
520 DENNIS S. KEELER
Proof. (1) ) (2) is [AV, Proposition3.2].
(2))(3)followsfromthefactthataveryampledivisorplusadivisorgenerated
byitssectionsisaveryampledivisor. GivenanydivisorH andaveryampledivisor
H0, choosem suchthat(cid:1) (cid:0)H(cid:0)H0 is generatedby globalsections for m(cid:21)m .
0 m 0
Then (cid:1) (cid:0)H (cid:0)H0+H0 =(cid:1) (cid:0)H is very ample for m(cid:21)m .
m m 0
(3) ) (4) is trivial.
(4))(1). ForanyampledivisorH andanyc(cid:21)0,onecanchoosem sothatfor
0
m(cid:21)m , wehaveN =(cid:1) (cid:0)cH is anample divisor. Then(1)followsimmediately
0 m
from Lemma 2.2.
A similar proposition holds for left (cid:27)(cid:0)1-ample divisors, with F and H replaced
by F(cid:27)(cid:0)m and (cid:27)mH. One deduces this easily from
Lemma 2.4 ([St3, p. 31]). A divisor D is right (cid:27)(cid:0)1-ample if and only if D is left
(cid:27)-ample.
Proof. Let D be right (cid:27)(cid:0)1-ample. Then for any coherent sheaf F, there exists an
m such that
0
Hq(X;O (D+(cid:27)D+(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1D)(cid:10)F(cid:27))=0
X
for q > 0 and m (cid:21) m . Since cohomology is preserved under automorphisms,
0
pulling back by (cid:27)m(cid:0)1, we have
Hq(X;O (D+(cid:27)(cid:0)1D+(cid:1)(cid:1)(cid:1)+(cid:27)(cid:0)(m(cid:0)1)D)(cid:10)F(cid:27)m)=0
X
for q >0 and m(cid:21)m . So D is left (cid:27)-ample.
0
It is often useful to replaceD with (cid:1) and (cid:27) with (cid:27)m to assume D and (cid:27) have
m
a desired property. Using standard techniques, one can also show
Lemma 2.5 ([AV, Lemma 4.1]). LetD beadivisor onX. Givenapositiveinteger
m, D is right (cid:27)(cid:0)1-ample if and only if (cid:1) is right (cid:27)(cid:0)m-ample.
m
3. The non-quasi-unipotent case
LetA1 (X)bethesetofdivisorsofX modulonumericalequivalence. Thatis,
Num
for divisors D and D0, one has D (cid:17)D0 2A1 (X) if and only if (D:C) =(D0:C)
Num
for all integral curves C (cid:26) X. We will use this de(cid:12)nition implicitly several times,
especiallythefactthatanonzeroelementofA1 (X)hasnonzerointersectionwith
Num
some curve. Further, one has that A1 (X) is a (cid:12)nitely generated free abelian
Num
group [K, p. 305, Remark 3]. Let P be the action of (cid:27) on A1 (X); hence
Num
P 2GL(Z‘) for some ‘.
A matrix is called quasi-unipotent if all of its eigenvalues are roots of unity. We
callanautomorphism(cid:27) quasi-unipotentif P is. The main goalof this sectionis to
show that a non-quasi-unipotent (cid:27) cannot give a (cid:27)-ample divisor.
First, we must review a useful fact about integer matrices.
Lemma 3.1. Let P 2GL(Z‘). Then P is quasi-unipotent if and only if all eigen-
values of P have absolute value 1. Thus if P is not quasi-unipotent, then P has an
eigenvalue of absolute value greater than 1.
Proof. The (cid:12)rst claim is [AV, Lemma 5.3]. For the second claim, the property of
P not being quasi-unipotent is reduced to saying P has an eigenvalue of absolute
value not 1. Since P has determinant (cid:6)1, P has an eigenvalue of absolute value
greater than 1.
CRITERIA FOR (cid:27)-AMPLENESS 521
The following lemma shows a relationshipbetween the spectral radius r=(cid:26)(P)
and the intersection numbers ((cid:27)mD:C), where D is an ample divisor.
Lemma 3.2. Let P be as described above with spectral radius r = (cid:26)(P). There
exists an integral curve C with the following property: If D is an ample divisor,
then there exists c>0 such that
((cid:27)mD:C)(cid:21)crm for all m(cid:21)0:
Proof. Let(cid:20)betheconegeneratedbynumericallye(cid:11)ectivedivisorsinA1 (X)(cid:10)R.
Num
Intheterminologyof[V],(cid:20)isasolidconesinceithasanonemptyinterior[K,p.325,
Theorem 1]. Since P maps (cid:20) to (cid:20), the spectral radius r is an eigenvalue of P and
r has an eigenvector v 2(cid:20) [V, Theorem 3.1].
Since v 2(cid:20)nf0g,there existsa curveC with (v:C)>0. Givenanample divisor
D, there is a positive ‘ so that ‘D(cid:0)v is in the ample cone [V, p. 1209]. Thus
‘((cid:27)mD:C)=‘(PmD:C)>(Pmv:C)=rm(v:C):
Taking c=(v:C)=‘, we have the lemma.
L
Now a graded ring B = B is (cid:12)nitely gradedif dimB <1 for all i. Such
i(cid:21)0 i i
a ring B has exponential growth (see [SZ]) if
(cid:18) (cid:19)
X 1
n
(3.3) limsup dimB >1:
i
n!1 i(cid:20)n
If B has exponential growth, it is neither right nor left noetherian [SZ, Theorem
0.1]. This fact combined with the intersection numbers above allow us to prove
Theorem 3.4. Let X be a projective scheme with automorphism (cid:27). If X has a
right (cid:27)(cid:0)1-ample divisor, then (cid:27) is quasi-unipotent.
Proof. Suppose that D is a right (cid:27)(cid:0)1-ample divisor. Let (cid:1) = D+(cid:27)D +(cid:1)(cid:1)(cid:1)+
m
(cid:27)m(cid:0)1D. By(2.3)and(2.5),wemayreplaceD with(cid:1) and(cid:27) with(cid:27)m andassume
m
that D is ample.
Let P be the action of (cid:27) on A1 (X). Suppose P is non-quasi-unipotent with
Num
spectralradiusr>1andchooseanintegralcurveC asinLemma3.2. LetI bethe
ideal sheaf de(cid:12)ning C in X. Since D is right (cid:27)(cid:0)1-ample, the higher cohomologies
of I((cid:1) ) = I (cid:10)O ((cid:1) ) and O ((cid:1) ) vanish for m (cid:29) 0. So one has an exact
m X m C m
sequence
0!H0(X;I((cid:1) ))!H0(X;O ((cid:1) ))!H0(C;O ((cid:1) ))!0:
m X m C m
For m (cid:29) 0, the Riemann-Roch formula for curves [Fl, p. 360, Example 18.3.4]
gives
dimH0(C;O ((cid:1) ))=((cid:1) :C)+a constant term:
C m m
Thus using the exact sequence and the previous lemma, there exists c>0 so that
dimH0(X;O ((cid:1) ))>crm
X m
for m (cid:29) 0. Thus the associated twisted homogeneous coordinate ring has expo-
nential growth and hence is not (right or left) noetherian [SZ, Theorem 0.1]. So D
cannot be right (cid:27)(cid:0)1-ample, by [AV, Theorem 1.4].
522 DENNIS S. KEELER
Remark 3.5. Onecangiveamoreelementary,computationalproofofTheorem3.4.
Indeed, examining the Jordan form of P gives an upper bound on ((cid:27)mD:C). Fur-
ther,usingthe full strengthof[V, Theorem3.1]andasymptoticestimates, onecan
improve the lower bound of Lemma 3.2. We then have
(3.6) c mkrm >((cid:27)mD:C)>cmkrm
1
form>0,where k+1 is the size ofthe largestJordanblockassociatedto r. Then
using estimates similar to those in the proof of [AV, Lemma 5.10], one can (cid:12)nd an
ample divisor H such that
((cid:1) (cid:0)H:(cid:27)mC)<0
m
for all m (cid:29) 0. This contradicts the fourth equivalent condition for right (cid:27)(cid:0)1-
ampleness in Proposition 2.3.
Even when an automorphism (cid:27) is not quasi-unipotent, one can form associated
twisted homogeneous coordinate rings. As might be expected, some of these rings
have exponential growth.
Proposition 3.7. Let X be a projective scheme with non-quasi-unipotent auto-
morphism (cid:27). Let D be an ample divisor. Then there exists an integer n >0 such
0
that for all n (cid:21) n , the ring B = B(X;(cid:27);O (nD)) has exponential growth and is
0 X
neither right nor left noetherian.
Proof. AgainchooseacurveC asinLemma3.2withidealsheafI. ByLemma2.2,
there exists n such that for all n(cid:21)n and q >0,
0 0
Hq(X;I(nD+N))=Hq(C;O (nD+N))=0
C
foranyampledivisorN. Inparticular,theabovecohomologiesvanishfornD+N =
nD+(cid:27)(nD)+(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1(nD) where m > 1. Then repeating the last paragraph
of the proof of Theorem 3.4 shows that B has exponential growth.
When X is a nonsingular surface, [AV, Corollary 5.17] shows that the above
proposition is true for n =1. Their proof makes use of the relatively simple form
0
of the Riemann-Roch formula and the vanishing of H2(X;O ((cid:1) )) when (cid:1) is
X m m
the sum of su(cid:14)ciently many ample divisors. The proof easily generalizes to the
singular surface case, but not to higher dimensions.
Question 3.8. Givenanon-quasi-unipotentautomorphism(cid:27) andampledivisorD
on a scheme X, must B(X;(cid:27);O (D)) have exponential growth?
X
There do exist varieties with non-quasi-unipotent automorphisms. If the canon-
ical divisor K is ample or minus ample, then any automorphism (cid:27) must be quasi-
unipotent (cf. (5.6)). So intuitively, one expects to (cid:12)nd non-quasi-unipotent au-
tomorphisms far away from this case, i.e., when K = 0. Further, there are strong
existence theorems for automorphisms of K3 surfaces (which do have K = 0).
Indeed, a K3 surface with non-quasi-unipotent automorphism is studied in [W].
Example 3.9. There exists a K3 surface with automorphism (cid:27) such that X has
no (cid:27)-ample divisors.
Proof. Wehler[W,Proposition2.6,Theorem2.9]constructsafamilyofK3surfaces
whose general member X has
Pic(X)(cid:24)=A1 (X)(cid:24)=Z2; Aut(X)(cid:24)=Z=2Z(cid:3)Z=2Z:
Num
CRITERIA FOR (cid:27)-AMPLENESS 523
(That is, Aut(X) is the free product of two cyclic groups of order 2.) The ample
generators H and H of A1 (X) have intersection numbers
1 2 Num
(H2)=(H2)=2; (H :H )=4:
1 2 1 2
Aut(X)hastwogenerators(cid:27) ;(cid:27) whoseactionsonA1 (X)canberepresented
1 2 Num
as two quasi-unipotent matrices
(cid:18) (cid:19) (cid:18) (cid:19)
1 4 (cid:0)1 0
(cid:27) = ; (cid:27) = :
1 0 (cid:0)1 2 4 1
p
However, the action of (cid:27) (cid:27) has eigenvalues 7(cid:6)4 3. So X has no (cid:27) (cid:27) -ample
1 2 1 2
divisor. Note that by Corollary 5.4 below, any ample divisor is (cid:27) -ample and (cid:27) -
1 2
ample.
4. The quasi-unipotent case
Now let (cid:27) be a quasi-unipotent automorphism with P its action on A1 (X).
Num
We will have several uses for a particular invariant of (cid:27).
De(cid:12)nition 4.1. Let k+1 be the rank ofthe largestJordanblock of P. We de(cid:12)ne
J((cid:27))=k.
Note that J((cid:27)) = J((cid:27)m) for all m 2 Znf0g. It may be that k is greater than
0, as seen in [AV, Example 5.18]. We will see in the next section that k must be
even, but this is not used here.
To prove Theorem 1.3, it remains to show that (for (cid:27) quasi-unipotent) if D is a
divisor such that (cid:1) is ample for some m, then D is right (cid:27)(cid:0)1-ample. So (cid:12)x such
m
aD. WemayagainreplaceD with(cid:1) and(cid:27) with(cid:27)n via(2.5),sothatD is ample
n
andP is unipotent, that is,P =I+N,where N is the nilpotent partofP. Inthis
case, k =J((cid:27)) is the smallest natural number such that Nk+1 =0.
We let(cid:17) denote numericalequivalenceandreserve=for linearequivalence. We
then have, for all m(cid:21)0,
(cid:18) (cid:19)
Xk
m
(4.2) (cid:27)mD (cid:17)PmD = NiD;
i
i=0
(cid:18) (cid:19)
Xk
m
(4.3) (cid:1) (cid:17) NiD:
m
i+1
i=0
Once a basis for A1 (X) is chosen, one can treatNiD as a divisor. Of course,
Num
thisrepresentationofNiDisnotcanonical. However,sinceamplenessandintersec-
tion numbers only depend on numerical equivalence classes, this is not a problem.
Lemma 4.4. Let (cid:27) be a unipotent automorphism with P = I +N and k = J((cid:27)).
If D is an ample divisor, then NkD 6(cid:17)0 in A1 (X).
Num
Proof. SinceNk 6=0,there existsadivisorE andcurveC suchthat(NkE:C)>0.
Choose ‘ so that ‘D (cid:0) E is ample. By equation (4.2) above, the intersection
numbers ((cid:27)m(‘D(cid:0)E):C) are given by a polynomial in m with leading coe(cid:14)cient
(‘NkD(cid:0)NkE:C)=k!. Since this polynomial must have positive values for all m,
we must have NkD 6(cid:17)0.
524 DENNIS S. KEELER
We now turn towards proving that for any divisor H, there exists m such that
0
(cid:1) (cid:0)H is ample, when (cid:27) is unipotent and D is ample. Then since D is ample,
m0
(cid:1) (cid:0)H is ample for m (cid:21) m . For certain H, this is true even if (cid:27) is not quasi-
m 0
unipotent.
Lemma 4.5. Let X be a projective scheme with automorphism (cid:27) (not necessarily
quasi-unipotent). Let D be an ample divisor and H a divisor whose numerical
equivalence class is (cid:12)xed by (cid:27). Then there exists an m such that (cid:1) (cid:0)H is ample.
m
Proof. Choose m such that D0 =mD(cid:0)H is ample. Let
(cid:1)0 =D0+(cid:27)D0+(cid:1)(cid:1)(cid:1)+(cid:27)j(cid:0)1D0:
j
Then (cid:1)0 (cid:17)m(cid:1) (cid:0)mH is ample and thus (cid:1) (cid:0)H is ample.
m m m
Proposition 4.6. Let X be a projective scheme with unipotent automorphism (cid:27).
Let D be an ample divisor and H any divisor. Then there exists an m such that
0
(cid:1) (cid:0)H is ample. Hence (cid:1) (cid:0)H is ample for m(cid:21)m .
m0 m 0
Proof. Let W (cid:26) A1 (X)(cid:10)R be the span of D;ND;:::;NkD. W is a k +1-
Num
dimensional vector space by Lemma 4.4. By equation (4.2), it contains the real
cone (cid:20) generated by S = f(cid:27)iDji 2 Ng. Using a lemma of Caratheodory [H, p. 45,
Lemma1],anyelementof(cid:20)canbewrittenasalinearcombinationofk+1elements
of S with nonnegative real coe(cid:14)cients. Thus for all m2N,
Xk
(cid:1)m (cid:17) fi(m)(cid:27)gi(m)D
i=0
where fi: N!R(cid:21)0 and gi: N!N. Expanding the (cid:27)gi(m)D above and comparing
the coe(cid:14)cient of D with equation (4.3), one (cid:12)nds that
Xk
f (m)=m:
i
i=0
Since f (m)(cid:21)0, for each m, there must be some j such that f (m)(cid:21)m=(k+1).
i j
NowchooselsuchthatlD(cid:0)H isampleandchoosem suchthatm =(k+1)(cid:21)l.
0 0
Then
fj(m0)(cid:27)gj(m0)D(cid:0)(cid:27)gj(m0)H
is in the ample cone for the given j. Set g = g (m ). The other f (m ) are
j 0 i 0
nonnegative. Then (cid:1) (cid:0)(cid:27)gH is in the ample cone as it is a sum of elements in
m0
the ample cone. Since it is a divisor, it is ample [K, p. 324, Remark 3].
We now prove the lemma by induction on q, the smallest positive integer such
that NqH (cid:17) 0. Since N is nilpotent, there is such a q for any H. The case q = 1
is handled by the previous lemma.
Now as (cid:27) (cid:17)I +N, we know (cid:27)(cid:0)m0((cid:27)gH (cid:0)H) is killed by Nq(cid:0)1. So there is an
m so that
1
Y =(cid:1) +(cid:27)(cid:0)m0((cid:27)gH (cid:0)H)
m1
is ample. Then as (cid:27) (cid:12)xes the ample cone,
(cid:1) (cid:0)(cid:27)gH +(cid:27)m0Y =(cid:1) (cid:0)H
m0 m0+m1
is ample.
CRITERIA FOR (cid:27)-AMPLENESS 525
We now immediately have by Propositions 2.3 and 4.6 and Theorem 3.4:
Theorem 4.7. Let X be a projective scheme with an automorphism (cid:27). A divisor
D is right (cid:27)(cid:0)1-ample if and only if (cid:27) is quasi-unipotent and D+(cid:27)D+(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1D
is ample for some m.
5. Corollaries
Thecharacterizationof(right)(cid:27)(cid:0)1-amplenesshasmanystrongcorollarieswhich
are now easy to prove, but were only conjectured before.
Corollary 5.1. Right(cid:27)-ampleandleft(cid:27)-ampleareequivalentconditions. Further,
(cid:27)-ampleness and (cid:27)(cid:0)1-ampleness are equivalent.
Proof. Let D be right (cid:27)(cid:0)1-ample. By Theorem 4.7, (cid:27) is quasi-unipotent and (cid:1)
m
is ample for some m. Then (cid:27)(cid:0)1 is quasi-unipotent and
(cid:27)(cid:0)(m(cid:0)1)(cid:1) =D+(cid:27)(cid:0)1D+(cid:1)(cid:1)(cid:1)+(cid:27)(cid:0)(m(cid:0)1)D
m
is ample. Applying the theorem again, we have that D is right (cid:27)-ample. Thus D
is left (cid:27)(cid:0)1-ample by Lemma 2.4. The same lemma gives the second statement of
the corollary.
Remark 5.2. CombinedwithTheorem4.7,this provesTheorem1.3andsowemay
refer to a divisor as being simply \(cid:27)-ample".
In[AV],left(cid:27)-amplenesswasshowntoimplytheassociatedtwistedhomogeneous
coordinateringisleftnoetherian. However,asnotedinthefootnoteof[AS,p. 258],
the paper says,but does not prove,that B is noetherian. This actually is the case.
Corollary 5.3. Let B = B(X;(cid:27);O (D)) be the twisted homogeneous coordinate
X
ring associated to a (cid:27)-ample divisor D. Then B is a (left and right) noetherian
ring, (cid:12)nitely generated over the base (cid:12)eld.
Analysis of the GK-dimension of B will be saved for the next section.
From the de(cid:12)nition of (cid:27)-ample, it is not obvious when (cid:27)-ample divisors even
exist. Theorem 4.7 makes the question much easier.
Corollary 5.4. A projective scheme X has a (cid:27)-ample divisor if and only if (cid:27) is
quasi-unipotent. In particular, every ample divisor is a (cid:27)-ample divisor if (cid:27) is
quasi-unipotent.
Thus,itisimportanttoknowwhenanautomorphism(cid:27)isquasi-unipotent. From
the bounds in equation (3.6), we obtain
Proposition 5.5. LetDbeanampledivisor. Then(cid:27)isquasi-unipotentifandonly
if for all curves C, the intersection numbers ((cid:27)mD:C) are bounded by a polynomial
for positive m.
Proposition 5.6. Let X be a projective scheme such that
(1) X has a canonical divisor K which is an ample or minus-ample divisor, or
(2) the Picard number of X, i.e., the rank of A1 (X), is 1.
Num
Then any automorphism (cid:27) of X is quasi-unipotent. Indeed, some power of (cid:27) is
numerically equivalent to the identity.
526 DENNIS S. KEELER
Proof. In the (cid:12)rst case, for K to be ample or minus-ample, it must be a Cartier
divisor. Thus the intersection numbers ((cid:27)mK:C) are de(cid:12)ned, where C is a curve.
Since K must be (cid:12)xed by (cid:27), some power of (cid:27) must be numerically equivalent to
the identity by equation (3.6). In the second case, the action of (cid:27) itself must be
numerically equivalent to the identity.
Thusformanyimportantprojectivevarieties,suchascurves,projectiven-space,
Grassmann varieties [Fl, p. 271], and Fano varieties [Ko, p. 240, De(cid:12)nition 1.1],
one automatically has that any automorphism must be quasi-unipotent.
Returning to corollariesof Theorem4.7, we see that building new (cid:27)-ample divi-
sors from old ones is also possible.
Corollary 5.7. Let D be a (cid:27)-ample divisor and let D0 be a divisor with one of the
following properties:
(1) (cid:27)-ample,
(2) generated by global sections, or
(3) numerically e(cid:11)ective.
Then D+D0 is (cid:27)-ample.
Proof. Takem suchthat(cid:1) isample and(cid:1)0 =D0+(cid:1)(cid:1)(cid:1)+(cid:27)m(cid:0)1D0 isrespectively
m m
ample, generated by global sections, or numerically e(cid:11)ective. Then (cid:1) +(cid:1)0 is
m m
ample and we again apply the main theorem.
Thefollowingcouldbeshowndirectlyfromthede(cid:12)nition,butalsousingasimilar
method to the proof above, one can see
Corollary 5.8. Let (cid:27) and (cid:28) be automorphisms. Then D is (cid:27)-ample if and only if
(cid:28)D is (cid:28)(cid:27)(cid:28)(cid:0)1-ample.
Note that (cid:28) need not be quasi-unipotent.
Finally, as in the case of ampleness, (cid:27)-ampleness is a numerical condition.
Corollary 5.9. Let D;D0 be numerically equivalent divisors and let (cid:27);(cid:27)0 be nu-
merically equivalent automorphisms (i.e., their actions on A1 (X) are equal).
Num
Then D is (cid:27)-ample if and only if D0 is (cid:27)0-ample.
Proof. As (cid:1) (cid:17) D0 +((cid:27)0)D0 +(cid:1)(cid:1)(cid:1)+((cid:27)0)m(cid:0)1D0 and ampleness depends only on
m
the numerical equivalence class of a divisor, the corollary follows from our main
theorem.
6. GK-dimension of B
As mentioned, our main goal of this section is to prove
Theorem 6.1. Let B = B(X;(cid:27);L) for some projective scheme X and (cid:27)-ample
invertible sheaf L.
(1) GKdimB is an integer. Hence B is of polynomial growth. In addition,
GKdimB is independent of the (cid:27)-ample L chosen.
(2) If (cid:27)m (cid:17)I for some m, then GKdimB =dimX +1.
(3) If X is an equidimensional scheme, then
k+dimX +1(cid:20)GKdimB (cid:20)k(dimX (cid:0)1)+dimX +1
where k =J((cid:27)) (cf. De(cid:12)nition 4.1) is an even naturalnumberdepending only
on (cid:27).