Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with
$\frac{3g+3}{2}<\deg L\le 2g-5$. Then $L$ is normally generated if $\deg
L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple
covering of genus $p$ curve $C'$ with $C\stackrel{\phi}\to C'$ and $D$ a
divisor on $C'$ with $4p<\deg D< \frac{g-1}{6}-2p$. Then $K_C(-\phi^*D)$
becomes a very ample line bundle which is normally generated. As an
application, we characterize some smooth projective surfaces.Comment: 7 pages, 1figur

Rock Kim

Seonja Kim

Young

English

arXiv

Kim, Seonja

Kim, Young Rock

Osaka City University Repository

Kim, S. and Kim, Y.R.
Osaka J. Math.
44 (2007), 685–690
PROJECTIVE NORMALITY OF ALGEBRAIC CURVES AND
ITS APPLICATION TO SURFACES
SEONJA KIM and YOUNG ROCK KIM
(Received February 2, 2006, revised October 24, 2006)
Abstract
Let L be a very ample line bundle on a smooth curve C of genus g with (3g +
3)=2 < deg L  2g   5. Then L is normally generated if deg L > maxf2g + 2  
4h1(C , L), 2g   (g   1)=6  2h1(C , L)g. Let C be a triple covering of genus p curve
C 0 with C ! C 0 and D a divisor on C 0 with 4p < deg D < (g   1)=6   2p. Then
KC (  D) becomes a very ample line bundle which is normally generated. As an
application, we characterize some smooth projective surfaces.
1. Introduction
We work over the algebraically closed field of characteristic zero. Specially the
base field is the complex numbers in considering the classification of surfaces. A
smooth irreducible algebraic variety V in Pr is said to be projectively normal if the
natural morphisms H 0(Pr , OPr (m)) ! H 0(V , OV (m)) are surjective for every non-
negative integer m. Let C be a smooth irreducible algebraic curve of genus g. We
say that a base point free line bundle L on C is normally generated if C has a
projectively normal embedding via its associated morphism L : C ! P(H 0(C , L)).
Any line bundle of degree at least 2g +1 on a smooth curve of genus g is normally
generated but a line bundle of degree at most 2g might fail to be normally generated
([8], [9], [10]). Green and Lazarsfeld showed a sufficient condition for L to be nor-
mally generated as follows ([5], Theorem 1): If L is a very ample line bundle on C
with deg L  2g + 1  2h1(C , L)  Cliff(C) (and hence h1(C , L)  1), then L is nor-
mally generated. Using this, we show that a line bundle L on C with (3g + 3)=2 <
deg L  2g   5 is normally generated for deg L > maxf2g + 2   4h1(C , L), 2g  
(g   1)=6   2h1(C , L)g. As a corollary, if C is a triple covering of a genus p curve
C 0 with C  ! C 0 then it has a very ample KC ( D) which is normally generated
for any divisor D on C 0 with 4p < deg D < (g   1)=6   2p. It is a kind of gener-
alization of the result that KC ( rg13) on a trigonal curve C is normally generated for
3r  g=2  1 ([7]).
2000 Mathematics Subject Classification. 14H45, 14H10, 14C20, 14J10, 14J27, 14J28.
This work was supported by the Korea Research Foundation Grant funded by the Korean Gov-
ernment (2005-070-C00005) for the first author. This work was supported by Hankuk University of
Foreign Studies Research Fund of 2007 for the second author.
686 S. KIM AND Y.R. KIM
As an application to nondegenerate smooth surface S  Pr of degree 21  e with
g(H ) = 1 + f , maxfe=2, 6e   1g < f   1 < (1   2e   6)=3 for some e, f 2 Z
1,
we obtain that S is projectively normal with pg = f and  2 f   e + 2  K 2S  (2 f +
e   2)2=(21   e) if its general hyperplane section H is linearly normal, where 1 :=
deg S   r + 1. We derive this application using the methods in Akahori’s paper ([2]).
We follow most notations in [1], [4], [6]. Let C be a smooth irreducible projective
curve of genus g  2. The Clifford index of C is taken to be Cliff(C) = minfCliff(L) j
h0(C , L)  2, h1(C , L)  2g, where Cliff(L) = deg L 2(h0(C , L) 1) for a line bundle
L on C . By abuse of notation, we sometimes use a divisor D on a smooth variety V
instead of OV (D). We also denote H i (V , OV (D)) by H i (V , D) and h0(V , L)  1 by
r (L) for a line bundle L on V . We denote KV the canonical line bundle on a smooth
variety V .
2. Main results
Any line bundle of degree at least 2g + 1 on a smooth curve of genus g is nor-
mally generated. If the degree is at most 2g, then there are curves which have a non
normally generated line bundle of given degree ([8], [9], [10]). In this section, we in-
vestigate the normal generation of a line bundle with given degree on a smooth curve
under some condition about the speciality of the line bundle.
Theorem 2.1. Let L be a very ample line bundle on a smooth curve C of genus
g with (3g +3)=2 < deg L  2g 5. Then L is normally generated if deg L > maxf2g +
2  4h1(C , L), 2g   (g   1)=6  2h1(C , L)g.
Proof. We have h1(C , L)  2, since 2g   5  deg L > 2g + 2  4h1(C , L). Sup-
pose L is not normally generated. Then there exists a line bundle A ' L( R), R >
0, such that (i) Cliff(A)  Cliff(L), (ii) deg A  (g   1)=2, (iii) h0(C , A)  2 and
h1(C , A)  h1(C , L) + 2 by the proof of Theorem 3 in [5]. Assume deg KC L 1 = 3.
Then jKC L 1j = g13 . On the other hand, L = KC ( g13) is normally generated. So
we may assume deg KC L 1  4 and then r (KC L 1)  2 since deg L > 2g + 2  
4h1(C , L). Let B1 (resp. B2) be the base locus of KC L 1 (resp. KC A 1). And let
N1 := KC L 1( B1), N2 := KC A 1( B2). Then N1  N2 since A = L( R), R > 0
and h1(C , A)  h1(C , L) + 2. Hence we have the following diagram,
where Ci = Ni (C).
If we set mi := deg Ni , i = 1, 2, then we have m2jm1. If N1 is birationally very
ample, then by Lemma 9 in [8] and deg KC L 1 < (g 1)=2 we have r (N1)  [(deg N1 
PROJECTIVE NORMALITY OF ALGEBRAIC CURVES 687
1)=5]. It is a contradiction to deg L > 2g + 2   4h1(C , L) that is equivalent to
deg KC L 1 < 4(h0(C , KC L 1)  1). Therefore N1 is not birationally very ample, and
then we have m1  3 since deg KC L 1 < 4(h0(C , KC L 1)  1).
Let H1 be a hyperplane section of C1. If jH1j on a smooth model of C1 is special,
then r (N1)  (deg N1)=4, which is absurd. Thus jH1j is nonspecial. If m1 = 2, then
r (KC L 1( B1 + P + Q))  r (KC L 1( B1)) + 1
for any pairs (P , Q) such that N1 (P) = N1 (Q) since jH1j is nonspecial. Therefore we
have r (L( P Q))  r (L) 1 for (P , Q) such that N1 (P) = N1 (Q), which contradicts
that L is very ample. Therefore we get m1 = 3. Suppose B1 is nonzero. Set P  B1
for some P 2 C . Consider Q, R in C such that N1 (P) = N1 (Q) = N1 (R) = P 0 for
some P 0 2 C1. Since jH1j is nonspecial, we have
r (KC L 1(Q + R))  r (N1(P + Q + R)) = r (H1 + P 0)
= r (H1) + 1 = r (KC L 1) + 1
which is a contradiction to the very ampleness of L . Hence KC L 1 is base point
free, i.e., KC L 1 = N1. On the other hand, we have m2 = 1 or 3 for m2jm1. Since
KC A 1( B2) = N2  N1 = KC L 1, we may set N1 = N2( G) for some G > 0.
Assume m2 = 1, i.e. KC A 1( B2) = N2 is birationally very ample. On the other
hand we have r (N2)  r (N1)+(degG)=2, since N2( G)= N1 and Cliff(N2)  Cliff(A) 
Cliff(L) = Cliff(N1). In case deg N2  g we have r (N2)  (2 deg N2   g + 1)=3 by
Castelnuovo’s genus bound and hence
Cliff(L)  Cliff(N2)  deg N2   4 deg N2   2g + 23 =
2g   2  deg N2
3

g   1
6
,
since N2 = KC A 1( B2) and deg A  (g   1)=2. If we observe that the condition
deg L > 2g   (g   1)=6   2h1(C , L) is equivalent to Cliff(KC L 1) < (g   1)=6, then
we meet an absurdity. Thus we have deg N2  g   1, and then Castelnuovo’s genus
bound produces deg N2  3r (N2)   2. Note that the Castelnuovo number (d, r ) has
the property (d, r )  (d   2, r   1) for d  3r   2 and r  3, where (d, r ) =
(m(m   1)=2)(r   1) + m, d   1 = m(r   1) + , 0    r   2 (Lemma 6, [8]). Hence
(deg N2, r (N2))      

deg N2   deg G, r (N2)  deg G2

 (deg N1, r (N1)),
because of 2  r (N1)  r (N2)   (deg G)=2. Since r (N1)  (deg N1)=4 and deg N1 <
(g   1)=2, we can induce a strict inequality (deg N1, r (N1)) < g as only the number
regardless of birational embedding from the proof of Lemma 9 in [8]. It is absurd.
Hence m2 = 3, which yields C1 = C2.
688 S. KIM AND Y.R. KIM
Let H2 be a hyperplane section of C2. If jH2j on a smooth model of C2 is special,
then r (N2)  (deg N2)=6. Thus the condition deg KC L 1 < 4(h0(C , KC L 1)  1) yields
the following inequalities:
2 deg N2
3
 Cliff(N2)  Cliff(N1)  deg N12 ,
which contradicts to N1  N2. Accordingly jH2j is also nonspecial.
Now we have r (Ni ) = (deg Ni )=3  p, i = 1, 2 where p is the genus of a smooth
model of C1 = C2. Therefore
deg N1
3
+ 2p = Cliff(N1)  Cliff(N2) = deg N23 + 2p
which is a contradiction that deg N1 < deg N2. This contradiction comes from the as-
sumption that L is not normally generated, thus the result follows.
Using the above theorem, we obtain the following corollary under the same as-
sumption:
Corollary 2.2. Let C be a triple covering of a genus p curve C 0 with C ! C 0
and D a divisor on C 0 with 4p < deg D < (g  1)=6  2p. Then KC ( D) becomes
a very ample line bundle which is normally generated.
Proof. Set d := deg D and L := KC ( D). Suppose L is not base point free,
then there is a P 2 C such that jKC L 1(P)j = gr+13d+1. Note that gr+13d+1 cannot be com-
posed with  by degree reason. Therefore we have g  6d +3p due to the Castelnuovo-
Severi inequality. Hence it cannot occur by the condition d < (g   1)=6   2p. Sup-
pose L is not very ample, then there are P , Q 2 C such that jKC L 1(P + Q)j = gr+13d+2.
By the same method as above, we get a similar contradiction. Thus L is very am-
ple. The condition d < (g   1)=6   2p produces Cliff(KC L 1) = d + 2p < (g   1)=6
since deg KC L 1 = 3d and h0(C , KC L 1) = h0(C 0, D) = d   p + 1. Whence deg L >
2g   (g   1)=6   2h1(C , L) is satisfied. The condition 4p < d induces deg KC L 1 >
4(h0(C , KC L 1)   1), i.e., deg L > 2g + 2   4h1(C , L). Consequently L is normally
generated by Theorem 2.1.
REMARK 2.3. In fact, we have a similar result in [8] for trigonal curve
C : KC ( rg13) is normally generated if 3r < g=2   1 ([7]). Thus our result could be
considered as a generalization which deals with triple coverings under the some con-
dition.
Let S  Pr be a nondegenerate smooth surface and H a smooth hyperplane sec-
tion of S. If H is projectively normal and h1(H , OH (2)) = 0, then q = h1(S, OS) =
PROJECTIVE NORMALITY OF ALGEBRAIC CURVES 689
0, pg = h2(S, OS) = h1(H , OH (1)) and h1(S, OS(t)) = 0 for all nonnegative integer t
([2], Lemma 2.1, Lemma 3.1). Using Theorem 2.1, we can characterize smooth pro-
jective surfaces with the wider range of degrees and sectional genera. Recall the defi-
nition of 1-genus given by 1 := deg S   r + 1.
Theorem 2.4. Let S  Pr be a nondegenerate smooth surface of degree 21  e
with g(H ) = 1 + f , maxfe=2, 6e 1g < f   1 < (1  2e  6)=3 for some e, f 2 Z
1
and its general hyperplane section H is linearly normal. Then S is projectively normal
with pg = f and  2 f   e + 2  K 2S  (2 f + e   2)2=(21  e).
Proof. From the linear normality of H , we get h0(H , OH (1)) = r and hence
h1(H , OH (1)) =   degOH (1)  1 + g(H ) + h0(H , OH (1))
=  21 + e   1 + g(H ) + h0(H , OH (1))
= g(H ) 1 = f .
Therefore we have h1(H , OH (1)) > deg((K H 
OH ( 1))=4) + 1 since f > e=2 + 1 and
degOH (1) = 21   e = 2g(H )   2   (2 f + e   2). Thus OH (1) satisfies degOH (1) >
2g(H ) + 2   4h1(H , OH (1)). The condition f   1 > 6e   1 implies degOH (1) >
2g   (g   1)=6   2h1(H , OH (1)). Also the condition f   1 < (1   2e   6)=3 yields
degOH (1) > (3g + 3)=2. Hence OH (1) is normally generated by Theorem 2.1, and
thus its general hyperplane section H is projectively normal since it is linearly normal.
Therefore S is projectively normal with q = 0, pg = h0(S, KS) = h1(H , OH (1)) = f > 1
since h1(H , OH (2)) = 0 from degOH (1) > (3g + 3)=2.
If we consider the adjunction formula then KS .H = 2 f + e   2 and 0 ! KS !
KS + H ! K H ! 0. Thus we have 0! H 0(S, KS)! H 0(S, KS + H )! H 0(H , K H )!
0, since H 1(S, KS) = q = 0. Assume jKS + H j has a fixed component B. Set p 2
B \ H , then p becomes a base point of jK H j since H 0(S, KS + H ) ! H 0(H , K H ) is
surjective, which cannot occur. Therefore KS + H is free from fixed components. Thus
for any irreducible curve C in S, we can choose effective D 2 jH + KSj such that D
does not contain C and then D.C  0, which implies H + KS is nef. Hence we get
KS .(H + KS)  0 and then
K 2S   KS .H =  2 f   e + 2.
Thus  2 f  e + 2  K 2S  (2 f + e 2)2=(21 e) by the Hodge index theorem K 2S H 2 
(KS .H )2. Hence the theorem is proved.
ACKNOWLEDGEMENT. We would like to thank referee for valuable comments
which improve our paper.
690 S. KIM AND Y.R. KIM
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[3] A. Beauville: Complex Algebraic Surfaces, Cambridge Univ. Press, Cambridge, 1983.
[4] P. Griffiths and J. Harris: Principles of Algebraic Geometry, Wiley-Intersci., New York, 1978.
[5] M. Green and R. Lazarsfeld: On the projective normality of complete linear series on an al-
gebraic curve, Invent. Math. 83 (1986), 73–90.
[6] R. Hartshorne: Algebraic Geometry, Graduate Text in Math. 52, Springer, New York, 1977.
[7] S. Kim and Y. Kim: Projectively normal embedding of a k-gonal curve, Comm. Algebra 32
(2004), 187–201.
[8] S. Kim and Y. Kim: Normal generation of line bundles on algebraic curves, J. Pure Appl.
Algebra 192 (2004), 173–186.
[9] H. Lange and G. Martens: Normal generation and presentation of line bundles of low degree
on curves, J. Reine Angew. Math. 356 (1985), 1–18.
[10] D. Mumford: Varieties defined by quadric equations; in Questions on Algebraic Varieties
(C.I.M.E., III Ciclo, Varenna, 1969), Ed. Cremonese, Rome, 1970, 29–100.
Seonja Kim
Department of Digital Broadcasting & Electronics
Chungwoon University
Chungnam, 350–701
Korea
e-mail: sjkim@chungwoon.ac.kr
Young Rock Kim
Department of Mathematics Education
Graduate School of Education
Hankuk University of Foreign Studies
Seoul, 130–791
Korea
e-mail: rocky777@hufs.ac.kr


Projective normality of algebraic curves and its application to surfaces

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Projective Normality Of Algebraic Curves And Its Application To Surfaces

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