Relations in the canonical algebras on surfaces

Relations in the Canonical Algebras on Surfaces.

KAZUHIROKONNO(*)

ABSTRACT- The degree bound for primitive generators and relations of the canonical ring of a minimal surface of general type are studied via Green's Koszul coho- mology, assuming that the fixed part of the canonical linear system does not contain any Francia cycles. Slight refinements of the results due to Ciliberto and Reid are given.

Introduction.

Let S be a non-singular, projective, minimal surface of general type defined over the complex number fieldC. Thecanonical ring of Sis the gradedC-algebra

R(S;K_S)ˆM^‡1

mˆ0

H⁰(S;mK_S):

This naive object has been studied by many authors in order to see what can be expected on the analogy of Max Noether's and Enriques-Petri's theorems for curves. We recall here some important results obtained so far. Cili- berto [4] showed, among other things, thatR(S;K_S) is generated in degrees 5 under some reasonable conditions. Green [8] considered the case that the canonical linear system is free from base points in any dimension by applying his theory of Koszul cohomology groups (see also [9]). Reid [19]

showed that it has the 1-2-3 property, that is, it is generated in degrees3 and related in degrees 6, when S is a regular surface with K_S²3, pg(S)2 which has an irreducible canonical curve on the canonical model.

(*) Indirizzo dell'A.: Department of Mathematics Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan.

E-mail: konno@math.sci.osaka-u.ac.jp

Partially supported by Grants-in-Aid for Scientific Research (B) (No. 16340008) by Japan Society for the Promotion of Science (JSPS).

Mendes Lopes [17] considered surfaces with vanishing geometric genera and showed thatR(S;KS) is generated in degrees4 provided that 2KSis free.

In this article, we study the degree bound for primitive homogeneous generators and relations ofR(S;K_S). The key point is the well-known re- sult essentially due to Francia [7] and Reider [22] that 2K_S is free except possibly whenpg(S)ˆ0 andK²_S4. We apply the machinery of Green's Koszul cohomology groups ([8], [9]) and show thefollowing:

THEOREM A. Let S be a minimal algebraic surface of general type such that2K_S is free. Then R(S;K_S)is generated in degrees5and re- lated in degrees10. If furthermore q(S)ˆ0, then R(S;K_S)is generated in degrees4and related in degrees8except when(p_g(S);K_S²)ˆ(2;1) and possibly(pg(S);K²_S)ˆ(0;3).

This is nothing more than what one naturally expects after Ciliberto's results on generators [4]. Recall thatR(S;K_S) needs a generator in degree 4 if thefixed part ofjK_Sjcontains a special configuration of curves called a Francia cycle, that is, an effective divisor E with p_a(E)ˆ E²ˆ1 or pa(E)ˆ2;E²ˆ0, and many such surfaces are constructed in [4]. Hence the bound in Theorem A is sharp for regular surfaces in this sense.

On the other hand, it is an interesting problem [19] to study whether thecanonical ring has the1-2-3 property if thefixed part ofjK_Sjdoes not contain any Francia cycles. We consider the problem for a regular surface whose canonical map is not composed of a pencil. However, partly because wedo not know much about thefixed loci, wehaveto imposesomeextra restrictions modeled on fibred surfaces.

THEOREM B. Let S be a minimal algebraic surface of general type with pg(S)2, q(S)ˆ0 and K²_S3. Let jKSj ˆ jMj ‡Z be the decom- position into its variable and fixed parts. Suppose thatjMjhas an irre- ducible member. If one of the following conditions(1)and(2)is satisfied, then R(S;K_S)is generated in degrees3and related in6.

(1) H¹(Z;OZ)ˆ0 (possibly Zˆ0).

(2) Z does not contain a Francia cycle and decomposes as ZˆD‡G₁‡ ‡G_n, where

(a) D is an effective divisor with K_SDˆ0 (possibly Dˆ0), Supp (D)\Supp (Z D)ˆ ;, and

(b) for each i2 f1;. . .;ng,Giis a chain connected curve such that K_SG_i>0,OGi( G_i)is nef, andOGi(G_j)is numerically trivial when j6ˆi.

If the fixed part supports at most exceptional sets of rational singular points, then we haveh¹(Z;OZ)ˆ0. Hence (1) can be regarded as a slight generalization of Reid's theorem [19] referred above. A curve Dischain connected(cfr. [21]), if either it is irreducible, or for any proper subcurve DDthere exists an irreducible componentADsuch that (D D)A>0.

The restrictions in (2) are for technical reasons and should be replaced by a simpler assumption, e.g., the intersection form is negative semi-definite on Supp (Z). Notethat, if a curveG_ias in (2) exists, then the intersection form is negative semi-definite on Supp (G_i) andG_i is nothing morethan thenu- merical cycleon it in the terminology in [20]. The proof of Theorem B is based on the hyperplane section principle as in [19] and a result for chain connected curves similar to those in [15] and [13].

Theorganization of thepaper is as follows. Thefirst two sections areof preparatory nature. In § 1, we recall and restate the fundamental tools, such as the duality theorem, the vanishing theorem and theK_p;1theorem, for Green's Koszul cohomology groups [9] in the form suitable for our purpose. We recall in § 2 some important notions treating the connected- ness of effective divisors and show Theorem 2.6 which is crucial for the proof of Theorem B, (2). In § 3, we show Theorem A by studying the bi- canonical maps. We need several existing results to manage some special cases. Especially when p_gˆqˆ0, our proof heavily depends on [17].

Theorem B will be shown in §§ 4, 5. At the end of § 4, we state a result for irregular surfaces similar to Theorem B, (1). In Appendix (§6), we give a few remarks on the relative canonical algebras for fibrations and singu- larities, in order to supplement the results in [15] and [13] about Reid's 1- 2-3 conjecture.

The author would like to thank the referee for his interest and stimu- lating suggestions.

1. Koszul cohomology.

In [8] and [9], Green developed the theory of Koszul cohomology groups which gives us a powerful tool for studying graded rings associated to line bundles. Although the results in [9] are stated for compact complex manifolds, it is not so hard to see that most of them hold also for singular objects if one makes suitable modifications. In this section, we restate three main results in [9] (the duality theorem [9, Theorem (2.c.6)], the vanishing theorem [9, Theorem (3.a.1)] and the Kp;1 theorem [9, Theo- rem (3.c.1)]) in the forms suitable for our purpose.

LetXbea projectivescheme(overC) of puredimensionn. Le tLis an invertible sheaf onXand takea subspaceWH⁰(X;L) giving us a linear system which is free from base points. Then W defines a morphism F_W :X!P(W). By pulling back the exact sequence 0!V¹_P(W)( 1)! W_CO_P(W) ! O_P(W)(1)!0; oneobtains 0! E !W_CO_X ! L !0;

which gives us

0!^^p

E !^^p

W O_X !^p^¹

E L !0 for any positive integer p. In particular, V_r

E ' L^_ if rˆdimW 1ˆ

ˆrk (E). Let M bean invertiblesheaf on X. Then tensoring M L^q (q2Z) to the above, one gets

0! Ep;q(M)!^^p

W M L^q! Ep 1;q‡1(M)!0;

(Seq)_p;q :

whereE_p;q(M)ˆV_p

E M L^q. Le t d_p;q:^^p

WH⁰(X;M L^q)!^p^¹

WH⁰(X;M L^(q‡1)) bethecompositeofV_p

WH⁰(X;M L^q)!H⁰(X;Ep 1;q‡1(M)) and the natural injectionH⁰(X;Ep 1;q‡1(M)),!V_{p 1}

WH⁰(X;M L^(q‡1)) ob- tained from (Seq)_p;qand (Seq)_{p 1;q‡1}, respectively. Thend_p;qd_{p‡1;q 1}ˆ0.

TheKoszul cohomology group is nothing but K_p;q(X;M;L;W)ˆKer (d_p;q)=Im (d_{p‡1;q 1})

'Cokern ^^p‡1

WH⁰(X;ML^{(q 1)})!H⁰(X;E_p;q(M))o : THEOREM 1.1 (Duality Theorem [9, Theorem (2.c.6)]). Let X be a Gorenstein projective scheme of pure dimension n andL;M invertible sheaves on X. If W H⁰(X;L)denotes a subspace of dimension r‡1in- ducing a linear system which is free from base points, then Kp;q(X;M;L;W)is dual to Kr n p;n‡1 q(X;vX M^_;L;W)provided that Hⁱ(X;M L^{(q i)})ˆHⁱ(X;M L^{(q i 1)})ˆ0 for 1in 1, where v_X denotes the dualizing sheaf.

PROOF. Wehavea natural injection K_p;q(X;M;L;W),! ,!H¹(X;Ep‡1;q 1(M)). Consider the coboundary map

d_i:Hⁱ(X;E_{p‡i;q i}(M))!H^i‡1(X;E_{p‡i‡1;q i 1}(M))

obtained from (Seq)_{p‡i‡1;q i 1}for 1in 1. IfHⁱ(X;M L^{(q i 1)})ˆ0 for 1in 1, then di is injective for 1in 1. Therefore, Kp;q(X;M;L;W),!H^{n 1}(X;E_{p‡n 1;q n‡1}(M)) and

0!H^{n 1}(X;Ep‡n 1;q n‡1(M))!Hⁿ(X;Ep‡n;q n(M))

!^p‡n^

WHⁿ(X;M L^{(q n)}) is exact. By the Serre duality theorem, the dual sequence of the latter is

^

p‡n

W^_H⁰(X;v_X M^_ L^{(n q)})!H⁰(X;v_X Ep‡n;q n(M)^_)

!H¹(X;vX Ep‡n 1;q n‡1(M)^_)!0:

SincewehaveV_p‡n

W^_'V_{r‡1 p n} Wand E_{p‡n;q n}(M)^_'^p‡n^

E^_ M^_ L^{(n q)} '^{r n p}^

E M^_ L^{(n q‡1)} byV_r‡1

W'CandV_r

E ' L^_, respectively, the above exact sequence is nothing morethan

^

r‡1 p n

WH⁰(X;vX M^_ L^{(n q)})!H⁰(X;Er n p;n q‡1(vX M^_))

!H¹(X;Er n p‡1;n q(v_X M^_))!0:

This shows that the kernel of Hⁿ(X;Ep‡n;q n(M))!V_p‡n W Hⁿ(X;M L^{(q n)}) is thedual spaceofK_{r n p;n‡1 q}(X;v_X M^_;L;W).

Hence K_p;q(X;M;L;W) is mapped injectively to K_{r n p;n‡1 q}(X;v_X M^_;L;W)^_. Similarly, ifHⁱ(X;M L^{(q i)})ˆ0 for 1in 1, then Kr n p;n‡1 q(X;vX M^_;L;W)!Kp;q(X;M;L;W)^_ is injective, since H^j(X;v_X M^_ L^{(n q j)}) is dual toH^{n j}(X;M L^{(q (n j))}) by theSerre

duality theorem. p

We remark that the duality theorem holds for Gorenstein curves without additional cohomological conditions. See also [13, Lemma 1.2.1] for curves on a smooth surface. The following two theorems can be shown as in [9] without major changes. Hence we only give the statements without proofs.

THEOREM1.2 (Vanishing Theorem [9, Theorem (3.a.1)]). Let X be an irreducible projective variety. If L;M are invertible sheaves on X and W H⁰(X;L) is a subspace, then Kp;q(X;M;L;W)ˆ0 provided that h⁰(X;M L^q)p.

For simplicity, wewriteKp;q(X;L) instead ofKp;q(X;L;L;H⁰(X;L)).

THEOREM 1.3 (The K_p;1 Theorem [9, Theorem (3.c.1)]). Let X be an irreducible projective variety,Lan invertible sheaf on X with h⁰(X;L)ˆ

ˆr‡1and put mˆdimF_L(X). Then the following hold.

(1) Kp;1(X;L)ˆ0for p>r m:

(2) Kr m;1(X;L)ˆ0unlessFL(X)is an m-fold of minimal degree.

(3) Kr m 1;1(X;L)ˆ0 unless either degFL(X)r‡2 m or F_L(X)lies on an(m‡1)-fold of minimal degree.

Theirreducibility ofXis not essential in the vanishing theorem, while it is an important assumption in the Kp;1 theorem, since we require the uniform position property for the zero cycle obtained as a general linear section ofF_L(X). See [8] and [9] for the detail.

We recall one more lemma from [8] which will be used frequently.

LEMMA1.4 ([8]). LetLbe an invertible sheaf on a projective scheme X and let k be the smallest positive integer such thatL^k is generated by global sections. Assume that the graded ring R(X;L)ˆL

m0H⁰(X;L^m) is generated in degreesd. Put

d0ˆminn2Z0fH⁰…X;L^k† H⁰…X;L^{m k†}† !H⁰…X;L^m† is surjec- tive for any m>n}

d1min

n2Z0

n^²

H⁰…X;L^k†H⁰…X;L^{m 2k†}†!H⁰…X;L^k†H⁰…X;L^{m k†}†

!H⁰…X;L^m†is exact at the middle term for any m<no Then R(X;L) is related in degrees maxfd‡d₀;d₁g. If d₀ is not a multiple of d, then R(X;L)is related in degreesmaxfd‡d₀ 1;d₁g.

PROOF. Though this is contained in the proof of [8, Theorem 3.11], we recall the argument for the later use.

Choosing a minimal set of homogeneous generators ofR(X;L), wehave a surjective homomorphismF:V!R(X;L), whereVis a polynomial ring overC. WeletVˆL

m0Vmbethedecomposition into thegraded pieces such thatFbecomes a homomorphism of graded algebras. The kernelIof F is a homogeneous ideal and we denote it as IˆL

m0I_m. Welet fj₁;. . .;j_Ngbea basis forV_k.

Assumethatn>maxfd‡d0;d1gand letF2Inbe a relation in degree n. Weshall show thatF0 modulo relations in lower degrees. LetGbe any monomial appearing inF. Sincen>d‡d₀, weseethatGcan bedi- vided by a monomial G₁ satisfying d₀<degG₁d‡d₀. Then, since degG₁>d₀,F(G₁) can be expressed as a sum of products of degreekand degreen kelements inR(X;L). Therefore, we can writeFˆP_N

iˆ1j_iFi

modulo relations in degrees maxfn k;d‡d0g, where the Fi's are homogeneous polynomials of degreen k. Then, sincen>d1, wecan find polynomials F_ij of degreen 2ksatisfyingF_iˆP

j_jF_ij andF_ijˆ F_ji, modulo relations in degrees <n. Then we obtain FˆP

ij_iF_iˆ

ˆP

ijj_ij_j(F_ij‡F_ji)ˆ0 modulo relations in lower degrees.

Ifd0is not a multipleofd, then the above argument works also when

nˆd₀‡d. p

2. Chain connected curves.

A non-zero effective divisor on a non-singular surface will be simply called a curve. In this section, we recall basic notions about the con- nectedness of curves from [5, Appendix] and [21] and collect some results for the later use. See also [20]. In some sense, this section is a re- consideration of [21].

Let Dbe a curve. For an integer k, Disnumerically k-connected if D₁D₂k holds for any decomposition DˆD₁‡D₂ where D₁, D₂ are curves. Note that a nef and big curve is necessarily 1-connected. In par- ticular, so is a canonical curve of a minimal surface of general type.

DEFINITION2.1. A numerically 2-connected curveEis called aFrancia cycle, if either (i)p_a(E)ˆ1 andE²ˆ 1, or (ii)p_a(E)ˆ2 andE²ˆ0.

A curve as in (i) will be sometimes called a ( 1) elliptic cycle.

If Supp (D) is connected and the intersection form is negative semi- definite on it, thenDis called thenumerical cycleifDis thesmallest curve such that Dis nef on its support. When the intersection form is negative definite, that is, Supp (D) is the exceptional set of a normal surface sin- gularity, thenumerical cycleis usually called thefundamental cycle. A curve D is chain connected ([21]), if either it is irreducible, or for any proper subcurveDDthere exists an irreducible componentADsuch that (D D)A>0. It is easy to see that numerical cycles and 1-connected curves are chain connected.

LEMMA2.2. For a chain connected curve D, the following hold.

(1) h⁰(D;O_D)ˆ1 and a non-zero element of H⁰(D;O_D)does not vanish identically on any component.

(2) If L is a nef line bundle on D, then H⁰(D; L)6ˆ0if and only if OD(L)' OD.

PROOF. We first show (2) by using an argument in [5, Appendix]. Let s2H⁰(D; L) be a non-zero element. If s vanishes identically on some components ofD, weletDbethebiggest subcurveofDon whichsvanishes identically. Thensinduces a non-zero element ofH⁰(D D; L D) which does not vanish identically on any component ofD D. This implies that L Dis nef onD D. On the other hand, by the chain connectedness ofD, there is an irreducible component AD D such that DA>0. Then deg ( L D)jA DA<0, which contradicts what wehavejust seen above. Hencesdoes not vanish identically on any component ofD. Then L is numerically trivial, sinceLis nef, and the nowhere vanishing sections induces an isomorphismO_D' O_D( L).

To show (1), weassumethath⁰(D;OD)>1. Then we can find a non-zero elements2H⁰(D;OD) which vanishes identically on some components of D. Takethebiggest subcurveDon whichsvanishes identically and copy thefirst half of theproof of (2) with Lbeing replaced byO_D. Then it will immediately lead us to a contradiction. Henceh⁰(D;O_D)ˆ1. Thelast as- sertion in (1) is clear from the above proof of (2). p Here we recall the following easy lemma whose proof can be found in [13, Lemma 2.2.1].

LEMMA2.3. Let D be a curve and L a line bundle on D. Take an irre- ducible component AD and letDbe a minimal curve containing A such that the restriction map H⁰(D;L)!H⁰(D;L)is surjective. Then one of the following holds.

(1) KD L is nef onD.

(2) A is of multiplicity one inDand KD L is nef onD A. Fur- thermore, deg (L (D A))j_A>degK_A and the image of the restriction map H⁰(D;L)!H⁰(A;L) contains that of the natural inclusion H⁰(A;L (D A))!H⁰(A;L).

If jLj has no base points on A\(D A) in (2), then h⁰(A;L) h⁰(A;L (D A))‡1.

PROPOSITION2.4. LetDbe chain connected andLa line bundle onD such that L KD is nef and deg (L KD)>0. If x2BsjLj, then there exists a subcurveDofDsatisfying the following properties.

(1) x is a non-singular point ofD.

(2) The restriction map H⁰(D;L)!H⁰(D;L)is surjective.

(3) O_A(L)'v_D O_A(x), where A denotes the unique irreducible component ofDthrough x.

(4) KD L is nef onD A.

PROOF. Wemay assumethat D is reducible. Take an irreducible componentAthroughxand letDˆD_Abea minimal subcurveofDsuch that ADandH⁰(D;L)!H⁰(D;L) is surjective.

Wefirst assumethat DˆA. WehavedegLj_Aˆdeg (L KD)j_A‡

‡degKA‡(D A)A. SinceL KDis nef andDis chain connected, we get degLj_A>degK_A. Sincex2BsjO_A(L)j,xis a non-singular point ofAand O_A(L)'v_A(x).

Wenext assumethatDˆD. SinceL K_Dis nef and deg (L K_D)>0, it follows from Lemma 2.3 thatAis of multiplicity oneinD,K_D Lis nef on D A and deg (L (D A))j_A>degKA. Notethat H⁰(D;L)!

!H⁰(D A;L) is surjective, becauseH¹(A;L (D A))ˆ0. Hence, we can assumethatx62A\(D A), because, otherwise, there exists another irreducible component B through x and then D_BD A by what we have just seen which enables us to argue with (B;D_B) instead of (A;D_A).

Thenx2BsjOA(L (D A))j. Sincedeg (L (D A))j_A>degKA,xhas to bea non-singular point ofAand weseethatO_A(L (D A))'v_A(x) which is equivalent to O_A(L)'v_D O_A(x). Notethat x is also a non- singular point of Dbyx62A\(D A).

Wefinally assumethat D6ˆA and D6ˆD. WehaveK_D Lˆ

ˆKD L (D D) onD. SinceDis chain connected, we can find an ir- reducible component BD such that (D D)B>0. For suchB, weget deg (KD L)j_B<0. Hence KD L is not nef on Dand wearein thesi- tuation of Lemma 2.3, (2). SinceH⁰(D;L)!H⁰(D A;L) is surjective, we may assumethat x62A\(D A) as in the previous case. Then we can concludethatxis a non-singular point ofAandOA(L (D A))'vA(x).

This showsOA(L)'vD OA(x). p

COROLLARY2.5. LetDbe a chain connected curve on a smooth surface Ssuch thatK_Sand Dare both nef onDandD²<0. IfLis a line bundle onDsuch thatL rK_Sis a nefQ-bundle for some rational numberr >1, thenH¹(D; L)ˆ0andjLjhas no base points.

PROOF. Wemay assumethat D is reducible. We have L KDˆ

ˆL rKS‡(r 1)KS DonD. He nce L KDis nef of positive degree.

By the Serre duality theorem, H¹(D;L)^_'H⁰(D;K_D L). So weget H¹(D;L)ˆ0, becauseDis chain connected.

Weshow that jLj is freefrom basepoints. Wefirst remark that the intersection form is negative definite on Supp (D) andDis thenu- merical cycle, since Supp (D) is connected,OD( D) is ne f andD²<0.

Supposenow thatxis a base point ofjLj. Then there exists a subcurveD and an irreducible component A as in Proposition 2.4. Wehave O_D(K_D L)ˆ O_D(rK_S L (r 1)K_S‡D). Since O_A(L K_D)' O_A(x), wehaveDAˆ(r 1)K_SA‡(L rK_S)A 1. SinceK_D Lis nef onD A, wehaveD(D A)(L rKS)(D A)‡(r 1)KS(D A). In sum, weget D²ˆDA‡D(D A)(L rK_S)D‡(r 1)K_SD 1. If DˆD or DˆA, then, since D²<0, wemust haveD² ˆ 1 and K_SDˆ0, which is im- possiblefor thereason of parity. AssumethatD6ˆDandD6ˆA. SinceD is chain connected, we can find an irreducible component B of D sa- tisfying (D D)B>0. Then 0DB>DB, which shows thatKD Lis of negative degree on B. Henceweget BˆA. Now, it follows from 0>DAˆ ˆ(L rK_S)A‡(r 1)K_SA 1 that de gLj_AˆK_SAˆ0 and 1ˆDAˆ ˆ(D A)A‡A². Thelast shows that A²ˆ 1 and (D A)Aˆ0, becauseA²<0 andAis of multiplicity oneinD. It is again

impossiblefor thereason of parity. p

LetDbe a chain connected curve on a non-singular surfaceS. Takean arbitrary subcurveA₁ ofDand putD₁ˆA₁. IfD₁6ˆD, then there is an irreducible component A₂ of D D₁ such that D₁A₂>0. Weput D2ˆD1‡A2. IfDi is defined and Di6ˆD, then we take an irreducible componentAi‡1ofD Disuch thatDiAi‡1>0, and putDi‡1ˆDi‡Ai‡1. In this way, we get a composition seriesfA_ig^N_iˆ1such thatDˆP_N

iˆ1A_i. It is said to be irreducible, if the first curveA₁ is also irreducible. We put X_iˆD D_{N i}for 0iN.

THEOREM2.6. Let D be a chain connected curve on a non-singular surface S with K²_S>0such that K_Sand D are both nef on D. Foraˆ1;2, let L_a be a line bundle on D such that L_a 2K_S is nef. Then the multi- plication map

H⁰(D;L1)H⁰(D;L2)!H⁰(D;L1‡L2)

is surjective, unless there exists a Francia cycle ED on which L₁and L₂ are both numerically equivalent to2K_S.

PROOF. SinceDis chain connected andOD( D) is nef, the intersec- tion form is negative semi-definite on Supp (D) andDis nothing morethan the numerical cycle on its support. Furthermore, for a subcurveDDwith K_SDˆ0, wehaveD²<0 by Hodge's index theorem, sinceK²_S>0.

If Dis irreducible, it is easy to check that the multiplication map is surjective unlessDitself is a Francia cycle andL1 L22KSonD, where thesymbolmeans the numerical equivalence. So we may assume thatD is reducible.

We first consider the extremal case that D²ˆ0. ThenO_D(D) is nu- merically trivial andDis 1-connected. We haveK_Dˆ(K_S‡D)j_Dwhich is nef in the present case. Hencep_a(D)1. Ifp_a(D)ˆ1, thenK_SDˆD²ˆ0, which is impossible by Hodge's index theorem becauseK²_S>0. Hence we havepa(D)2. SinceLa 2K_Dˆ(La 2K_S) 2Dj_Dis nef, the assertion follows from [13, Theorem I].

WeassumethatD²<0. Then the intersection form is negative definite on Supp (D). We follow Laufer's argument [15]. We take an irreducible composition series fAig^N_iˆ1 forDsatisfyingDA1 <0. SinceLa 2KS and Darenef onD, wecan show thatH¹(D;La)ˆ0 and therestriction map H⁰(X_{N i‡1};L_a)!H⁰(X_{N i};L_a) is surjective. This in particular implies that H⁰(D;L_a)!H⁰(X_k;L_a) is surjective for 0<k<N,aˆ1;2. Furthermore, jL_ajis free from base points by Corollary 2.5. We have the exact sequence

0!H⁰(A_i;L X_{N i})!H⁰(X_{N i‡1};L)!H⁰(X_{N i};L)!0 for LˆL1, L2, L1‡L2. Then we see that H⁰(XN i‡1;L1) H⁰(X_{N i‡1};L2)!H⁰(X_{N i‡1};L1‡L2) is surjective provided that so are H⁰(X_{N i};L₁)H⁰(X_{N i};L₂)!H⁰(X_{N i};L₁‡L₂) and

H⁰(A_i;L₁ X_{N i})W_2;iL

W_1;iH⁰(A_i;L₂ X_{N i})

!H⁰(A_i;L₁‡L₂ X_{N i});

(2:1)

whereW_a;iˆImfH⁰(D;La)!H⁰(A_i;La)g. By induction, in order to show that H⁰(D;L₁)H⁰(D;L₂)!H⁰(D;L₁‡L₂) is surjective, it suffices to show that (2.1) is surjective for anyi, 1iN. SinceL_a 2K_Sis nef, we havedeg (La KS)j_AidegKSj_Ai0. IfLa KS is of degree zero onAi, thenA_iis a ( 2)-curve and (2.1) is clearly surjective, becauseLaj_Aiis trivial and BsjLaj ˆ ;. This enables us to assume that deg (La K_S)j_Ai>0 for aˆ1;2 in what follows. WedenotebyD_a;ia minimal subcurveofDcon- tainingA_isuch thatH⁰(D;L_a)!H⁰(D_a;i;L_a) is surjective.

Weassumethat degL₁j_Ai degL₂j_Ai and consider the multiplication map m_2;i:H⁰(Ai;L1 XN i)W2;i!H⁰(Ai;L1‡L2 XN i). Weshall

show that eitherm_2;iis surjective or there is a Francia cycleDcontaining Ai and L22KS on D. By Theorems 1.1 and 1.2, m_2;i is surjective if h⁰(A_i;K_Ai‡L₂ L₁‡X_{N i})dimW_2;i 2. Henceweonly haveto check that

dimW_2;i

p_a(A_i)‡2; if K_Ai‡L₂ L₁‡X_{N i}is special;

pa(Ai)‡1‡deg (L2 L1)j_Ai‡(D Di 1)Ai A²_i; otherwise:

8<

(2:2) :

Assumethat W_2;iˆH⁰(A_i;L₂). Then dimW_2;iˆdegL₂j_Ai‡1 p_a(A_i).

Theinequality dimW_2;ip_a(A_i)‡2 is equivalent to deg (L₂ K_S)j_Ai A²_i 3, which fails only whenK_SA_iˆ1,A²_i ˆ 1 and degL₂j_Aiˆ2, that is,Ai is a Francia cycleof arithmetic genus oneon whichL22KS. The second inequality in (2.2) is equivalent to deg (L1 KS)j_Ai DAi‡

‡D_{i 1}A_i2, which always holds under our assumptions.

Assumethat W_2;i6ˆH⁰(A_i;L₂) and put D_i:ˆD_2;i. Since H⁰(D;L₂)!

!H⁰(X_{N i‡1};L₂) is surjective, we have D_iX_{N i‡1}. In particular, we haveDiˆDonly wheniˆ1. It follows from Lemma 2.3 that we have ei- ther (i) Di (L2 KS) is ne f on Di, or (ii)Ai is of multiplicity onein Di, D_i (L2 K_S) is ne f on D_i A_i and O_Ai(L2 (D_i A_i)) is non-special;

furthermore,

(2:3) dimW2;i1‡h⁰(Ai;L2 (Di Ai))

ˆpa(Ai)‡deg (L2 KS)j_Ai‡(D Di)Ai DAi; sincejL₂jhas no basepoints.

Weexcludethepossibility (i). AssumethatDi (L2 KS) is ne f onDi. Then we haveD_iAdeg (L2 K_S)j_AK_SA0 and, hence, (D D_i)A0 holds for any irreducible componentAofD_i. SinceDis chain connected, this is possibleonly ifD_iˆD. IfD_iˆD, then we have (D (L₂ K_S))D0 and it would followD²0, which is absurd. Hence (i) is impossible, and we are in thecase(ii). In particular, wemust have(D Di)Ai>0 except wheniˆ1, D1ˆD.

Weuse(2.3) to examine(2.2). IfK_Ai‡L2 L1‡X_{N i}is special onA_i, it suffices to check that deg (L₂ K_S)j_Ai‡(D D_i)A_i DA_i2, which is direct. If it is non-special onA_i, then a sufficient condition for (2.2) is that 2pa(Ai)‡(D Di)Ai 2DAi‡Di 1Ai‡deg (L1 2KS)j_Ai3. This does not hold in thefollowing cases:

(a) iˆ1,D1ˆD,DA1ˆ 1,pa(A1)ˆdeg (L1 2KS)j_A1ˆ0.

(b) i2,pa(A_i)ˆDA_iˆdeg (L1 2K_S)j_Aiˆ0, (D D_i)A_iˆD_{i 1}A_iˆ1.

In either case,L₁,L₂ and 2K_S are of the same degree onA_i.

Consider the case (a). Recall that D (L2 KS) is ne f on D A1. Since L2 2KS and D are both nef, we see that every component of D A₁is a ( 2)-curve,D(D A₁)ˆ0 andL₂2K_SonD. Consider the cohomology long exact sequence for

0! O_{D A1}( A1)! O_D! O_A1!0:

Wehaveh⁰(D;OD)ˆ1 and H⁰(D;OD)!H⁰(A1;OA1) is non-trivial. This shows that H⁰(D A1;O( A1))ˆ0. Recall that we have O_{D A1}(L2)' ' O_{D A1}. It follows from the exact sequence

0!H⁰(D A1;L2 A1)!H⁰(D;L2)!H⁰(A1;L2)

andH⁰(D A1;L2 A1)'H⁰(D A1;O_{D A1}( A1))ˆ0 that dimW2;1 ˆ

ˆh⁰(D;L₂)ˆ2K_SD D(K_S‡D)=2ˆ(3=2)K_SA₁ D²=2. Hence the in- equality in (2.2) becomes (1=2)(K_SA₁ D²) DA₁3. This does not hold only whenKSA1ˆ D²ˆ DA1 ˆ1, that is,Dis a ( 1) elliptic cycle on whichL22K_S. Weshow that such a ( 1) elliptic cycleDis 2-connected.

LetDˆA‡Bbe any effective decomposition ofD. Wecan assumethatA1

is contained inA. ThenBconsists of ( 2)-curves and it follows thatB²is a negative even integer. Recall thatO_{D A1}(D) is numerically trivial. In par- ticular, wehave0ˆDBˆAB‡B². Hence we getAB2 and see thatDis 2-connected.

Consider the case (b). (2.2) is now dimW_2;i A²_i ˆK_SA_i‡2. Wehave D_iA_iˆ 1 and (D_i A_i)D_ideg (L₂ K_S)j_{Di Ai}K_S(D_i A_i)0. Since 0>D²_i ˆ(D_i A_i)D_i‡A_iD_iK_S(D_i A_i) 1 1, weget D²_i ˆ 1.

Then the equalities hold everywhere. Since K_S(D_i A_i)ˆ0,D_i A_icon- sists of ( 2)-curves. Furthermore, we see thatL2 2KS on Di. Wecan show that D_i is 2-connected as in (a). Then h⁰(D_i;ODi)ˆ1 and weget dimW_2;iˆh⁰(D_i;L₂)ˆ(3=2)K_SA_i‡1=2 similarly as in (a). Thedesired inequality dimW_2;iK_SA_i‡2 holds except whenK_SA_iˆ1, that is,D_iis a Francia cycleof arithmetic genus oneon whichL₂2K_S.

Now weexchangetheroÃleof L1, L2, and consider the multiplication mapm_1;i:W_1;iH⁰(A_i;L2 X_{N i})!H⁰(A_i;L1‡L2 X_{N i}) for each in- dexifor which we failed to show the surjectivity ofm_2;i. Sincewecan as- sumeD_1;iˆD_2;iby thefollowing lemma, theassertion follows. p LEMMA2.7. Let S and D be as in the previous theorem. If E is a sub- curve with E²ˆ 1, then the restriction map H⁰(D;L)!H⁰(E;L) is surjective for any line bundle L on D such that L 2K_Sis nef.

PROOF. We assume that the restriction map is not surjective and show that this leads us to a contradiction. Consider the cohomology long exact

sequence for

0! OD E(L E)! OD(L)! OE(L)!0:

SinceH¹(D;L)ˆ0, wemust haveH¹(D E;L E)6ˆ0. By theduality, we getH¹(D E;L E)^_ˆH⁰(D E;D (L K_S)). Lets2H⁰(D E;D

(L KS)) be a non-zero element. If Z is thebiggest subcurveon which s vanishes identically (possibly Zˆ0), then D (L KS) Z is nef on D E Z. In particular, wehave(D Z)(D E Z) deg (L K_S)j_{D E Z}0. Wehave

0(D Z)(D E Z)ˆ(D Z)² E(D Z)ˆ(D E Z)²‡E(D Z) E²: Since(D Z)²0 and (D E Z)²<0, wegetE²E(D Z)0. Then weget (D E Z)²ˆ 1, E(D Z)ˆ0 and (D Z)(D E Z)ˆ0, becauseE²ˆ 1. This implies thatL KSis of degree zero onD E Z and it follows that K_S(D E Z)ˆ0, because L 2K_S is nef. This is

impossiblefor thereason of parity. p

3. General degree bounds.

From now on, Sdenotes a non-singular projective minimal surface of general type. It is known that 2K_Sis free whenp_g(S)>0 orp_gˆ0,K²_S5 (see [7], [22], [2], [3]). In what follows, we always assume that 2K_Sis free.

PutrˆP2(S) 1ˆK_S²‡x(OS) 1. Notethat wehaverˆ1 if and only if pgˆqˆ0, K²_Sˆ1, that is, S is a numerical Godeaux surface. Since we haveassumed that 2K_S is free, we have r2 and therefore exclude Godeaux surfaces from our considerations. We also remark that the image of the bi-canonical map is necessarily a surface when 2K_S is free.

Our main strategy in studyingR(S;KS) is theuseof theKoszul coho- mology. The following two lemmas are easy applications of Theorems 1.1, 1.2 and 1.3, which combined with Lemma 1.4 can cover a major part of Theorem A.

LEMMA3.1. Suppose that2K_Sis free. Then the multiplication map H⁰(S;2KS)H⁰(S;(m 2)KS)!H⁰(S;mKS)

is surjective in the following cases:

(1) m>7.

(2) mˆ7and K_S²‡pg(S) q(S)3.

(3) mˆ6and K_S² q(S)‡2.

(4) mˆ5, q(S)ˆ0and the bi-canonical image of S is not a surface of minimal degree.

PROOF. Since S is a minimal surface of general type, we have H¹(S;iKS)ˆ0 whe n i6ˆ0;1. Of course H¹(S;iKS)ˆ0 holds also for iˆ0;1 whe n S is regular. Put eˆ0;1 according to m is even or odd.

Themultiplication map in question is surjective if and only if K_{0;(m e)=2}(S;eK_S;2K_S;H⁰(S;2K_S))ˆ0. By the proof of Theorem 1.1, the natural map

K_{0;(m e)=2}(S;eK_S;2K_S;H⁰(S;2K_S))

!K_{r 2;3 (m e)=2}(S;(1 e)K_S;2K_S;H⁰(S;2K_S))^_ is injective (resp. surjective) when H¹(S;(m 4)K_S)ˆ0 (re sp.

H¹(S;(m 2)K_S)ˆ0). Therefore, when m6 or mˆ5, q(S)ˆ0, the problem is reduced to showing that the Koszul sequence

^

r 1

H⁰(S;2K_S)H⁰(S;(5 m)K_S)!^r^²

H⁰(S;2K_S)H⁰(S;(7 m)K_S)

!^r^³

H⁰(S;2K_S)H⁰(S;(9 m)K_S) is exact at the middle term. Ifm8, thenH⁰(S;(7 m)KS)ˆ0 and we have (1). If mˆ7, then we have H⁰(S;(5 m)K_S)ˆ0, H⁰(S;(7 m)K_S)'C, and theKoszul map V_{r 2}

H⁰(S;2K_S)!V_{r 3}

H⁰(S;2K_S)H⁰(S;2K_S) is injective whenr 30, which gives (2). Ifmˆ6 andp_g(S)ˆh⁰(S;K_S) r 2, then the exactness follows from Theorem 1.2, which gives (3). As- sumethatmˆ5 andq(S)ˆ0. Recall that thebi-canonical imageis a surface, since2K_Sis free. Then Theorem 1.3 shows that the above sequence is exact at the middle term, unless the bi-canonical image is a surface of minimal degree.

Hence (4). p

Quitesimilarly, wecan show thefollowing:

LEMMA3.2. Assume that2KSis free. Then the Koszul sequence

^²

H⁰(2K_S)H⁰((m 4)K_S)!H⁰(2K_S)H⁰((m 2)K_S)!H⁰(mK_S) is exact at the middle term in the following cases:

(1) m>9;

(2) mˆ9and K_S²‡pg(S) q(S)4;

(3) mˆ8and K_S²q(S)‡3:

Another strategy is the hyperplane section principle. Assume that p_g(S)>0 and letD2 jK_Sjbe a general member. PutLˆK_SjD. Consider the cohomology long exact sequence for

0! O_S(mK_S D)! O_S(mK_S)! O_D(mL)!0

for any non-negative integer m. Since Sis a minimal surface of general type, we have H¹(S;mK_S D)ˆH¹(S;(m 1)K_S)ˆ0 except when q(S)>0 and mˆ1;2. Hence the restriction map H⁰(S;mK_S)!

!H⁰(D;mK_S) is surjective form0 whenq(S)ˆ0 and form6ˆ1;2 when q(S)>0. Consider the homomorphism of gradedC-algebras

R(S;KS)ˆM

m0

H⁰(S;mKS)!R(D;L)ˆM

m0

H⁰(D;mL):

(3:4)

induced by the restriction maps. The kernel is the homogeneous ideal of R(S;K_S) generated by a single element in degree one, that is, a section defining D. Therefore, for example, in order to show that R(S;K_S) is generated in degrees3 and related in degrees6, it is sufficient to show thesamethings for theimageR⁰(D;L) ofR(S;K_S)!R(D;L). SinceDis Gorenstein, Theorem 1.1 works well, when we study R⁰(D;L) via the Koszul cohomology groups.

Now, we are going to show Theorem A. Our basic assumption is that 2K_S is free. Though a considerable part inevitably overlaps with [4] and [24], we do not exclude it for the convenience of readers, unless it involves too much. Among other things, we shall freely use basic inequalities in the surfacegeography:K²_S>0,x(O_S)>0,K²_S2pg(S) 4, andK²_S2pg(S);

whenq(S)>0 (see [6]).

We study the exceptional cases in (2), (3), (4) of Lemma 3.1.

(a) Supposethat K²_S‡p_g qˆ2. Then we have K_S²ˆ2, p_gˆq;

K_S²ˆ1, pgˆq‡1. If q(S)ˆ0, then (pg;K_S²)ˆ(0;2), (1;1). If q(S)>0, thenK_S²ˆ2 andpgˆqˆ1, becauseK_S² 2pg(S) holds for irregular sur- faces of general type.

LEMMA3.3. Assume that(p_g;q;K_S²)ˆ(0;0;2),(1;0;1),(1;1;2). Then the following hold:

(1) corankfH⁰(S;2K_S)H⁰(S;5K_S)!H⁰(S;7K_S)g ˆ1.

(2) corankfH⁰(S;2KS)H⁰(S;4KS)!H⁰(S;6KS)g ˆpg(S).

(3) corankfH⁰(S;2KS)H⁰(S;3KS)!H⁰(S;5KS)g ˆ3q(S).

Furthermore, the Koszul sequence

^²

H⁰(S;2KS)H⁰(S;(m 4)KS)

!H⁰(S;2K_S)H⁰(S;(m 2)K_S)!H⁰(S;mK_S) is exact at the middle term for m8when q(S)ˆ1; for arbitrary m when q(S)ˆ0.

PROOF. Le t

0! E !H⁰(S;2K_S) O_S ! O_S(2K_S)!0

betheexact sequenceobtained by theevaluation map. If m4, then we have H¹(S;(m 2)K_S)ˆ0 and weseethat thecokernel of H⁰(S;2KS) H⁰(S;(m 2)KS)!H⁰(S;mKS) is isomorphic to H¹(S;E O_S((m 2)K_S)) which is dual to H¹(S;E^_ O_S((3 m)K_S)).

Since V₂

E ' O_S( 2K_S), wehaveE^_ O_S((3 m)K_S)' E^_V₂ E O_S((5 m)K_S)' E O_S((5 m)K_S). So, wehaveonly to calculate h¹(S;E O_S((5 m)K_S)). It follows from the exact sequence

H⁰(S;2KS)H⁰(S;(5 m)KS)!H⁰(S;(7 m)KS)!H¹(S;E OS((5 m)KS))

!H⁰(S;2KS)H¹(S;(5 m)KS)!H¹(S;(7 m)KS) that wehave

H¹(S;E OS(5 m)KS)' H⁰(S;(7 m)KS) whenmˆ6;7;

H⁰(S;2KS)H¹(S;OS) whenmˆ5:

(

Hence we get (1)-(3). As to thelast assertion, notethekernel of H⁰(S;2K_S)H⁰(S;(m 2)K_S)!H⁰(S;mK_S)isH⁰(S;E O_S((m 2)K_S)).

SinceV₂

E ' O_S( 2K_S) and wehavetheexact sequence 0!^²

E O_S((m 4)K_S)!^²

H⁰(S;2K_S) O_S((m 4)K_S)

! E OS((m 2)KS)!0;

we have only to check whetherH¹(S;(m 6)K_S) vanishes or not. p WhenK_S²ˆpgˆ1 andqˆ0, it is known [1] that thecanonical model of S is isomorphic to a complete intersection of type (6;6) in theweighted projective spaceP(1;2;2;3;3). Therefore,R(S;K_S) is generated in degrees 3 and related in degrees6. We study the other two cases.

LEMMA 3.4. Assume that K_S²ˆ2and pgˆqˆ0. If2KS is free, then R(S;KS)is generated in degrees4and related in degrees8.

PROOF. Weshall show thatH⁰(S;3K_S)H⁰(S;4K_S)!H⁰(S;7K_S) is surjective. Note that 3K_Sis free (see, e.g., [22]). By Theorem 1.1, it suffices to show that

^⁵

H⁰(S;3KS)!^⁴

H⁰(S;3KS)H⁰(S;3KS)!^³

H⁰(S;3KS)H⁰(S;6KS) is exact at the middle term. By Theorem 1.3, we have the claim unless the tri-canonical imageof S is a surface of degree 5 in P⁶. Sincewehave (3K_S)²ˆ18 which is not a multiple of 5, we are done. This and Lemma 3.3 show thatR(S;K_S) is generated in degrees4. Then puttingdˆ4,d₀ˆ7, d1ˆ0 and kˆ2, Lemma 1.4 shows that it is related in degrees 10ˆ4‡7 1. Sincepgˆ0, we have no generators in degree 1. If there is a relation of degree 9 or 10, then any monomial appearing the relation can be divided by a monomial of degree 5 or 6. Then, using Lemma 3.3, the proof of Lemma 1.4 shows that such a relation can be reduced to relations in

degrees8. p

LEMMA3.5. Assume that K_S² ˆ2, pgˆqˆ1. Then R(S;KS)is gener- ated in degrees5and related in degrees10.

PROOF. Surfaces with such numerical invariants are extensively studied in [12]. We freely use the results there. The Albanese map f :S!BˆAlb(S) is a fibration of genus 2. Then fv_S is an in- decomposable, nef locally free sheaf of rank two and of degree one. Since it has no locally free quotient of degree zero, we have an exact sequence

0! OB!fvS! OB(P)!0;

for a pointP2B. By [12],Sis obtained as the minimal resolution of a finite doublecovering surfaceofPˆP_B(fv_S) which has at most rational double points and with branch locus inj6D 2Gj, whereDis a tautological divisor and Gis thefibreofP!BoverP. HenceKSis induced byD. Furthermore, we havethedecomposition H⁰(S;mK_S)'H⁰(P;mD)H⁰(P;(m 3)D‡G) into the(1)-eigen spaces under theaction of thecovering transformation group.

Wefirst claim that theinvariant partL

m0H⁰(P;mD) is generated in degrees 4. Weshow that H⁰(P;2D)H⁰(P;(m 2)D)!H⁰(P;mD) is surjective for m5. By the duality theorem, it suffices to show that

H¹(P;(6 m)D‡KP)ˆ0. Sincethecanonical bundleof P is given by 2D‡G, wearedone. As to theanti-invariant part, weconsider

M

m 3 iˆ1

H⁰(P;iD)H⁰(P;(m 3 i)D‡G)!H⁰(P;(m 3)D‡G):

Notethat theimageofH⁰(D)H⁰((m 4)D‡G) consists of thosesections vanishing on D, whilst that of H⁰((m 3)D)H⁰(G) consists of sections vanishing onG. In particular, whenmˆ5, this implies that we need a new generator in degree 5 which does not vanish at D\G. Ifm6, then the imageof H⁰(2D)H⁰((m 5)D‡G) contains a section not vanishing at D\G. This and Lemma 3.3 imply thatR(S;K_S) is generated in degrees5.

It remains to show thatR(S;K_S) is related in degrees10. By Lem- ma 1.4 with dˆ5,d₀ ˆd₁ ˆ7 and kˆ2, we see that it is related in de- grees11ˆ5‡7 1. Assume that we have a relation in degree 11. LetG be a monomial of degree 11 appearing in the relation. Then it is easy to see thatGcan bedivided by a monomialG1of degree 8 or 9. Hence, as in the proof of Lemma 1.4, we can show that any relations in degree 11 are in-

duced by relations in lower degrees. p

(b) Supposethat K_S²q(S)‡1 but K_S²‡pg q3. By Noether's in- equality and the fact that K_S² 2pg holds for irregular surfaces, we get pgˆ2;qˆ0;K²_Sˆ1. It is known [11] that thecanonical model of a surface withp_gˆ2;qˆ0;K_S² ˆ1 is isomorphic to a hypersurface of degree 10 in P(1;1;2;5). In particular,R(S;K_S) is generated in degrees5 and related in degrees10.

(c) Supposethatq(S)ˆ0,K_S²‡pg3,K²_S2 and that thebi-canonical imageofSis a surface of minimal degreer 1 inP^r. By theclassification [18], a surface of minimal degree is either (a)P² (rˆ2), or (b) a quadric surfacein P³ (rˆ3), or (c) the Veronese surface in P⁵ (rˆ5), or (d) a rational normal surfacescroll. In particular, it is ruled by straight lines except when it is the Veronese surface. Ifkdenotes the degree of the bi- canonical map (onto the image), then

4K_S²ˆk(K_S²‡p_g 1):

Wedividethecaseinto sub-cases according tok.

(c1) If kˆ2, then K²_Sˆp_g 1. It follows (K_S²;p_g)ˆ(2;3). Then it is shown in [10] that thecanonical model ofSis isomorphic to a hypersurface of degree 8 inP(1;1;1;4). In particular,R(S;KS) is generated in degrees 4 and related in degrees8.

(c2) Ifkˆ3, then we haveK²_Sˆ3pg 3. Assumethat thebi-canonical image is ruled by straight lines and letDbethepull-back toSof a line.

Then, since the bi-canonical map is of degree 3, we must have 2K_SDˆ3, which is absurd. Assumethat thebi-canonical imageis theVeronese surfaceinP⁵. Then we havep_g(S)ˆ2,q(S)ˆ0 andK²_Sˆ3. Furthermore, wehavea divisorHwith 2KSˆ2Hcoming from a lineonP². Notethat we have H6ˆKS, because h⁰(S;KS)ˆ2 but h⁰(S;H)h⁰(P²;OP(1))ˆ3. In particular,Sis not simply connected and the unramified double coveringS~ ofSassociated to the 2-torsion divisorK_S His a surfacewithp_gˆ5 and K²ˆ6. Now,S~is on the Noether line and, therefore, has a unique pencil of curves of genus 2 (see, [10]). By the uniqueness, the pencil is preserved by theaction of thecovering transformation group ofS~ !S. This implies that Salso has a pencil of curves of genus 2. Then the bi-canonical map of S cannot be of odd degree, since it factors through the relative bi-canonical map which is of degree 2. Hence such a surface does not exist.

(c3) If kˆ4, then we have p_g ˆ1. Weassumethat K_S² 5 and show that this leads us to a contradiction. We haver6 and it follows that the bicanonical image is ruled by lines. Then we have a penciljDjof curves with 2KSDˆ4, that is, KSDˆ2. Hodge's index theorem shows that 4ˆ(K_SD)²K²_SD². SinceK_S²5, wegetD²ˆ0. ThenjDjis a pencil of curves of genus 2. This is impossible by Xiao's theorem [23, TheÂoreÁme5.6]

which in particular implies thatSdoes not have a pencil of curves of genus 2 whenK_S²5 and the degree of the bi-canonical map is 4. HenceK_S²4.

LEMMA 3.6. Let S be a minimal regular surface of general type with p_g(S)ˆ1,2K_S²4. Then R(S;K_S)is generated in degrees4and re- lated in degrees8.

PROOF. Sincepgˆ1, wehavea uniquecanonical curveD2 jK_Sj. We show that themultiplication map H⁰(S;2K_S)H⁰(S;3K_S)L

H⁰(S;K_S) H⁰(S;4K_S)!H⁰(S;5K_S) is surjective. For this purpose, it is sufficient to show thatH⁰(D;2L)H⁰(D;3L)!H⁰(D;5L) is surjective, whereLˆKSj_D (see the argument around (3.4)). By Theorem 1.1, it is surjective when

K^_S² 1

H⁰(D;2L)H⁰(D;L)!^K^^{2S 2}

H⁰(D;2L)H⁰(D;3L)

isinjective. Sincepgˆ1 andqˆ0, wehaveh⁰(D;L)ˆ0. Therefore,R(S;KS) is generated in degrees4.

In order to see thatR(D;L) (hence alsoR(S;K_S)) is related in degrees