Maximal Mordell-Weil lattices of fibred surfaces with <span class="mathjax-formula">$p_g=q=0$</span>

Maximal Mordell-Weil lattices of fibred surfaces with p

ˆ q ˆ 0.

SHINYAKITAGAWA(*)

Dedicated to Professor Yoshihara Hisao on his sixtieth birthday

ABSTRACT -Slope inequalities are given for fibred regular surfaces with geometric genus zero according as the Clifford index of a general fibre.

In the case where the slopes attain the minimums, such fibred surfaces are rational, whose constructions are described. Furthermore, the max- imal Mordell-Weil lattices of the fibred rational surfaces are completely determined.

1. Introduction.

We shall work over the complex number field C. LetX be a smooth projective surface withpgˆqˆ0, wherepgandqrespectively denote as usual the geometric genus and the irregularity ofX. Letf :X!P¹ be a relatively minimal fibration whose general fibreFis a smooth projective curve of genusgand of Clifford indexc. After Shioda ([13]) introduced and developed the theory of the Mordell-Weil lattice, several attempts have been made to clarify the Mordell-Weil lattices for higher genus fibrations.

For example, the cases where cˆ0;1 and 2 are respectively studied in [12], [11] and [8], where the maximal Mordell-Weil lattices are completely determined (see also [7]). The present article is an extension of them.

THEOREM1.1. Keep the same notation as above. Assume that c3.

Let r be the Mordell-Weil rank of f :X!P¹. Then the following hold:

(*) Indirizzo dell'A.: Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-16, Toyonaka, Osaka 560-0043, Japan.

E-mail: [email protected]

Supported by The 21st Century COE Program named ``Towards a new basic science: depth and synthesis.

(1) When g>(c‡2)(c‡3)=2,

r2(c‡2)(g‡c‡1)=(c‡1):

If the right-hand side is an integer, then there exists f which attains the equality except when c is odd, g is a multiple of(c‡1)and g<(c‡1)².

(2) When(c‡1)(c‡2)=2g(c‡2)(c‡3)=2, r3g (c‡3)(c 4)=2:

For all g, there exists f which attains the equality.

Furthermore, in both cases, X is a rational surface if r attains the above maximum.

We consider fibred rational surfaces whose Mordell-Weil ranks attain the maximums as in Theorem 1.1. Then we have an explicit description of the NeÂron-Severi group NS(X). Thus, for such fibrations, the Mordell-Weil lattices are completely determined and the corresponding Dynkin dia- grams are expressed in terms ofcandg. In particular, for the case (1), they are certain extensions ofE8(see Theorem 3.7).

In order to show Theorem 1.1, we apply the same method as in [8], where we take a reduction (Y;G) of (X;F) and give a lower bound for (K_Y‡G)². In Theorem 2.3, we have two slope inequalities, which give upper bounds of the Mordell-Weil rank as in Theorem 1.1. Ifrattains the bound, then any fibres off must be irreducible. In the first of §3, we show such fibrations are indeed constructed from (Y;G) with the minimal (K_Y‡G)²under the suitable conditions.

Wheng<(c‡1)(c‡2)=2, which Theorem 1.1 does not cover, we have a loose bound r((2c‡8)g‡6c‡8)=(c‡2) in the same way as in Theorem 1.1. The proof of Theorem 2.3 shows that the case where g<(c‡1)(c‡2)=2 is complicated. Ifc4 andgˆ(c‡1)(c‡2)=2, then there in fact exist two kinds of fibred rational surfaces with the minimal slopes whose structures are distinct for the plane curve models ofF. One of them is a plane curve of degree (c‡4) having (c‡2) double points as its singularity. AnotherGis near to a multi-anti-canonical divisor onYwhich is obtained by blowingP²up at five points.

Acknowledgments. The author would like to express his heartfelt gratitude to Professor Kazuhiro Konno for his valuable advice, guidance and encouragement. Thanks are also due to Professors Hisashi Usui and Viet Nguyen Khac for interesting discussions. The author thanks Pro- fessor Atsushi Noma for sending me a preprint [10].

2. Slope inequalities

LetXbe a smooth projective surface withpgˆqˆ0 andf :X!P¹a relatively minimal fibration of curves of genus g2. We briefly review basic notation and results of such fibred surfaces while considering the adjoint bundle KX‡F according to [8, §1]. It is known that K_X=P¹ˆKX‡2F is nef and we have (KX‡2F):Cˆ0 for an irreducible curve C onX if and only ifC is a ( 2)-curve contained in a fibre off. Since p_gˆqˆ0, we have that K_X‡FˆK_X=P¹ F is also nef and h⁰(X;K_X‡F)ˆg. Remark that X is automatically a rational surface when (K_X‡F)²<2g 2 (see [8, Lemma 1.1]).

LEMMA 2.1 [cf. [8, Lemma 1.2]]. Let C be an irreducible curve on X such that(KX‡F):Cˆ0. If (KX‡F)²>0, then C is a smooth rational curve satisfying one of the following:

(1) C is a( 2)-curve contained in a fibre.

(2) C is a( 1)-section, i.e., a( 1)-curve with F:Cˆ1.

Suppose that there exists a ( 1)-curveEwith (K_X‡F):Eˆ0 and let m₁:X!X1 be its contraction. SinceF:Eˆ1,F1:ˆ(m₁)F is smooth on X1. Furthermore, we havem₁(KX1‡F1)ˆKX‡F. If there exists a ( 1)- curveE₁ with (K_X1‡F₁):E₁ˆ0, then, by contracting it, we get the pair (X₂;F₂) withF₂ smooth andK_X2‡F₂ pulls back toK_X‡F. We can con- tinue the procedure until we arrive at a pair (X_n;F_n) such that we cannot find a ( 1)-curve En with (KXn‡Fn):Enˆ0. We put Y:ˆXn and G:ˆFn. Ifm:X!Ydenotes the natural map, thenm(KY‡G)ˆKX‡F and GˆmF is a smooth curve isomorphic to F. The original fibration f :X!P¹ corresponds to a pencilL_f jGjwith at most simple (but not necessarily transversal) base points. When (K_X ‡F)²>0,K_X‡F is nef and big. This implies that,Y is the minimal resolution of singularities of the surface Proj(R(X;KX ‡F)), which has at most rational double points by Lemma 2.1, where R(X;K_X‡F)ˆL

n0H⁰(X;n(K_X‡F)). Therefore, such a model is uniquely determined. We call the pair (Y;G) thereduction of (X;F).

We consider the rational map F_jKX‡Fj:X!P^{g 1} defined by jK_X‡Fj. If jK_X‡Fj is composed of a pencil, thenf :X!P¹ is a hy- perelliptic fibration (cf. [8, Lemma 1.3]). Furthermore, we have the following:

PROPOSITION 2.2 [cf. [8, Lemma 1.3 and Proposition 1.1]]. Assume

that0<(KX‡F)²2g 5. ThenF_jKX‡Fjdefines a birational morphism

onto the image, which factors through the reduction of (X; F).

Before proceeding further, we recall here two important invariants of a smooth projective curveCof genusg(C)4. The gonality gon(C) ofCis given as the minimum of the degrees of surjective morphisms ofCtoP¹. The existence theorem on special divisors (e.g. [1]) implies that gon(C)(g(C)‡3)=2. The Clifford index ofCis defined as

Cliff(C)ˆminfdegL 2h⁰(L)‡2 jL2Pic(C); h⁰(L)>1; h¹(L)>1g:

These two invariants are closely related to each other. It is known that Cliff(C)ˆgon(C) 2 or gon(C) 3 and in the latter case we have infinitely manyg¹_gon(C)'s onC. For this and further properties, see [2].

We now return to the situation we are interested in. The slope off is defined byl_f ˆK²_X=P1=degfK_X=P1. We have the following lower bounds:

THEOREM2.3. Let X be a smooth projective surface with pgˆqˆ0 and f :X!P¹a relatively minimal fibration of genus g and of Clifford index c3. Let(Y;G)denote the reduction of(X;F). Then the following hold:

When g>(c‡2)(c‡3)=2;

l_f 5‡(c 1)=(c‡1) (2c‡4)=g:

(2:1)

Ifl_f attains the equality in(2:1), then Y is the Hirzebruch surfaceS_dof degree d.

When(c‡1)(c‡2)=2g(c‡2)(c‡3)=2, l_f 5‡(c‡3)(c 4)=(2g):

(2:2)

If l_f attains the equality in (2:2), then there exists a blow-down y₀:(Y;G)!(P²;G₀)such that G₀is one of the following:

(i) a curve of degree(3c‡6)=2with four(c‡2)=2-ple points and a c=2-ple point, where c is even and gˆ(c‡1)(c‡2)=2,

(ii) a curve of degree(3c‡5)=2with five(c‡1)=2-ple points, where c is odd, c5and gˆ(c‡1)(c‡2)=2,

(iii) a curve of degree nine with four triple points, where cˆ4and gˆ16,

(iv) a curve of degree (c‡4) with (n‡1) double points, where gˆ(c‡1)(c‡4)=2 n and 1nc‡1.

In particular, X is a rational surface iflf attains the equality in(2:1) or(2:2).

In order to show Theorem 2.3, from gl_f ˆ(K_X‡F)²‡4(g 1) and Proposition 2.2, we only have to prove the following:

PROPOSITION2.4. LetY be a smooth rational surface andGa smooth irreducible curve of genusgand of Clifford indexc3onY. Assume that (K_Y ‡G):E1for any( 1)-curvesEonY. Then the following hold:

When g>(c‡2)(c‡3)=2;

(KY‡G)²2c(g c 1)=(c‡1):

If the equality sign holds, then Y ˆS_d. When(c‡1)(c‡2)=2g(c‡2)(c‡3)=2;

(K_Y‡G)²g‡(c² c 4)=2:

If(KY‡G)²attains the equality, then there exists a blow-downy0:(Y;G)!

!(P²;G0)such that G0is a curve with the same degree and singularity as in (i)-(iv)ofTheorem 2:3.

First of all, in the caseY ˆP², it is well-known thatGis a smooth plane curve of degree (c‡4). Therefore, we have gˆ(c‡2)(c‡3)=2 and (KY‡G)²ˆ(c‡1)²ˆg‡(c² c 4)=2. Next, in the case YˆS_d, we describe explicitly as follows:

LEMMA 2.5. Let G be a smooth irreducible curve of genus g and of Clifford index c3onS_d. LetD0be a section withD²₀ˆ d andGa fibre of S_d. Assume thatD₀:G2when dˆ1andD₀:GG:G when dˆ0. Then

G(c‡2)D0‡((c‡2)d=2‡1‡g=(c‡1))G;

where the symbolmeans the linear equivalence of divisors. Furthermore, (KSd‡G)²ˆ2c(g c 1)=(c‡1) holds and (c;g;d) satisfies one of the following:

(i) c is even, g(c‡1)², g is divided by (c‡1) and 0d 2(g‡c‡1)=((c‡1)(c‡2)).

(ii) c is odd, g(c‡1)², g is divided by (c‡1)=2 and 0d 2(g‡c‡1)=((c‡1)(c‡2))with d2g=(c‡1) mod 2.

(iii) c is even,(c‡1)(c‡4)=2gc(c‡1), g is divided by(c‡1) and dˆ1.

(iv) c is odd, (c‡1)(c‡4)=2g(c‡1)(2c‡1)=2, g is an odd multiple of(c‡1)=2and dˆ1.

PROOF. Under the assumptions,G:Gˆgon(G)ˆc‡2 holds (see [9]).

Remark thatD₀:G0 sinceGis irreducible. The rest of statements follow

from a standard calculation. p

LetVbe a smooth rational surface which is neitherP²norS_dandDa smooth irreducible curve of genus g7 onV. Assume that Cliff(D)3 and (K_V‡D):E1 for any ( 1)-curvesEonV. Then we can find at least one base-point-free pencil of rational curves onV. We choose among them a pencil jRj of rational curves with R²ˆ0 in such a way that a:ˆ(K_V‡D):Ris minimal. We callatheminimal ruling degree of(V;D).

Note that we haveD:Rˆa‡25 sinceK_V:Rˆ 2. Letc:V!P¹be the morphism defined byjRj. We take a relatively minimal model ofVwith respect tocand consider the image ofD. Then we perform a succession of elementary transformations ([5]) at singular points of the image curve to arrive at a particular relatively minimal model (V^#;D^#), called a #-mini- mal model in [6], enjoying several nice properties which we collect below.

The natural mapy:(V;D)!(V^#;D^#) is a minimal succession of blowing- ups which resolves the singular points ofD^#. We assume thatV^#'S_dand D^#(a‡2)D0‡bG. Let pi, 1in, be the singular points ofD^# in- cluding infinitely near ones, and let m_i be the multiplicity of D^# atp_i. Assume for simplicity thatm₁m₂ m_n 2. SincejRjis chosen so that (K_V‡D):Ris minimal, we can assume that the following are satisfied (see [5] and [6]):

(#1)b(a‡2)dwhend>0, andba‡2 whendˆ0, (#2)ba‡2‡m1whendˆ1,

(#3)m1(a‡2)=2 andm1minf(a‡2)=2; b (a‡2)gwhendˆ1.

We say that (V^#;D^#) is of general type if 2b (a‡2)d2(a‡2).

Otherwise, i.e., whendˆ1 and 2b<3(a‡2), the pair is called ofspecial type. If this is the case, by contracting the minimal section, we get a model ofDwhich is a plane curve of degreebwith a (b a 2)-ple point andn other singular points of respective multiplicitiesmi(b a 2).

At first, consider the case where (V^#;D^#) is of general type. Set b⁰ˆb (d‡2)(a‡2)=2, where b⁰maxf0;(d 2)(a‡2)=2g, and where b⁰2Zorb⁰2Z[1=2] according asais even or odd. Then we have

D^#(a‡2)D₀‡(b⁰‡(d‡2)(a‡2)=2)G:

(2:3)

LEMMA 2.6. Keep the notation and assumptions as above. Then the following hold:

When g(a‡1)²,

(K_V‡D)²>a(g a 1)=(a‡1):

If a is even and g<(a‡1)², then

(KV‡D)² 2a(g 1)=(a‡2):

When the equality sign holds here, b⁰ˆ0and m_nˆ(a‡2)=2.

If a is odd and g<(a‡1)², then

(KV ‡D)²2g(a 1)=(a‡1)‡2;

When the equality sign holds, b⁰ˆ0and mn ˆ(a‡1)=2.

PROOF. From a standard calculation, we have (K_V‡D)² ˆ

ˆ2a(a‡b⁰) Pⁿ

iˆ1(m_i 1)²and

2gˆ2(a‡1)(a‡1‡b⁰) Xⁿ

iˆ1

m_i(m_i 1):

(2:4)

Therefore, we have (K_V ‡D)² 2(m₁ 1)

m₁ (g (a‡1)(a‡1‡b⁰)) 2a(a‡b⁰)

ˆXⁿ

iˆ1

(m_i 1) 1 m_i m₁

;

which is non-negative. Hence (K_V‡D)²2(m₁ 1)

m₁ (g (a‡1)(a‡1‡b⁰))‡2a(a‡b⁰):

(2:5)

If the equality sign holds here, thenm₁ˆ ˆm_n. On the other hand, 2(m1 1)

m₁ (g (a‡1)(a‡1‡b⁰))‡2a(a‡b⁰) 2a

a‡1g 2a

ˆ2(m1 a 1)(g (a‡1)(a‡1‡b⁰)) m1(a‡1)

is positive from (#3) and (2:4). Thus (K_V ‡D)²>2a(g a 1)=(a‡1) holds.

In the rest of the proof, we assume thatg<(a‡1)². We restrict our- selves to the case wherea is even, since the other case is quite similar.

Then (#3) and (2:4) imply that 2(m1 1)

m₁ (g (a‡1)(a‡1‡b⁰))‡2a(a‡b⁰) 2a

a‡2(g‡b⁰ 1)

ˆ2(2m₁ a 2)(g (a‡1)(a‡1‡b⁰)) m1(a‡2)

is non-negative. Therefore we have

(KV ‡D)²2a(g‡b⁰ 1)=(a‡2) (2:6)

from (2:5). Thus (KV ‡D)²2a(g 1)=(a‡2). If the equality sign holds,

thenb⁰ˆ0 andm1ˆ ˆmnˆ(a‡2)=2. p

Next, we consider the case where (V^#;D^#) is of special type. Let m0ˆb (a‡2). Then we have

D^#(a‡2)D0‡(a‡2‡m0)G; 2m0<(a‡2)=2:

(2:7)

LEMMA 2.7. Keep the notation and assumptions as above. Then the following hold:

When(a‡1)(a‡4)=2ga(a‡1)or(a‡1)(2a‡1)=2according as a is even or odd,

(KV‡D)²>2a(g a 1)=(a‡1):

If a(a‡3)=2g<(a‡1)(a‡4)=2and a is even, then (KV‡D)²g‡(a² a 4)=2:

When the equality sign holds, (m0;n)ˆ(2;(a‡1)(a‡4)=2 g) or (a;n;m5)ˆ(6;5;3).

If g<a(a‡3)=2and a is even, then

(K_V‡D)²2g(a 2)=a‡4:

When the equality sign holds, m_nˆa=2and n>4‡4=(a 2).

If(a‡1)(a‡2)=2g<(a‡1)(a‡4)=2and a is odd, then (K_V‡D)²g‡(a² a 4)=2:

When the equality sign holds, (m₀;n)ˆ(2;(a‡1)(a‡4)=2 g) or (n;m₄)ˆ(4;(a‡1)=2).

If g<(a‡1)(a‡2)=2and a is odd, then

(KV ‡D)²2g(a 1)=(a‡1)‡1:

When the equality sign holds, mnˆ(a‡1)=2and n>4.

PROOF. From a standard calculation, we have (KV‡D)² ˆ a²‡2(m0 1)a Pⁿ

iˆ1(m_i 1)²; 2g ˆ (a‡1)(a‡2m₀) Pⁿ

iˆ1m_i(m_i 1):

(2:8)

In the same way of (2:5), we have (K_V‡D)²2(m₀ 1)

m0 g‡a²‡(1 m₀)a 2m²₀‡2m₀

m0 :

(2:9)

If the equality sign holds here, thenm0ˆm1ˆ ˆmn. By an argument similar to Lemma 2.6, we have (KV ‡D)²>2a(g a 1)=(a‡1).

Wheng(a²‡a‡4m0)=2, 2(m0 1)

m₀ g‡a²‡(1 m0)a 2m²₀‡2m0

m₀ g‡a² a 4

2

(2:10)

ˆ(2g a² a 4m0)(m0 2) 2m₀

is non-negative. Remark that (a²‡a‡4m₀)=2 increases as m₀ grows.

Consider the case where ga(a‡3)=2 or (a‡1)(a‡2)=2 according as a is even or odd. From (2:10), (#3) and (2:9), we have (K_V ‡D)²g‡(a² a 4)=2. Furthermore, ifm0ˆ2, then (K_V‡D)² ˆ g‡(a² a 4)=2 and nˆ(a‡1)(a‡4)=2 g. When m₀6ˆ2 and (K_V ‡D)²ˆg‡(a² a 4)=2 hold, m₀ˆm₁ˆ ˆm_n and 2gˆ

ˆa²‡a‡4m0. Hence, then, we havem0ˆ ˆmnˆa=2 or (a‡1)=2 sincega(a‡3)=2 or (a‡1)(a‡2)=2 according to the parity ofa. These and (2:8) implynˆ4‡4=(a 2) or 4 according asais even or odd.

In the rest of the proof, we may consider the case whereg<a(a‡3)=2 andais even, since the other case is similar. Recall thatm₀a=2. When ga(a‡1‡m₀)=2,

2(m₀ 1)

m0 g‡a²‡(1 m₀)a 2m²₀‡2m₀ m0

2a 4 a g‡4

ˆ(2g a(a‡1‡m₀))(2m₀ a) am₀

is non-negative. Sincea(a‡3)=2a(a‡1‡m0)=2 for anym0, it follows from (2:9) that (KV‡D)²2g(a 2)=a‡4. If (KV‡D)² attains the equality, thenm₀ˆ ˆmnand 2m₀ aˆ0. p

As a corollary of Lemmas 2.6 and 2.7, we have the following.

LEMMA2.8. Let V be a smooth rational surface which is neitherP²nor S_d and D a smooth irreducible curve of genus g7on V. Assume that (K_V‡D):E1for any( 1)-curves E on V and the minimal ruling de- gree of(V;D)is a3. Then the following hold:

When g(a‡1)(a‡4)=2,

(KV‡D)²>2a(g a 1)=(a‡1):

When(a‡1)(a‡2)=2g<(a‡1)(a‡4)=2, (K_V‡D)²g‡(a² a 4)=2:

If (KV‡D)² attains the equality, then there exists a blow-down y0:V !P²contracting( 1)-curves E_isuch that D is one of the following:

(i) D((3a‡6)=2)y₀O_P²(1) ((a‡2)=2)P³

iˆ0E_i (a=2)E4, where a is even and gˆ(a‡1)(a‡2)=2,

(ii) D((3a‡5)=2)y₀O_P²(1) ((a‡1)=2)P⁴

iˆ0E_i, where a is odd, a5and gˆ(a‡1)(a‡2)=2,

(iii) D9y₀O_P²(1) 3P³

iˆ0E_i, where aˆ4and gˆ16, (iv) D(a‡4)y₀O_P²(1) 2Pⁿ

iˆ0E_i, where nˆ(a‡1)(a‡4)=2 g) and(a‡1)(a‡2)=2g<(a‡1)(a‡4)=2:

Furthermore,Cliff(D)ˆa if(V;D)is one of them.

When g<(a‡1)(a‡2)=2,

(KV‡D)² 2a(g 1)=(a‡2) if a is even;

2g(a 1)=(a‡1)‡1 otherwise:

( (2:11)

PROOF. Wheng(a‡1)(a‡4)=2, from Lemmas 2.6 and 2.7, we have (K_V‡D)² >2a(g a 1)=(a‡1). When g<(a‡1)(a‡2)=2, the in- equalities (2:11) also follow immediately. Suppose that (V;D) satisfies one of (i)-(iv). Then we have Cliff(D)ˆafrom [8, Proposition 2.2], sincea3 andD²>(a‡2)².

For the case where (a‡1)(a‡2)=2g<(a‡1)(a‡4)=2, we may restrict ourselves to the case whereais even, since the other case is similar and simpler. Wheng(a²‡3a‡4)=2, we have

2a(g 1)=(a‡2)g‡(a² a 4)=2;

where the equality sign holds if and only if gˆ(a²‡3a‡4)=2. Thus Lemmas 2.6 and 2.7 imply the statement for g>(a²‡3a‡4)=2. Fur- thermore, if

(g;(K_V‡D)²)ˆ((a²‡3a‡4)=2;a²‡a); ((a‡1)(a‡2)=2;a²‡a 1) and (V^#;D^#) is of special type, then (m0;n)ˆ(2;(a‡1)(a‡4)=2 g) from Lemma 2:7.

In what follows, we consider the case where (V^#;D^#) is of general type.

At first, we concentrate on the case where gˆ(a²‡3a‡4)=2. Then we only have to determine the pair (V;D) satisfying (KV‡D)²ˆa²‡a. It follows from Lemma 2.6 that such a pair satisfies b⁰ˆ0 and m_n ˆ(a‡2)=2. Furthermore we havenˆ4 4=afrom (2:4). Thusaˆ4 and nˆ3 follow. Therefore D6yD₀‡(6‡3d)yG 3P³

iˆ1E_i and we haved2 becauseyD0:Dˆ6 3d0. Remark thatD^#has no singular points on the minimal sectionD0ofV^#in the case wheredˆ2. Hence the case wheredˆ1 covers the cases wheredˆ0;2.

We next consider the case where gˆ(a‡1)(a‡2)=2. It follows from (2:6) that

(K_V‡D)²a²‡a‡2a(b⁰ 1)=(a‡2)>a²‡a 1

when b⁰1. Hence we can assume b⁰ˆ0. If D^# has at least four (a‡2)=2-ple points, then (2:4) shows g<(a‡1)(a‡2)=2. In the same way as in Lemma 2.6, we have (K_V‡D)²a²‡a‡2 n⁰ for (V;D) whose D^# has just n⁰ (a‡2)=2-ple points with n⁰ˆ0;1;2;3. If (K_V ‡D)²ˆa²‡a‡2 n⁰ holds, then m_n⁰_‡1ˆ ˆm_nˆa=2. Fur- thermore, (KV‡D)² attains the minimum a²‡a 1 at n⁰ˆ3. In fact, (2:4) and (g;(KV‡D)²)ˆ((a‡1)(a‡2)=2;a²‡a 1) imply thatD^#has exactly three (a‡2)=2-ple points and a a=2-ple point as its singularity.

Thus

D(a‡2)yD₀‡(d‡2)(a‡2)

2 yG a‡2 2

X³

iˆ1

E_i a 2E₄

and we haved2 sinceyD₀:Dˆ(a‡2)(1 d=2)0. We can show that

the case wheredˆ1 covers the cases where dˆ0;2 similarly as in the

previous case. p

Keep the situation as in Lemma 2.8. Furthermore, we consider the case where 4c‡1a2c 2. When g(c‡1)(c‡2)=2, we have (KV‡D)² >g‡(c² c 4)=2 from Lemma 2.8. We also have (KV‡D)² >2c(g c 1)=(c‡1) when

g<(a‡1)(a‡4)=2 (a c)(2c 2 a)=(c 1) org>(c‡1)(a‡1). For (V;D) satisfying

(KV‡D)²2c(g c 1)=(c‡1) (2:12)

and

(a‡1)(a‡4)=2 (a c)(2c 2 a)=(c 1)g(c‡1)(a‡1);

(2:13)

we have the following:

LEMMA 2.9. Keep the notation and assumptions as above. Then Cliff(D)ˆa.

PROOF. Assume that (V;D) satisfies (2:12). Then it follows from Lemma 2.6 that (V;D) has a #-minimal model of special type. We recall (2:7). First of all, we show that (V;D) attains maximum ofnatm₀ˆ2 as follows:Take a smooth irreducible curveB2 jD‡Pⁿ

iˆ1(m_i 2)E_ijfor a fixed (V;D). We consider (V;B) and its#-minimal model (V^#;B^#). Remark that B^# has just n double points as its singularity. Then (KV‡B)²ˆ

ˆ(KV‡D)²‡Pⁿ

iˆ1mi(mi 2) and 2g(B)ˆ2g‡Pⁿ

iˆ1(mi‡1)(mi 2). These and (2:12) imply that

2c

c‡1g(B) 2c (KV‡B)² 1 c‡1

Xⁿ

iˆ1

(c mi)(mi 2):

Sincec(a‡2)=2>m₀m₁ m_n 2, the right-hand side is non- negative. Hence (V;B) also satisfies (K_V‡B)²2c(g(B) c 1)=(c‡1).

From (2:12) and (2:8) with m₁ˆ2, we have n(a c)(2c 2 a‡2(2 m0))=(c 1). In the end, atm0ˆ2,

n(a c)(2c 2 a)=(c 1):

(2:14)

We now return to the proof of Lemma 2.9. We have D²4‡2c‡2g 2g=(c‡1) n from (2:12) and (KV‡D)² ˆ

ˆ2(K_V‡D):D‡K_V² D²ˆ4g 4‡8 n D². Therefore (2:13) and (2:14) imply that

D² (a‡2)²4(a‡2 (a c)(2c 2 a)=(c 1))>0:

Hence Lemma 2.9 follows from [8, Proposition 2.2]. p PROOF OFPROPOSITION2.4. By Lemma 2.5 and the argument just before Lemma 2.5, we can assume thatYis a smooth rational surface which is nei- therP²norS_d. Now the assumption implies that the minimal ruling degreea of (Y;G) is greater than or equal toc. From Lemma 2.9, in the case where a>c, we have (K_Y‡G)²>maxfg‡(c² c 4)=2;2c(g c 1)=(c‡1)g when g(c‡1)(c‡2)=2. For the case where aˆc, Lemma 2.8 implies

Proposition 2.4. p

3. Mordell-Weil lattices

We briefly review the theory of the Mordell-Weil lattice due to Shioda.

Let X be a smooth projective surface withpgˆqˆ0 andf :X!P¹ a relatively minimal fibration of genusg. We assume that it has a section. Via f, we can regardXas a smooth projective curve of genusgdefined over the rational function field KˆC(P¹). Let J_F be the Jacobian variety of a generic fibreF off. The Mordell-Weil group offis the group ofK-rational pointsJ_F(K). It is a finitely generated abelian group, sinceXis a regular surface. The rank of the group, which we denote byr, is called theMordell- Weil rank. It follows from [13, Theorem 3] thatris given by

rˆr(X) 2 X

t2P¹

(v_t 1);

wherer(X) denotes the Picard number, that is, the rank of NS(X), andvt

denotes the number of irreducible components of the fibref ¹(t). In par- ticular, we haverˆr(X) 2 iff has irreducible fibres only.

There exists a natural one-to-one correspondence between the set ofK- rational pointsF (K) and the set of sections off. ForP2F (K) we denote by (P) the section corresponding toPwhich is regarded as a curve onX.

We specify a section (O) corresponding to the origin ofJ_F(K) and call it thezero section. Shioda's main idea in [13] is to regard J_F(K) as a Eu- clidean lattice endowed with a natural pairing induced by the intersection

form onH²(X). LetTbe the subgroup of NS(X) generated by (O) and all the irreducible components of fibres off. With respect to the intersection pairing, the sublatticeTis called thetrivial sublattice and its orthogonal complement LˆT^?NS(X) is called the essential sublattice. Via the natural isomorphism of groupsJ_F(K)'NS(X)=Tin [13, Theorem 3], we obtain a symmetric bilinear formh;ionJ_F(K) which induces the struc- ture of a positive-definite lattice onJ_F(K)=J_F(K)_tor (see [13, Theorem 7]). The lattice (J_F(K)=J_F(K)_tor;h;i) is called theMordell-Weil latticeof the fibrationf :X!P¹. Shioda shows that if all the fibres off are irre- ducible, then the Mordell-Weil lattice off is isomorphic toL , where the opposite lattice L is defined from L by putting the minus sign on the intersection pairing onL(see [13, Theorems 3 and 8]).

As to fibred surfaces withpgˆqˆ0 of Clifford indexc, we can bound the Mordell-Weil rank by Theorem 2.3 in the same way as in [8, Lem- ma 3.1] (see also [12, Theorem 2.8], [11, Proposition 2.2] and [7, Theo- rem 3.4]):

LEMMA 3.1. Let X be a smooth projective surface with pgˆqˆ0, f :X!P¹a relatively minimal fibration of genus g and of Clifford index c3. Let r be the Mordell-Weil rank of f . Then the following hold:

(1) r2(c‡2)(g‡c‡1)=(c‡1)when g>(c‡2)(c‡3)=2.

(2) r3g (c‡3)(c 4)=2when(c‡1)(c‡2)=2g(c‡2)(c‡3)=2.

If r attains the maximum, then all fibres of f are irreducible and the reduction of(X;F)is obtained by F_jKX‡Fj. In particular, X is a rational surface when r attains the maximum.

In what follows, we restrict ourselves to the case when r attains the maximum. Namely, we letf :X!P¹be a fibred rational surface of genus g and of Clifford index c3 whose Mordell-Weil rank attains the max- imum value given in Lemma 3.1. Suppose that we are given such a fibra- tionf :X!P¹and let (Y;G) be the reduction of (X;F). Thenf gives us a pencilL_f jGjall of whose members must be irreducible. Note also that all the base points of Lf must be transversal ones since K_X=P¹ is ample.

Conversely, if we are given a pencil L jGj which has only transversal base points and all fo whose members are irreducible, then we get a fi- bration with the desired properties by blowing up BsL.

Let the situation be as in (1) of Lemma 3.1. Then Y ˆS_d from Theorem 2.3. We show that the case (c‡1)(c‡2)dˆ2(g‡c‡1) is impossible. This can be seen as follows:If (c‡1)(c‡2)dˆ2(g‡c‡1),

then D0:Gˆ0. Hence the pull-back to X of D0 is a rational curve con- tained a fibre of f, which is inadequate in our situation. Therefore, we have (c‡1)(c‡2)d<2(g‡c‡1).

Let (S_d;G) be as in Lemma 2.5 with g>(c‡2)(c‡3)=2 and (c‡1)(c‡2)d<2(g‡c‡1). ThenGis very ample. Hence we can find a pencilL jGjwhose members are all irreducible and which has exactlyG² transversal base points. In fact, we can take it as a Lefschetz pencil (see [4]). Then the fibration f :X!P¹ obtained by Lhas Mordell-Weil rank which attains the equality as in (1) of Lemma 3.1.

DEFINITION 3.2. Let (c;g;d) be as in (i)-(iv) of Lemma 2:5 with g>(c‡2)(c‡3)=2 and d<2(g‡c‡1)=((c‡1)(c‡2)). A fibration f :X!P¹of genusgand of Clifford indexc3 obtained by blowing upS_d as above is called a fibration of type (c;g;d;0).

Let the situation be as in (2) of Lemma 3.1. From Theorem 2.3, there exists the blow-down y₀:(Y;G)!(P²;G₀) constracting (n‡1) ( 1)- curves E₀;E₁;. . .;E_n such thatG₀ is as in (i)-(iv) of Theorem 2.3, where nˆ4 for (i) or (ii), and where nˆ3 for (iii). Putp_jˆy₀(E_j). Ifp₁ is an infinitely near point ofp0, then the pull-back toXofE0 E1is a rational curve contained in a fibre offfrom (E0 E1):Gˆ0, which contradicts the irreducibility off. Furthermore, in the quite same argument, we have the following:

PROPOSITION3.3. Keep the notation and assumptions as above. If

n1, then a configuration of (n‡1) points p0;. . .; pn satisfies two

conditions as follows: The first condition is that p₀;. . .; p_n expect the

c=2-ple point p₄ as in (i) of Theorem 2.3 are not infinitely near

points. For each case of (i)-(iv), the second condition is the following:

(i) Any three points of p0;. . .;p3are not colinear, though p4may be an infinitely near point.

(ii) Any four points of p₀;. . .;p₄are not colinear.

(iii) p₀;. . .;p₃ are in general position, that is, any three points of them are not colinear.

(iv) Any (c‡4)=2or(c‡5)=2points of p0;. . .;pn are not colinear according as c is even or odd.

We use the following theorem in order to show the next lemma.

THEOREM 3.4 ([10]). Let SP^{deg (S) g(S)‡1} be a nondegenerate, line- arly normal, smooth regular surface of degree deg (S) and of sectional genus g(S). Let x₀;. . .;x_k be distinct (k‡1) points of S for kdeg (S) 2g(S) 2. Let s be a blowing up at x₀;. . .;x_k of S and E_jˆs ¹(x_j). Set HˆsO_S(1) P^k

iˆ0E_i. Then H is very ample if and only if for all j with1jk‡1, any distinct j points offx0;. . .;x_kgdo not lie on any irreducible reduced curve of degree j on S.

LEMMA3.5. Let D0 be a plane curve with the same degree and singu- larity as in(i)-(iv) of Theorem2:3. Assume that a configuration of sin- gular points p0;. . .;pn of D0 satisfies the two conditions as in Proposi- tion3.3. Let y₀:(V;D)!(P²;D₀) denote the composite of blow-ups at n‡1points p₀;. . .;p_n. Then D is very ample.

PROOF. PutEjˆy₀¹(pj). At first, we consider the case (i). We remark thatj K_Vjis free from base points, becausep0;. . .;p4are in almost gen- eral position (see [3, Theorem 1 of III in p. 39]). Consider the surface ob- tained by blowingP²up atp₀;. . .;p₃in general position. Then the anti-bi- canonical map is an embedding toP¹⁵. LetSbe the embedded surface toP¹⁵ ands:V!Sthe blow-down contractingE4. Since no line is onS,

Hˆs( 2KS) E4ˆ6y₀O_P²(1) 2X³

iˆ0

Ei E4

is very ample from Theorem 3.4. Therefore DˆH‡(c=2 1)( KV) is also. We can treat the case (ii) in the same way by setting Hˆ4y₀O_P²(1) P⁴

iˆ0E_i. For the case (iii), it is obvious.

Let the situation be as in (iv). Since the case wherecis even is similar and simpler, we restrict ourselves to the case wherecis odd. LetSbeP² embedded by jO_P²((c‡5)=2)j. Then irreducible reduced curves whose degree is at most (n‡1) come from only lines on P². Therefore Theo- rem 3.4 implies thatHˆy₀O_P²((c‡5)=2) Pⁿ

iˆ0E_iis very ample. Hence we only have to show that

jD Hj ˆ c‡3

2 y₀O_P²(1) Xⁿ

iˆ0

E_i

is free from base points. If any (c‡3)=2 points of p₀;. . .;p_n are not co- linear, then we can see similarly thatD His also very ample by applying

Theorem 3.4. Thus we suppose thatp0;. . .;p_(c‡1)=2lie on a linel0;...;(c‡1)=2. Furthermore, we concentrate on the case wherenˆc‡1, since the other cases are similar and simpler. Remark that any p_i, (c‡3)=2ic‡1 are not lie on l0;...;(c‡1)=2 from the second condition (iv) in Proposition 3.3.

Letl_ibe a line throughp_i. In particular, we denote the line throughp_iand p_i‡(c‡1)=2 byli;i‡(c‡1)=2, 1i(c‡1)=2. Consider the strict transform of l0‡^(c‡1)=2P

iˆ1 li;i‡(c‡1)=2

by y0. It is disjoint from the strict transform of l0;...;(c‡1)=2ifl₀is distinct froml0;...;(c‡1)=2. Furthermore,jD Hjis free from base points on E₀ since we can choosel₀ for any directions at p₀. In the same way, jD Hj is also on the whole of y0l0;...;(c‡1)=2. Next, consider l0;...;(c‡1)=2‡ ^c‡1P

iˆ(c‡3)=2li. Then jD Hj is free from base points on Vny0l0;...;(c‡1)=2 since we can chooselifor any directions atpi. p From Lemma 3.5, for any cases as in Proposition 3.3, we can show the existence of the fibrations similarly as in the previous case. Hence we can take a pencilL jDjenjoying the desired properties, as a Lefschetz pencil for example. In particular, L has (3g (c‡3)(c 4)=2 n) transversal base points.

DEFINITION 3.6. Let (V;D) be as in Lemma 3:5. If V is obtained by blowing up n‡1 points ofP², the corresponding fibrationf :X!P¹ of genus g and of Clifford index c as above is called a fibration of type (c;g;1;n).

3.1 ±Type(c;g;d;0)

Here we determine the Mordell-Weil lattices for fibrationsf :X!P¹ of type (c;g;d;0). For this purpose, we use the following notation. We denote the pull-backs toXofD₀andGby the same symbols. Furthermore, we denote bye1;e2;. . .;erthe disjoint ( 1)-sections off coming from the base points of Lf, where rˆ2(c‡2)(g‡c‡1)=(c‡1). Then we have NS(X)'ZD₀ZGL^r

iˆ1Ze_iand Fˆ(c‡2)D₀‡ (c‡2)d

2 ‡1‡ g c‡1

G X^r

iˆ1

e_i: (3:15)

We takeeras the zero section (O). The sublatticeT(c;g;d;0) NS(X) gener- ated by er andF is the trivial sublattice. LetL(c;g;d;0) be the orthogonal complement ofT_(c;g;d;0). Then the Mordell-Weil lattice (J_F(K);h;i) is iso- morphic toL_(c;g;d;0) .

Recall then that the degreedof the Hirzebruch surface is an invariant of the fibration.

THEOREM3.7. For a fibration of type(c;g;d;0), the lattice L_(c;g;d;0) is isomorphic to a positive-definite unimodular lattice of rank rˆ2(c‡2)(g‡c‡1)=(c‡1) whose Dynkin diagram is given by the following:

Figure 1 in the case when(c‡2)d=2 g=(c‡1)0(modc‡2) Figure 2 in the case when(c‡2)d=2 g=(c‡1)1(modc‡2) Figure 3 in the case when(c‡2)d=2 g=(c‡1)1‡`(modc‡2)with 1`c 1,

Figure 4 in the case when(c‡2)d=2 g=(c‡1)c‡1(modc‡2)

Fig. 1

Fig. 2

Fig. 3

Fig. 4

In particular, L_(c;g;d;0) depends on only 2g=(c‡1) mod (c‡2) in the case when c is odd, and on the combination of g=(c‡1) mod (c‡2)and the parity of d in the case when c is even. Here the numbers in the circles denote the self-parings of elements, and a line between two circles shows that the paring of the corresponding two elements is equal to( 1). Fur- thermore, L_(c;g;d;0) is an odd lattice in the case when c is odd(see Table1) and the parity of the lattice is the same as that of(d‡g‡1)or of(g‡1) respectively in the case when c2 mod 4 (see Tables 2 and 3) or c0 mod 4 (see Tables4and 5). In particular, even and odd lattices both occur for a fixed g(c‡1)²in the case when c2 mod 4.

Table 1. ±L(c;g;d;0) is an odd lattice in the case whenc1 mod 2.

2g=(c‡1) modc‡2 Dynkin diagram

0 Figure 1

1 Figure 3 with`ˆ(c 1)=2

2 Figure 4

2i‡1 (iˆ1;2;. . .;(c 3)=2) Figure 3 with`ˆ(c 1)=2 i

2j (jˆ2;3;. . .;(c 1)=2) Figure 3 with`ˆc‡1 j

c Figure 2

c‡1 Figure with`ˆ(c‡1)=2

Table 2. ±L(c;g;d;0) is an even lattice in the case whenc2 mod 4:

g=(c‡1) modc‡2 d Dynkin diagram

0 odd Figure 3 with`ˆc=2

1 even Figure 4

h (hˆ2;4;. . .;c=2 1) odd Figure 3 with`ˆc=2 h

i (iˆ3;5;. . .;c=2) even Figure 3 with`ˆc‡1 i

c=2‡1 odd Figure 1

c=2‡1‡j (jˆ1;3;. . .;c=2 2) even Figure 3 with`ˆc=2 j

c=2‡1‡k (kˆ2;4;. . .;c=2 1) odd Figure 3 with`ˆc‡1 k

c‡1 even Figure 2

Table 3. ±L(c;g;d;0) is an odd lattice in the case whenc2 mod 4:

g=(c‡1) modc‡2 d Dynkin diagram

0 even Figure 1

h (hˆ1;3;. . .;c=2 2) odd Figure 3 with`ˆc=2 h

i (iˆ2;4;. . .;c=2 1) even Figure 3 with`ˆc‡1 i

c=2 odd Figure 2

c=2‡1 even Figure 3 with`ˆc=2

c=2‡2 odd Figure 4

c=2‡2‡j (jˆ1;3;. . .;c=2 2) even Figure 3 with`ˆc=2 1 j

c=2‡2‡k (kˆ2;4;. . .;c=2 1) odd Figure 3 with`ˆc k

Table 4. ±L(c;g;d;0) is an even lattice in the case whenc0 mod 4:

g=(c‡1) modc‡2 d Dynkin diagram

1 even Figure 4

h (hˆ1;3;. . .;c=2 1) odd Figure 3 with`ˆc=2 h

i (iˆ3;5;. . .;c=2‡1) even Figure 3 with`ˆc‡1 i

c=2‡1 odd Figure 1

c=2‡1‡j (jˆ2;4;. . .;c=2 2) even Figure 3 with`ˆc=2 j

c=2‡1‡k (kˆ2;4;. . .;c=2) odd Figure 3 with`ˆc‡1 k

c‡1 even Figure 2

Table 5. ±L(c;g;d;0) is an odd lattice in the case whenc0 mod 4:

g=(c‡1) modc‡2 d Dynkin diagram

0 even Figure 1

h (hˆ0;2;. . .;c=2 2) odd Figure 3 with`ˆc=2 h

i (iˆ2;4;. . .;c=2) even Figure 3 with`ˆc‡1 i

c=2 odd Figure 2

c=2‡2 even Figure 3 with`ˆc=2 1

c=2‡2 odd Figure 4

c=2‡2‡j (jˆ2;4;. . .;c=2 2) even Figure 3 with`ˆc=2 1 j

c=2‡2‡k (kˆ2;4;. . .;c=2 2) odd Figure 3 with`ˆc k

PROOF. Let us keep the notation as above. In particular,Fis given by (3:15) and (O)ˆer. Take the following elements fromL(c;g;d;0):

j_{r 1}ˆG X^c‡2

iˆ1

e_i; j_iˆe_i e_i‡1 (1ir 2; i6ˆr c 1):

We takej_randj_{r c 1}fromL_(c;g;d;0) according to the following rule:

(1) Ifc1ˆd=2 g=((c‡1)(c‡2))2Z, then put j_rˆD0‡c1G e1; j_{r c 1}ˆer c 1 er c: (2) Ifc2ˆd=2 g=((c‡1)(c‡2)) 1=(c‡2)2Z, then put

j_rˆD0‡c2G; j_{r c 1}ˆer c 1 er c:

(3) Ifc3ˆd=2 g=((c‡1)(c‡2)) (1‡`)=(c‡2)2Z, then put j_rˆD0‡c3G‡ X^{r 1}

iˆr `

e_i; j_{r c 1}ˆer c 1 er c:

(4) Ifc₄ˆd=2 g=((c‡1)(c‡2))‡1=(c‡2)2Z, then put j_rˆD0‡c4G e1 …F‡(O)†; j_{r c 1}ˆF‡(O) er c:

Here the numbering of thej_i's corresponds to that of the vertices in Figures 1, 2, 3, 4 according to 2g=(c‡1) mod (c‡2) withd2g=(c‡1) mod 2 in the case whencis odd and to the combination ofg=(c‡1) mod (c‡2) and the parity ofdin the case whencis even.

Then these together withF, (O) clearly form a basis for NS(X) overQ in either case. While we see that fj₁;j₂;. . .;j_rg forms a Z-basis for L_(c;g;d;0) , we divide our argument between the case (4) and the other cases.

At first, we restrict ourselves to the case (3), since the cases (1) and (2) are quite similar. Consider the matrix representing the base change from (D₀;G;e₁;e₂;. . .;e_r) to (j_r;j_{r 1};j₁;j₂;. . .;j_{r 2};F;(O)). Then it is easy to see that, off the (r‡1)-th row, it is an integral triangular matrix all of whose diagonal entries are equal to one, and we have

Fˆer 1‡(c‡2)j_r‡ 2g

c‡1‡2‡`

j_{r 1}‡ 2g

c‡1‡1‡` X^c‡2

kˆ1

kj_k

‡^rX^{` 1}

kˆc‡3

(r‡`(c‡2) k)j_k‡ X^{r 2}

kˆr `

((c‡3)(r k) c 2)j_k (O):

For (4), we only have to note that the matrix representing the base

change from (D0;G;e1;e2;. . .;er) to (j_r‡F‡(O);j_{r 1};j₁;j₂;. . .;j_{r c 2}; er c 1;er c;. . .;er 1;(O)) is an integral triangular matrix, all of whose diagonal entries are equal to one, and we have

Fˆ er c 1‡(c‡2)(j_r‡F‡(O))

‡ 2g

c‡1j_{r 1}‡X^c‡2

kˆ1

2gk

c‡1‡c‡2 k

j_k‡^{r c}X³

kˆc‡3

(r c 2 k)j_k X^{r 1}

kˆr c

e_k (O);

j_{r c 1} F (O)ˆ e_{r c}; j_iˆ e_i‡1‡e_i (iˆr c;. . .;r 2):

Hence in either casefj₁;j₂;. . .;j_rgforms aZ-basis forL(c;g;d;0) and we obtain the corresponding Dynkin diagrams.

As to the last statement, we consider the case whenc2 mod 4 only, since the other cases are similar. We note that the self-pairing numbers of j_i's are always even except forj_r, while the parity of that ofj_rvaries even if we fixg(c‡1)². Consider for example the case wheng is divisible by (c‡1)(c‡2). Ifdis even thenj_r²ˆ2g=((c‡1)(c‡2))‡1 is odd, and ifd is odd thenj_r²ˆ2g=((c‡1)(c‡2))‡(c‡2)=2 is even with`ˆc=2. p

3.2 ±Type(c;g;1;n)

We shall determine the Mordell-Weil lattices for fibrationsf :X!P¹ of type (c;g;1;n) with (c‡1)(c‡2)=2g(c‡2)(c‡3)=2. Whenn0, the reductionYis obtained by blowingP²up atp₀;. . .;p_ncorresponding to singular points ofG0as in Theorem 2.3. Furthermore, the configuration is as in Proposition 3:3. Lete_ibe the inverse image onXofp_i(iˆ0;. . .;n).

We denote bylthe pull-back toXof a line onP²and lete_i,n‡1ir, be the disjoint ( 1)-sections of f coming from base points of L_f. Then NS(X)'ZlL^r

iˆ0Ze_i. We take the last ( 1)-sectione_ras the zero section (O). Then (O) andF generate the trivial sublatticeT_(c;g;1;n)NS(X). Let L_(c;g;1;n)be the essential sublattice obtained as the orthogonal complement ofT_(c;g;1;n).

We first consider the case whereG₀is as in (iv) of Theorem 2.3. Then Fˆ(c‡4)l 2Xⁿ

iˆ0

e_i X^r

iˆn‡1

e_i:

THEOREM3.8. Let(c‡1)(c‡2)=2g(c‡2)(c‡3)=2. For a fibra- tion of type(c;g;1;n)with nˆ(c‡1)(c‡4)=2 g, the lattice L(c;g;1;n) is

isomorphic to a positive-definite odd unimodular lattice of rank rˆ3g (c‡3)(c 4)=2whose Dynkin diagram is as in Figure5.

Fig. 5

PROOF. Take the following elements fromL_(c;g;1;n), whose numbering corresponds to that of the vertices in Figure 5:

j_iˆe_{i 1} e_i (1ir 1; i6ˆn‡1);

j_n‡1ˆe_n e_n‡1 e_n‡2; j_rˆ l‡ X^{r 1}

iˆr c 4

e_i:

Then we can show that the above elements form aZ-basis forL(c;g;1;n)in the same way as in the case (3) in the proof of Theorem 3.7. p Secondly, we shall determine the Mordell-Weil lattices for fibrations of type (c;(c‡1)(c‡2)=2;1;4). Ifcis even, then

Fˆ3c‡6

2 l c‡2 2

X³

iˆ0

ei c 2e4

X

c²‡5c‡9

iˆ5

ei; (O)ˆe_c²_‡5c‡9: THEOREM3.9. For a fibration of type(c;(c‡1)(c‡2)=2;1;4)with an even number c, the lattice L(c;(c‡1)(c‡2)=2;1;4) is isomorphic to a positive- definite odd unimodular lattice of rank c²‡5c‡9whose Dynkin diagram is as in Figure6.

Fig. 6

PROOF. Take the following elements from L(c;(c‡1)(c‡2)=2;1;4), whose numbering corresponds to that of the vertices in Figure 6:

j_iˆe_{i 1} e_i (iˆ1;2;3;6;7;. . .;c²‡5c‡8);

j₄ˆe₃ e₄ e₅; j₅ˆe₄ ^c=2‡4X

iˆ5

e_i; j_c²_‡5c‡9ˆl e₀ e₁ e₂:

Then we can show that the above elements form a Z-basis for L(c;(c‡1)(c‡2)=2;1;4) in the same way as in the case (3) in the proof of Theo-

rem 3.7. p

Next, we consider the case wherec is odd,gˆ(c‡1)(c‡2)=2,dˆ1 andnˆ4. Then

Fˆ3c‡5

2 l c‡1 2

X⁴

iˆ0

e_i ^c2X^‡5c‡9

iˆ5

e_i; (O)ˆe_c²_‡5c‡9:

THEOREM3.10. For a fibration of type(c;(c‡1)(c‡2)=2;1;4)with an odd number c, the lattice L(c;(c‡1)(c‡2)=2;1;4) is isomorphic to a positive-de- finite odd unimodular lattice of rank c²‡5c‡9whose Dynkin diagram is as in Figure7.

Fig. 7

PROOF. Take the following elements from L(c;(c‡1)(c‡2)=2;1;4), whose numbering corresponds to that of the vertices in Figure 7:

j_iˆe_{i 1} e_i (1ic²‡5c‡8; i6ˆ5; (2c²‡7c‡15)=2);

j₅ˆe4

X

(c‡9)=2 iˆ5

e_i; j_(2c²_‡7c‡15)=2ˆe_(2c²_‡7c‡13)=2 (F‡(O));

j_c²_‡5c‡9ˆl e0 e1 e2 e_(2c²_‡7c‡15)=2:

Then we can show that the above elements form a Z-basis for L(c;(c‡1)(c‡2)=2;1;4) in the same way as in the case (4) in the proof of Theo-

rem 3.7 p

In the last, we shall determine the Mordell-Weil lattices for fibrations of type (4;16;1;3). Then

Fˆ9l 3X³

iˆ0

e_i X⁴⁸

iˆ4

e_i; (O)ˆe48:

THEOREM3.11. For a fibration of type(4;16;1;3), the lattice L_(4;16;1;3)

is isomorphic to a positive-definite even unimodular lattice of rank 48 whose Dynkin diagram is as in Figure 8.

Fig. 8

PROOF. Take the following elements fromL_(4;16;1;3), whose numbering corresponds to that of the vertices in Figure 8:

j_iˆei 1 ei (iˆ1;2;3;5;6;. . .;47); j₄ˆe3 e4 e5 e6; j₄₈ˆl e0 e1 e2:

Then we can show that the above elements form aZ-basis forL_(4;16;1;3)in the same way as in the case (3) in the proof of Theorem 3.7. p

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