A Discriminant and an Upper Bound for W2 for Hyperelliptic Arithmetic Surfaces

Compositio Mathematica 115: 37–69, 1999. 37

c 1999 Kluwer Academic Publishers. Printed in the Netherlands.

A discriminant and an upper bound for ! for hyperelliptic arithmetic surfaces

IVAN KAUSZ ? Mathematisches Institut der Universitat¨ zu Koln,¨ Weyertal 86-90, 50931 Cologne, Germany e-mail: [email protected]

Received 13 November 1996; accepted in ﬁnal form 1 September 1997

g K

Abstract. We deﬁne a natural discriminant for a hyperelliptic curve X of genus over a ﬁeld as a

g +

canonical element of the 8 4 th tensor power of the maximal exterior product of the vectorspace

v K X

of global differential forms on X .If is a discrete valuation on and has semistable reduction v

at v , we compute the order of vanishing of the discriminant at in terms of the geometry of the v reduction of X over . As an application, we ﬁnd an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface. Mathematics Subject Classiﬁcations (1991): 11G30, 14G40, 14H10. Key words: hyperelliptic curves, arithmetic surfaces, Arakelov theory.

0. Introduction

In the present paper we introduce a natural discriminant for hyperelliptic curves of

> g + genus g 2. It will be deﬁned as a canonical element of the 8 4 th tensor power of the maximal exterior product of the vectorspace of global differential forms on

the curve. To fix ideas, let us first consider the analogous case of an elliptic curve K E over a field , given, say, by the Weierstraß equation

2 3 2

+ a xy + a y = x + a x + a x + a :

y 1 3 2 4 6

2 K

It is well known that while its discriminant depends on the special equation,

12 0 1 12

= x= y + a x + a 2 H E;

the element E=K d 2 1 3 is a

E=K

E v K O K

genuine invariant of . Moreover, if is a discrete valuation of and v

is the associated discrete valuation ring, we may regard E=K as a rational section

! S = O E ! S

=S E =S

E of the bundle on Spec ,where : is the minimal

E S !

=S regular model of over and E is the relative dualizing sheaf. Assuming

semistable reduction at v , there is a simple geometric interpretation of the order of

vanishing of E at : It is the number of singular points of the geometric special

! S

ﬁbre of E . Our main Theorem (3.1) is a generalization of this fact to the case =S of hyperelliptic curves. Given a semistable curve X with smooth hyperelliptic

? Partially supported by the DFG.

INTERPRINT J.V.: PIPS Nr.: 148874 MATHKAP comp4166.tex; 8/08/1998; 8:53; v.7; p.1

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generic ﬁbre X=K , it gives a formula for the order of vanishing of the discriminant

0 1 8g 4

2 H X; X v

X det of at in terms of the geometry of the

X=K

special fibre. On the way to this formula we give an explicit construction of a S regular model of X over a finite extension of as the ramified double covering of a pointed prestable curve of genus zero. As an application, we find an upper bound 2

for the Arakelov self-intersection ! of the relative dualizing sheaf on semistable hyperelliptic arithmetic surfaces (Corollary 7.8). The search for an upper bound for 2

! has been initiated by A. N. Parshin [Par1]. He observed that in the geometric case such a bound follows from the Bogomolov–Miyaoka–Yauinequality between Chern-classes of an algebraic surface and that in the arithmetic case it would lead to the positive answer of various diophantine questions, as for instance the

abc-conjecture. The most naive arithmetic analogue of the Bogomolov–Miyaoka– Yau inequality seems to be wrong: There are curves of genus two which provide counterexamples (cf. [BMM]). In [MB], Moret-Bailly therefore formulates as a 2

hypothesis a shape of an upper bound for ! which still implies the same arithmetic consequences as given by Parshin. Our bound has the shape of Moret-Bailly’s hypothesis. Unfortunately, it has two shortcomings. Firstly, it involves the choice of a metric on the relative dualizing sheaf of the universal curve over the moduli

space of curves of genus g , which we cannot make explicit. Secondly (and this is more serious), it is very special to the hyperelliptic situation. In particular it does not involve the discriminant of the number ﬁeld. This work owes most to Chapter 4 of [C-H], where Cornalba and Harris describe

the structure of the boundary of the moduli space of hyperelliptic curves of

g + genus g and give an expression of the 8 4 th power of the Hodge bundle

in terms of the boundary components. The existence of a canonical element in

g +

0 1 8 4

H X; g

det for a hyperelliptic curve of genus seems to be well

X=K = known (see for example [U] in the case of g 2), though I don’t know any ref- erence in the general case. Our construction of a regular model for a hyperelliptic curve has been inspired by work of E. Horikawa [Hor] and U. Persson [Per]. An 2

upper bound for ! of the form of our Corollary 7.8 had been previously established = in the case g 2 and with respect to the Arakelov metric by J.-B. Bost in a letter to B. Mazur [Bost2], using explicit formulas given in [Bost1]. In fact it was this letter and Mazur’s answer to it [Maz], which was the starting point of our investigation.

1. Hyperelliptic semistable curves

V ; E ;c V E By a graph we understand a triple ,where and are disjoint sets (the

set of vertices and the set of edges respectively) and c (the coincidence relation) is

cE V

a rule that to each edge E associates a subset consisting of one or two

2 N n

vertices. For example, for every n we deﬁne the cyclic graph of length

C =fV g ; fE g ;cE =fV ;V g i 2 Z=nZ:

n i i i i i+

2Z=nZ i2Z=nZ i 1 for

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We adopt the usual notions from graph theory such as ﬁniteness, connectedness, morphism of graphs, subgraph, quotient of a graph by a group action. A connected graph that has no cyclic subgraph will be called a tree.

DEFINITION 1.1. A marked graph is a graph together with a distinguished subset of its set of vertices (the set of ‘marked vertices’). A marked graph is called semistable (stable), if it is connected and if from every unmarked vertex there start at least two (three) edges.

For example, the (unmarked) cyclic graph C1 is semistable (the edge which starts

and ends at the unique vertex, is counted twice).

V ; E ;c

DEFINITION 1.2. Let = beagraphand an automorphism of .An

E = E

edge E of will be called direction-reversing with respect to ,if ,

E =fV ;V g V =V c 1 2 and 1 2.

For example, the unique edge of the cyclic graph C1 is direction-reversing with

respect to the identity. Let be a (marked) graph and an involution (i.e. an

automorphism of order 2) of .Wedenoteby the (marked) graph that is

obtained from by omitting all edges which are direction-reversing with respect

to .

DEFINITION 1.3. An involution of a marked graph is called hyperelliptic,if

it leaves marked vertices ﬁxed and if the quotient is a tree. A marked graph is called hyperelliptic, if it admits a hyperelliptic involution.

PROPOSITION 1.4. Let be a ﬁnite semistable marked graph which is not an

unmarked cyclic graph. Then there is at most one hyperelliptic involution on .

Proof.Let and be two hyperelliptic involutions on .Let be a vertex and

V 6= V

assume ﬁrst that . It is then easy to see that there exists a cyclic subgraph

V C

C containing and that is mapped onto itself by any hyperelliptic involution. V

Now C contains a vertex 0 which is marked or from which there starts an edge

V = V

not belonging to C . In both cases it is easy to see that necessarily 0 0 .

But the action of (and )on is completely determined by its action on one

V = V V = V

of its vertices, so we have . Now let us assume .Then

0 0

V =V

, because otherwise, interchanging the role of and in the previous

argument, we would obtain a contradiction. This shows that and act identically

on the set of vertices of . It is now easy to see that and act identically also on 2

the set of edges of . g

Recall (cf. [Kn]) that an n-pointed prestable curve of genus over a scheme

S X ! S n s S ! X

consists of a proper ﬂat morphism : and sections i : of S such that for each geometric point s of the following holds:

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40 IVAN KAUSZ X

(1) The geometric ﬁbre s is a reduced curve and has only ordinary double points

as singularities.

P = s s X

i s (2) The points i are distinct regular points of .

H X ; O =g: X

(3) dim s

n X; P

i= ;:::;n An -pointed prestable curve i 1 over an algebraically closed ﬁeld

is called stable (semistable), if it is connected and if on each smooth rational X

component of X the number of points that are double points of or are among

P n

the i is at least three (two). More generally, an -pointed prestable curve over an arbitrary scheme is called stable (semistable), if each of its geometric ﬁbres is stable (semistable).

To every prestable curve X over an algebraically closed ﬁeld we associate in the

usual way a graph X , whose vertices correspond to the irreducible components and

whose edges correspond to the double points of . We provide X with a canonical

marking by requiring that a vertex be marked if and only if the corresponding > irreducible component of X has genus 1. It is then clear that the marked graph

associated to a semistable (stable) curve is semistable (stable) in the above sense. n

Let X be an -pointed prestable curve over an algebraically closed ﬁeld. Let

p 2 X X p ! X be a double point. Let X be the partial normalization of at and

~ 1

;p 2 X p Z ' P p the points of the preimage of .Let and choose three points

1 2 k

;z ;z 2 Z n +

z1 2 3 .The( 1)-pointed prestable curve, obtained by taking the union

X Z p z i = ; z i

of and , identifying i with ( 1 2) and taking as marked points 3 and

X p

those coming from X , will be called the modiﬁcation of at .

Y; P ;::: ;P n

Let 1 n be an -pointed prestable curve over an algebraically

closed ﬁeld k of characteristic 2. Recall (cf. [H-M], Section 4) that a morphism

X ! Y

: is called an admissible double covering, if the following holds

(1) X is a prestable curve and is a ﬁnite morphism of degree two.

P i = ;::: ;n

(2) is ramiﬁed over every point i ( 1 )andetale´ over any other

smooth point of Y .

2 Y q q =fpg (3) If q is a double point, then either is etale´ over or and

there are isomorphisms

O ' k [[x; y ]]=x y ;

X;p

O ' k [[u; v ]]=u v ;

Y;q

k [[u; v ]]=u v ! k [[x; y ]]=x; y such that induces the morphism which

2 2

x v y maps u to and to .

We can now deﬁne the central notion of this paper.

6= X=k DEFINITION 1.5. Let k be a ﬁeld of characteristic 2andlet be a semistable curve.

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AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 41 X=k (1) Assume ﬁrst that k is algebraically closed. Then we call hyperelliptic,if

the following holds

(a) The associated marked graph X is hyperelliptic.

0 X

(b) Let X be the (pointed) prestable curve that is the modiﬁcation of

at each of its double points which corresponds to an edge of X which

is direction-reversing with respect to the hyperelliptic involution of X .

! Y Then there exists an admissible double covering X onto a pointed

prestable curve Y of genus 0 that carries marked points into marked points.

X=k

(2) In general, we call X=k hyperelliptic,if is hyperelliptic in the sense of

X = X k k k

1, where k and is some algebraic closure of .

Let be a hyperelliptic semistable curve. The hyperelliptic involution of X

induces a canonical involution on the set of components and on the set of double X

points of X . A double point of is called direction-reversing, if the corresponding

edge of X is direction-reversing with respect to .

6= X=k

LEMMA 1.6. Let k be an algebraically closed ﬁeld of characteristic 2.Let

Y=k n Y

be a prestable curve of genus g and a connected -pointed prestable curve

X ! Y

of genus 0. Let : be an admissible double covering. Then all components

X n >

of X are smooth and is connected if and only if 1 in which case we have

= g + X g > n 2 2. If, in addition, is semistable and 2, then it is hyperelliptic without direction-reversing double points.

Proof. It is clear that the components of X are smooth, since by deﬁnition of

admissible double coverings, X has singular points only when two different com-

= X ! Y X

ponents meet. If n 0then is etale´ and it follows that is isomorphic

q Y n > Y

to Y .Let 1. Then, by induction on the number of components of ,itis

g g + = n easy to see that X is connected of genus with 2 2 . The last statement of

the lemma is clear. 2

k 6= X=k

Assume k algebraically closed with char 2andlet be a hyperelliptic

> semistable curve of genus g 2. Denote by the canonical involution of its set

of double points. Let p be a non-direction-reversing double point. Then the partial

fp; pg

normalization of X at is the disjoint sum of two connected components

X X g X i = ; i

1, 2.Let i be the genus of ( 1 2). There are two cases:

p 6= p g = g + g + p

.Then 1 2 1 and we call to be of type l ,where

= fg ;g g2f ;::: ;[g = ]g:

l : min 1 2 0 1 2

p = p g = g + g p l =

.Then 1 2 and we call to be of type l ,where :

g ;g g2f ;::: ;[g= ]g

min f 1 2 1 2 .

0 k

Direction-reversing points will also be called to be of type 0.If is not algebraical-

X k

ly closed and k an algebraic closure, then all points of lying over a ﬁxed

X p singular point p of are of the same type. We call to be of the corresponding type.

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2. The discriminant of a hyperelliptic curve >

Throughout this section, we make the following assumptions: g is an integer 2.

K 6= g + X =K

is a ﬁeld of characteristic 2 with at least 2 2elements. K is a smooth

hyperelliptic curve of genus g such that there exists a ﬁnite morphism of degree

X P

two from K onto the projective line .

F X x y

PROPOSITION 2.1. (1) The function ﬁeld of K has generators and and

= f x f x 2 K [x]

deﬁning relation y ,where is a separable polynomial of +

degree 2 g 2. ~

2 ~

= K ~x; y~ y~ = f ~x f ~x 2 K [~x]

(2) If F ,where and is separable of degree +

2 g 2, then there is an invertible matrix

a b

2 Gl2

c d

g +

2 K x =ax~ + b=cx~ + d y = e=cx~ + d y~

and an element e such that and . X

Proof. (1) It is well known (cf. [Art1] Ch. 16.7) that the function ﬁeld of K

x[y ]=y f x f x 2 K [x]

is K for some square-free polynomial of degree

+ g + f x= g + a 2 K f a 6=

2 g 1or2 2. If deg 2 1, choose an element with 0

> g +

(here we use the assumption that #K 2 2) and make the transformation

g +

1 2 ~

= =~x a+a y = ~x a y~ y~ = f ~x

x 1 , , to get an equation ,where

f ~x = + f x g +

deg 2 g 2. Thus we may assume that has degree 2 2. Let

K K X

be an algebraic closure of .Since K is smooth by hypothesis, the curve

X = X K g

: K is also regular and of genus and its function ﬁeld is described by K

= f x f x K [x] the same equation y . Therefore, by loc. cit., is square-free in

and thus separable.

~x F

(2) It follows from the assumptions that K is of index two in and of genus

ax~ +b

~x=K x x =

zero. By loc. cit. we conclude that K and therefore that for

cx~ +d

an invertible matrix

a b

2 K :

Gl2

c d +

g 1

= e=cx~ + d ~y

It is easy to see that then we have necessarily y for some

2 K 2

e .

f f x 2 K [x] Recall that the discriminant of a nonzero polynomial of degree

d is deﬁned as

2d 2

f =A a a 2 K ;

i j

i6=j

A 2 K a ;::: ;a f

where is the leading coefﬁcient and 1 d are the zeroes of in

K y = f x X

an algebraic closure of .Let be an equation for K as in the above

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AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 43

x=y i = proposition. Recall further the well-known fact that the differentials x d

;::: ;g H X ; =

=K X =K

0 1 form a basis of X . We deﬁne an element

K K

0 1 8g 4

H V = X ;

of the one-dimensional vector-space K by set-

X =K

ting K

g +

8 4

g 1

x x

d x d

=D ; ^ ^

y y

g +

4 4

f =

where D : 2 . (The reason for the power of two in the deﬁnition of

D will be given in Section 6.) By the following result, is a canonical element of V

PROPOSITION 2.2. (1) is independent of the special choice of the equation

= f x

y .

0 0

K =K X = X K K

(2) Let be an arbitrary ﬁeld-extension and K : .Then

= =K

X is the image of under the canonical mapping

O ^

0 g +

0 1 8 4

K = V ! V X : ; H

=K X

K K

Proof. It is a straightforward calculation to show that is invariant under trans-

formations as described in Proposition 2.1 (2). This proves the ﬁrst statement. As to 2 the behavior under base change: This follows directly from the construction of .

3. Statement of the main theorem, ﬁrst reductions

K v

In this section, R denotes a discrete valuation ring, its quotient ﬁeld, the

v K = Z k

induced discrete valuation on K (normalized in the sense that ), the

R X=R X

residue ﬁeld of .Let be a prestable curve with smooth generic ﬁbre K and

p X X p p

let be a singular point of the special ﬁbre k of .Themultiplicity of is

R X=R = O =F F X;p

deﬁned as the length of the -module p : 1,where 1 denotes

X;p =R

the ﬁrst Fitting ideal of the -module O .Therelative dualizing sheaf X;p

! X=R j j X ,! X

K =R

X of is the unique invertible subsheaf of (where :

X =K K 1

is the canonical immersion), which coincides with on the smooth part of

X=R

X=R .

0 =R

The behaviour of these invariants under base change is as follows: Let R

0 0

X = X R

be a discrete valuation ring dominating R and set .Thenwehave

p =ep p X ! X

i , where the sum is over the points of the ﬁbre over of

e R =R ! 0 0

and is the ramiﬁcation index of . The relative dualizing sheaf X is

! X ! X

simply the pull back of X under the projection .

~ X=R

Now assume k to be algebraically closed and let be a second prestable

X ! X f R p =

curve and : a birational projective -morphism. Then we have i

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44 IVAN KAUSZ

p X p

, where the sum is over the singular points of k mapping onto . Therefore,

! X X p

if X happens to be the minimal desingularization of ,then is simply

p X

the number of double points of k lying above (cf. [MB], proof of Theorem 2.4).

= ! f !

On the other hand, we have X . This follows from [Art2], Corol- X=R

lary (3.4)(ii) and the fact that X has only rational singularities.

k 6= k X

Assume char 2and not necessarily algebraically closed and that K

is hyperelliptic and satisﬁes the assumptions made at the beginning of Section 2.

=K X=R

Then we can consider X as a rational section of the ‘line bundle’

g +

0 8 4

M = H X; ! R

X on Spec anddenotebyord the order of

s 2 R

vanishing of in the closed point Spec . In other words, we set

=v a;

ords

2 K = a 2 M K M

where a , 0 and 0 is a generator of . R

After these preliminaries we can formulate our main theorem

R; K; k ; v 2 R

THEOREM 3.1. Let be a discrete valuation ring with 2 .Let

X=R X

be a semistable curve with smooth hyperelliptic generic ﬁbre K of genus

g > X

2 and assume that there exists a ﬁnite morphism of degree two from K onto

1 X=R

the projective line P .If is either the minimal regular or the stable model of

X R

K over , then the following holds

X X=R

(1) The special ﬁbre k of is hyperelliptic in the sense of Section 1.

(2) Let be the hyperelliptic involution on the set of double points of k .Thenin

a given -orbit all double points have the same multiplicity, which we call the

multiplicity of that -orbit.

(3) With the notation introduced at the end of Section 1,let

0 0

A0 the number of double points of type 0,

A = i

i the number of -orbits of double points of type ,

B = j

j the number of double points of type

i = [g = ];j = ;::: ;[g= ] 0 ;::: ; 1 2 1 2 , where we count double points ( -

orbits) with multiplicities. Then we have the equation

P P

[g [g= ] ]

1=2 2

= g A + A g ii + + B g j j:

i j

ords 2 1 4

= j = 0 i 0 1 The proof of the theorem will be given in Section 5. In the present section we restrict ourselves to prove some ﬁrst reductions.

LEMMA 3.2. (a) It sufﬁces to show the theorem under the additional assumption

that the residue ﬁeld k is algebraically closed.

X=R R

(b) Assume that k is algebraically closed and let be a stable -curve ~

satisfying the assumptions of the theorem. Denote by X=R the minimal regular

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AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 45

~ X=R

model of X . Assume that the conclusions of the theorem hold true for and that

~ X

k has no direction-reversing double points. Then the conclusions of the theorem

hold true also for X=R .

R =R

0 0

X R

discrete valuation rings and let X be the minimal regular model of .Ifthe

0 X conclusions of the theorem hold true for X then also for . Proof. Part (a) and (c) of the lemma are easy consequences of Proposition 2.2

and the remarks we made at the beginning of this section. To prove part (b), ﬁrst

X ! Y k observe that by assumption, there exists an admissible double covering k

g + Y P n

onto a 2 2 -pointed genus-0-curve k .Inwhatfollows,a -chain (of length )

~ ~

X X Z k of k is a closed connected subscheme of that is maximal with the property

1 P

that all the irreducible components of Z are isomorphic to and meet the rest

X Z

of k in exactly two points. The graph of is then linear. It is well known that ~

~ 1 1

X P Z P X X

k k

arises from k by contracting all the -chains of .Let be a -chain of

~ ~

X Z ;::: ;Z p ;p 2 X

n k

and let 1 be its successive components and 1 2 k the points

Z X

where meets the rest of k . Denote by the hyperelliptic involution on the set

~ X

of components and double points of k . We distinguish three cases

p = p Z = Z i = ;::: ;n

n+ i

Type-0-case: 1 2.Wehavethen i 1 for 1

= m +

and it follows that n 2 1 is odd because otherwise the double point

\ Z p = Z

= n = n+

1 2 1 2 1 would be direction-reversing. The middle component

Z Z Y

1 is the only one of which is ramiﬁed over k . 0

1 ~

p 62 fp ;p g P Z X

Type- -case: 1 1 2 . Then there is a -chain of k which is

0 0 0

Z Z ;::: ;Z Z =Z

disjoint to and has successive components such that i . n

1 i

2f ;::: ;[g = ]g n +

Thereexistsan l 1 1 2 such that all the 1 -orbits of double

X Z [ Z l

points of k contained in , are of type .

p = p Z

Type- -case: 1 1. Then all components of are ﬁxed by .There

l 2f ;::: ;[g= ]g n + X

exists an 1 2 such that all the 1 double points of k lying

on , are of type l .

~ 1

Z X P

Let be the set of all components of k which are isomorphic to and meet

~ X

the rest of k in exactly two points, except those, which are the middle component

X X

of some type-0-chain. Let be the pointed prestable curve obtained from k by k

contracting all components belonging to Z and marking one ramiﬁcation point on

g +

each middle component of type-0-chains. On the other hand, let Y be the 2 2 -

k Y

pointed prestable genus-0-curve which is obtained from k by contracting all

images of components belonging to Z . It is clear that restricts to a hyperelliptic

X X X

involution on ,that is the modiﬁcation of k in all of its direction-reversing

0 0

X ! Y X ! Y k

double points, and that k induces an admissible double covering

k k X

carrying marked points into marked points. This proves that k is hyperelliptic.

P Z

In the type and type cases a -chain is contracted to a ( -orbit of) double

X n +

point(s) of k , which is of multiplicity 1 and is of the same type as the ( -orbits

Z m +

of) double points lying on Z .If is of type 0, the 1 -orbits of double points

Z X

contained in , are contracted to a double point of k , which is of multiplicity

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46 IVAN KAUSZ

n 1 and of type 0. It follows that statements (2) and (3) of the theorem also hold 2 for X=R .

4. Construction of minimal models

Let R; K; k ; v be as in the assumptions at the beginning of the last section. Except

for Lemma 4.1, we assume k to be algebraically closed. Assume furthermore that

2 2 .

X =K

LEMMA 4.1. Let K be a smooth hyperelliptic curve. There exists a discrete

0 0 0

K k

valuation ring R with quotient ﬁeld , residue ﬁeld , and normalized discrete

0 0

= X K v R R X K

valuation , ﬁnite over and dominating , such that K is a

x; y

regular proper model of the function ﬁeld K associated to the equation + 2g 2

2 Y

y = f x = A x a ;

: i =

i 1

0 0 0

A 2 R ;a 2 R i = ;::: ; g + a 6= a v a a 2 N

i j i j

where i for 1 2 2, and 2 0

i 6= j fa j i = ;::: ; g + g > a 2 k a

i i for and # i 1 2 2 3 being the residue class of .

Proof. By Proposition 2.1 we may assume (after passing to some ﬁnite extension

g +

2 2 2

K X y = A x a A 2 K i

of )that K belongs to an equation for some =

i 1

a 2 K

and pairwise different i . After a simple transformation we may even assume

a 2 R i fa ;::: ;a g > a

i g + i

i for all and # 2 2 2. If the number of different equals 2, we

a = = a 6= a = = a

r r +

can write (after some renumbering): 1 1 2g 2,where

r > m = v a a v a a 6 i 6

3andwhere : 1 2 is minimal among the values 1 i 1

t 2 R k r

r 1). Let be a local parameter. We may assume that has at least elements.

b 2 R b a a =t R

Therefore we ﬁnd an element such that i 1 is invertible in for

g +

m 1

= ;::: ;r x = t b + =x~ +a ;y = =x~ y~

i 1 1. After the transformation 1 1 1

fa ;::: ;a g > +

we can make the additional assumption that # 1 2g 2 3. Passing to the

K t=K v A 2 Z v a a 2 N j

extension we achieve ﬁnally 2 and i 2 0 for all

= v A

1 2

6= j x = x;~ y = t y~ i . After the transformation the equation has all the

required properties. 2

X =K

For the rest of this section let K be the regular proper model of the function +

2 2g 2

K x; y y = f x = A x a A

ﬁeld associated to the equation : i ,where =

i 1

a v

and the i satisfy the conditions (with respect to ) listed in the above lemma. Our

purpose is to give a very explicit description of the minimal regular model X=R of

X g +

K and to show that it is semistable. First we associate a 2 2 -pointed prestable +

2g 2 1

Y f x= x a Y = P

curve to the polynomial i as follows: Let : and

R = i 1 0

P Y 2 P Y

let 1 the closure of the point 1:0 in . The principal divisor on

0 K 0

2g 2

Y f x 2 K Y f x = P g + P

Y 1

,deﬁnedby is div i 2 2 ,where = 0 0 0 i 1

comp4166.tex; 8/08/1998; 8:53; v.7; p.10

P Y a 2 P Y i = ;::: ; g + Y i i is the closure of :1 in 1 2 2). By deﬁnition,

0 K 0

is obtained from Y0 by successively blowing up closed points of the special ﬁbre,

P P

i i

where the i meet, until the strict transform of becomes regular. By abuse of

P Y i 2 f ;::: ; g + ; 1g

notation, we denote the strict transform of i in 1 2 2 by

P Y; P ;::: ;P g +

i + the same symbol . It is then clear that 1 2g 2 is a 2 2 -pointed

Y = P

prestable curve of genus zero with generic ﬁbre K .

Y Y Y

The graph associated to the special ﬁbre k of is a ﬁnite tree that can

a 2 R m R

be described directly in terms of the elements i :Let be the maximal

R n 2 N r fa ;::: ;a g ! R=m

n +

ideal of and for 0 let : 1 2g 2 be the natural

a m T

mapping that associates to i its residue class modulo . We deﬁne a graph

T V T =q V

> n

as follows. The vertices of are the elements of the set n 0 ,where

n 1

V = fV 2 R=m j r V > g T V; V

n # 2 . The set of edges of consists of pairs ,

0 0

V 2V V 2V n > V 7! V

where n , 1 for some 0and under the canonical map

0 0

V !V c cV; V = fV; V g

+ n n 1 . The coincidence relation ,ﬁnally,isdeﬁnedby .

It is easily seen that is a ﬁnite tree and that it is canonically isomorphic to Y .

V V

T has a canonical vertex 0, the unique element of 0. It corresponds to the

Y Y

irreducible component of k which is the strict transform of the special ﬁbre of 0.

We deﬁne a canonical partial ordering on the set of vertices of T by setting

0 0

V 6 V , V 2V ;V 2V ; m

: n where

m > n; V 7! V V !V : n and under m

The vertex V0 is the absolute minimum with respect to this partial ordering. It will T

be convenient to associate to each vertex V of the following list of numbers

nV = n; V 2V ;

: where n

V = r V ;

' : #

V 'V 'V ;

V 2V +

n 1

V >V

1ifn and are odd

V = ; C :

0otherwise

= 'V ; V ;V ;::: ;V = V f V n

: i where 0 1 = i 1

are the vertices of the linear subgraph of T V; that connects V0 and

V = C V f V :

B : 2

V Y nV 'V

If is an irreducible component of k ,wedenoteby , etc. the numbers

V associated to the corresponding vertex of T . For example, is the number of

1 Quing Liu has brought to my attention that the construction of this tree already appears in [Bosch].

comp4166.tex; 8/08/1998; 8:53; v.7; p.11

48 IVAN KAUSZ

P i = ;::: ; g + V Y i

sections 1 2 2 , meeting the irreducible component of k .The

V C V f V

geometrical meaning of B , and is revealed by the lemma below.

f Y f x 2

LEMMA 4.2. (1) Let denote the principal divisor on deﬁned by +

2g 2

K Y f V V f = P P +

: 1 V Y

. Then we have i .

k =

i 1 irred +

2g 2

C V V C = P +

: V Y

(2) The divisor : i is regular as a subscheme

k =

i 1 irred

of Y .

B V V B g + P +

(3) is integral for any and, denoting by the divisor 1 1

B V V Y B = C f L = O B V

on , we have 2 . In particular, : Y is a square

O C Y

root of Y in the Picard group of .

f V Y

Proof. The statement for the horizontal part of is obvious. Let k be

v K Y

an irreducible component and denote by V the corresponding valuation of .

v x a = nV ;va a

i i

It is easy to see (cf. proof of Lemma 5.1) that V min ,

a 2fa ;::: ;a g T V +

where 1 2g 2 represents the vertex of that corresponds to .From

f this, the statement about the vertical part of follows immediately. Parts (2) and

(3) of the lemma are now easy to verify. 2

The proposition below describes a well-known construction of ramiﬁed double

coverings and lists some of its properties. C

PROPOSITION 4.3. Let Y be a regular integral Noetherian separated scheme,

Y L O F

an effective divisor on , an invertible Y -module and a global section of

2 1 1

L C F L L !O C ,!O Y , whose divisor is . induces a morphism Y ,

A = O L O Y

which on the sheaf : Y induces the structure of an -algebra. Let

= A

X : Spec . Then the following holds

X ! Y

(1) The structure morphism : is ﬂat and ﬁnite of degree 2.

Y 6= X (2) If C is regular and all residue ﬁelds of have characteristic 2,then is

regular.

V C V = W

(3) Let C be regular and a connected component of .Then 2 for a

X V prime divisor W on , which as a scheme is mapped isomorphically onto

by .

0 0

f Y !

(4) Let Y be another regular integral Noetherian separated scheme and :

0 0 0

Y f Y 6 C C = f C L = f L

a morphism transversal to C i.e. .Let : , : .

0 2

F = f F L O

As above, the section : of induces an Y -algebra structure on

0 0 0

0 1

A = O L A = X Y

: Y . We have Spec . Y Proof. One can restrict to the afﬁne case where the veriﬁcation is straight-

forward. 2

C L Returning to our special situation, let Y , , be as in Lemma 4.2 and deﬁne

F 2 Y; L O C

to be the image of the canonical section of Y under the iso-

O C !L = O B C = B +f Y morphism Y 2 induced by the relation 2 .The

comp4166.tex; 8/08/1998; 8:53; v.7; p.12

AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 49

0 1

X = O L

construction in the proposition gives us a scheme : Spec Y together

X ! Y

with a ﬁnite ﬂat morphism : of degree two.

0 0

X =R R PROPOSITION 4.4. X is regular and is a proper ﬂat -curve whose spe-

cial ﬁbre is a normal crossing divisor and whose generic ﬁbre is isomorphic to the

X V Y K

hyperelliptic curve .Let be an irreducible component of the special ﬁbre k

Y W = V g = = C:V g >

V V of .Weset V : and : 1 2 1.Then is integral and 2

and

g = C V = W = W V

(1) V 2 if and only if 1. In this case, 2 for an

0 0 0 0

X X X

exceptional divisor W of that meets the rest of the special ﬁbre of

V k

in exactly two points.

g = W = W q W W

V V; V; V;i (2) If V 1,then 1 2,wherethe are prime divisors

1 0

P X

isomorphic as schemes to meeting the rest of in at least two points.

k k

g > W k V

(3) If V 0,then is a prime divisor that as a scheme is a regular -curve

g g = W X

V V

of genus V .If 0,then meets the rest of in at least two points.

k 0

Proof. It follows from Lemma 4.2 and Proposition 4.3, that X is a regular

0 X proper ﬂat R -curve. Using the local description of coming from the construc-

tion as a double covering of X , it is easy to see that the special ﬁbre is a normal 0

crossing divisor. By 4.3.4. the generic ﬁbre of X is integral with function ﬁeld

K x[y ]=y f x X V

and is therefore isomorphic to K .Nowlet be an irre- Y

ducible component of k .

V = nV T If C 1then is odd and from the description of the tree it follows

V Y g = V:V = V

easily that cuts the rest of k in exactly two points. Therefore 2 1

0 0

W = W W

2. By Proposition 4.3.3, V 2 for a prime divisor which as a scheme

V V 0

V P X

is isomorphic to = and meets the rest of in exactly two points. It follows

k k

0 0 0

W :W = W

that 1. By Castelnuovo’s criterion, is exceptional.

V V V

C V = W X V

= Y

If 0, then by Proposition 4.3, V (as a scheme) and

W ! V C:V V

V is ﬁnite of order two, ramiﬁed over exactly points of .If

C:V > C:V = W +

0, then by Hurwitz’ formula 2 genus of V 2. In particular, g

V is integral.

T V

One checks easily that if V corresponds to an extremal vertex of , is at

C:V = V

least two. Thus, if 0, then does not correspond to an extremal vertex

Y 2

and cuts the rest of k in at least two points. 0

The above proposition shows in particular that the exceptional divisors of X

X V Y

are exactly the components of which dominate a component of k with

0 0

V = X ! X X

C 1. Let be the blow-down of all exceptional divisors of .Itis X

clear now that X is the minimal regular model of K and is semistable.

Z g + Y k

Let k denote the 2 2 -pointed curve that is obtained from by contracting

V Y C V = V

all components of k with 1 (observe that these do not carry marked points).

comp4166.tex; 8/08/1998; 8:53; v.7; p.13

50 IVAN KAUSZ

X ! Z X ! Y

k k

COROLLARY 4.5. The morphism k induced by is an admissible

k X

double covering and k is hyperelliptic without direction-reversing double points.

Proof. By Lemma 1.6, it sufﬁces to show the ﬁrst part of the corollary. It is clear

X ! Z q 2 Z

k k

that k has properties (1) and (2) of an admissible covering. Let

V ;V Z q q

be a double point and 1 2 k the two components meeting in .If is the

V Y Y ! Z

k k

image of a component of k under the blow-down map , then both

V X ! Z V q k

the preimage of 1 under k and that of 2 is ramiﬁed over .Otherwise,

X ! Z q X ! Z

k k k k is etale´ over . Therefore satisﬁes also condition (3) of

admissible coverings. 2

5. Proof of Theorem 3.1

By the Lemmas 3.2 and 4.1, we may assume that k is algebraically closed and that

X=R X

is the minimal regular model of the hyperelliptic curve K associated to an

equation + 2g 2

2 Y

y = f x = A x a ;

: i =

i 1

A a

where and the i satisfy the conditions listed in Lemma 4.1. We have already

X X

seen in Corollary 4.5 that the special ﬁbre k is hyperelliptic. Since is regular, X

all the double points of k have multiplicity one, so the second statement of the

theorem is trivially fulﬁlled and the formula for ord s is all that remains to be

shown. For this, some preparation is necessary.

X Y T X X

We keep the notation of the previous section. Thus the objects K , , , ,

etc., associated with the above equation, are deﬁned. We will employ some abuse of

X X V 2VT T notation to describe vertical divisors on Y , and :If isavertexof ,

we denote by the same symbol the corresponding prime divisor on Y . Its pull-back

X ! Y W

under the projection : will be denoted V , as in Proposition 4.4. If

C V 6= W X ! X

1, the image of V under the blow-down map will again be

W Y

denoted by V . Finally, we will not distinguish between vertical divisors on

V T

and elements of Z .

V T

F 2 V T D 2 Z D V = F

LEMMA 5.1. For let F be deﬁned by :

n F; V

inf .

2 R F 2 V T

(1) For any b there exists an such that the vertical part of the

x b 2 R [x] K Y D

principal divisor of is precisely F . If, on the other

F 2 V R=m b 2 fa ;::: ;a g

g +

hand, n is represented by 1 2 2 ,then

x b > D

divvert F .

h 2 R [x] r b ;::: ;b 2

(2) For any primitive polynomial of degree there exist 1 r

fa ;::: a g + 1 2g 2 with

comp4166.tex; 8/08/1998; 8:53; v.7; p.14

AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 51

h 6 x b :

divvert divvert i =

i 1

V T v

Proof. First we will introduce some notation. For a vertex of we let V

Y Y

be the valuation on K that corresponds to the vertical prime divisor on

V h= v h V h 2 K Y

2V T

associated to .Sowehavedivvert V for any .

V = V V v

If 0, we will omit the index from V . There will be no confusion arising

hx = c x 2 K [x] K Y

> i

from this, since for a polynomial i 0 we have

v hx = v c

V i >

0 min i 0 . V

From the construction of Y it follows that if is an irreducible component of

Y V U R

k , a smooth point of has an open afﬁne neighborhood that is -isomorphic

[ ] U x = t + a

to some open part of R such that over we have the equation ,

t 2 R n = nV a 2fa ;::: ;a g +

where isalocalparameter, ,and 1 2g 2 represents

V T hX 2 R [x] v hx =

as a vertex of . Therefore for any the equation V

ht x + a

v holds.

2 R n >

We prove the ﬁrst statement of the lemma. Given b ,welet 0be

F b R ! R=m V

maximal such that the image of under the mapping lies in n .

v x b=n F; V V T

We have to show that V inf for any vertex of . For this,

a 2fa ;::: ;a g V +

let 1 2g 2 be a representative of . By the above remark, we have

v x b=v t x + a b= nV ;va b

V min . Therefore the claim follows

0 0

F; V = fV 6 V j V bg

from the fact that inf max is represented by .

F T b 2fa ;::: ;a g +

On the other hand, given a vertex of let 1 2g 2 bearepresen-

V a 2fa ;::: ;a g +

tative. Clearly, for any vertex with representative 1 2g 2 ,wehave

n F; V 6 nV ;va b D 6 x b

inf min . Therefore F divvert . The proof of the second statement of the lemma is more involved. We need to

introduce further notation.

0 0 0

V T nV =nV V

If V and are two vertices of with 1and ,thenwe

0 0

V V V

will say that V is a predecessor of and that is a successor of . Each vertex V

except V0 has exactly one predecessor. The set of successors of 0 is precisely

V h 2 K Y V 6= V T v h

1.Let and 0 avertexof . Then we denote by V the

v h v V V h V

difference V ,where is the predecessor of .

2 R [x] V T I claim that for any primitive polynomial h and any vertex of we

have

h V = V

deg if 0

v h 6

V ( )

v h ; 0

V otherwise

V 2S V

V V

where S denotes the set of successors of .

2 R [x]

Admitting the claim for a moment, we conclude that given a primitive h

r I I f ;::: ;rg

V V

2V T

of degree , there exists a family V of subsets 1 with

I = f ;::: ;rg (i) V0 1 .

comp4166.tex; 8/08/1998; 8:53; v.7; p.15

52 IVAN KAUSZ

I I V 6 V V

(ii) V for .

I = [ 0 I V 2VT S V 6= ;

2S V

(iii) V for all with .

I > v h V >V V (iv) # V for all 0.

From properties (i)–(iii) it follows that

[

I = f ;::: ;rg:

V 1

V 2V T

V maximal

a 2fa ;::: ;a g b = a

g + i V

We choose a representative V 1 2 2 and set for all maximal

V T i 2 I

vertices of and all V .

V >V v x b = i

It is easy to see then that for arbitrary we have V =

0 i 1

I > v h v = v V < V < < V = V

V V V n

# V .Since ,where are

j =

j 1 0 1

V V

successive vertices of the linear subgraph of T connecting 0 and , it follows that

> v h: v x b

V V i = i 1

This proves the second statement of the lemma.

V 6= V a 2

It remains the task to prove the claim . Assume ﬁrst that 0.Let

fa ;::: ;a g V n = nV V +

1 2g 2 a representative for and .Let be the predecessor

V m = v h m = v h V

of and set for abbreviation : , : V . By the remark we made

h h 2 R [x] V

at the beginning of the proof, there are primitive polynomials V , such

that

n 1

x; ht x + a=t h

n m

ht x + a=t h x:

m 0 m

t v h=m m = v h tx htx=t h x

V V

Since obviously ,wehave V .

i i

0 0

x 2 t; x x= > h h k xt x

i V

Let 0 be maximal such that V .Then =

i 0

k x 2 R [x] k x 62 x; t i 2 i

for some polynomials i such that 0 foratleastone 0

i i

f ;::: ;g h k txx k txx tx=t

i i

0 . It follows that V and that

i= =

i 0 0

v h= h x= k txx

v i

is primitive. Therefore V and . In particular, = i 0

we have

h x 6 v h; V

deg V (a)

hx 2 k [x] h x tR [x]

where denotes the residue class of V modulo .

V T V ;::: ;V V

Now let be an arbitrary vertex of and let 1 s be the successors of .

i = ;::: ;s b 2fa ;::: ;a g V

g + i

For 1 we choose elements i 1 2 2 representing and set

c =b a=t c 2 R c 6= c m i 6= j

i i i j

i : .Then and mod for . Similarly as above,

i = ;::: ;s =v hx = v h tx + c

V V i

we have for 1 the equations i : and i

comp4166.tex; 8/08/1998; 8:53; v.7; p.16

AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 53

i i

q q

h x = u t x c u 2 R [x]

q q i

V for certain . Therefore the residue =

q 0

c c k h x

i i V

class i of in is an at least -fold zero of and it follows

v h 6 h x: V

V deg (b)

i = i 1

Clearly, (a) and (b) together imply the claim.

V T

2 Z K V = B V +nV

LEMMA 5.2. Deﬁne K by : . We consider the

x=y X !

=R global differential d on as a rational section of the dualizing sheaf X .

Its divisor on X is

d x

=g P + K V W : V

div 1 1

V 2V T

V 6=

C 1

t 2 R V 2 V T

Proof.Let be a local parameter. Let n be a vertex of with

C V 6= a 2 fa ;::: ;a g V q +

1andlet 1 2g 2 be a representative for .Let be a Y

smooth point of k that lies on the irreducible component which corresponds to

Y U q

V . By construction of , there is an open afﬁne neighborhood of which is

R R [ ] A = U; O

-isomorphic to an open part of Spec such that in Y we have

n 0

= t + a X

the equation x . By construction of ,wehave

1 1 2

U; O ' A[ ]= h; =U; O L

f V n

2 R [ ] t f t + a

where h is the primitive polynomial and where over

B V

0 1

= U y = t x=y =

U we have the equation . It follows that d

B V +nV

= U A

t d . By choosing small enough we may assume that is gen-

0 h erated by h and and we conclude (similarly as in the proof of Proposition 6.2 of

= ! 0 j =

Section6)thatd is an invertible section of X . The statement

U =R

x=y

for the vertical part of divd follows. For the horizontal part it is enough to X=R look at the restriction of d x=y to the generic ﬁbre of . But there the result is

classical (and easy to see). 2

Lemmas 5.3 and 5.4 below are of purely combinatorial nature. Nevertheless,

Lemma 5.4 is the key to our proof of the theorem.

I i 2 I f N ! Z

LEMMA 5.3. Let be a ﬁnite index-set. For any let i : 0 be increasing

mappings.

n n I

x I x 2 N

2N i2N (1) There is a sequence n of -tupels with the properties

0 i 0

n 1

2 I n 2 N ;x 6 x

(a) for any i and any ; i

0 i

n 2 N x = n

2I (b) for any we have i ;

0 i

comp4166.tex; 8/08/1998; 8:53; v.7; p.17

54 IVAN KAUSZ

f x f y j y 2 N ; y = n : =

i i i i (c) min i max min

i 0

i2I i2I

i2I

n I

f x N

i2I n2N (2) If the i are strictly increasing and a sequence in satisﬁes

i 0 0

n 2 N

the properties (a)–(c) of 1 , then for any 0 we have

x = f f x ; i

i min

n i

i2I

1 n

i 2 I x = x +

where n is the unique index with 1.

n n

The proof of Lemma 5.3 is straightforward, so we omit it. In the next lemma,

V T

D 2 Z w D D V

2V T

for ,wedenoteby the number minV .

F V T

i> LEMMA 5.4. There exists a sequence i 1 in with the following pro-

perties

m 2 N ;wK + D w K +

(1) For any F is maximal among the numbers

i =

0 i 1

D G ;::: ;G m V T m

G ,where runs through all -tupels of elements of .

i =

i 1 1

2VT

(2) For V set

V = ; 'V

' 2 1 if is even,

V =

'V = ;

1 2 otherwise.

V 6= V ; V m F > V

Then for all 0 is the number of indices with m .For

i > V F = V

0 we have i 0.

1 V

m 2 N K + D 2 Z

(3) Let . The mapping F takes its minimal value

i =

i 1

m 1

w K + D F m

F at the vertex .

i =

i 1

m 2 N nF

(4) For any , the number m is even.

N = fv a a j i 6= j g N = j

Proof. By induction on max i . The case 0is

V ;::: ;V V i = ;::: ;r T

trivial. Let 1 r be the elements of 2.For 1 we denote by the

T V T

full subtree of , whose vertices are greater or equal to i . is isomorphic to the

x a b =t 2 R [x] t

j i tree associated to the polynomial j ,where is a local

b 2 fa ;::: ;a g V 2 R=t R j

g + i

parameter, i 1 2 2 represents and where the index

i i

2 i

a b t n ' K i

runs through all values for which j mod .Let , , etc. be the objects

n ' K

associated to T , analogous to , , etc. For example, we have

V = nV ;

n 2

; K V = K V +'V

i 2

D V = D V

F 2

2VT VT for V; F .

comp4166.tex; 8/08/1998; 8:53; v.7; p.18

AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 55

i i

F V T

By induction hypothesis, there are sequences j in that have the

f N ! Z

properties 1–4 of the lemma. We deﬁne functions i : 0 by setting

0 0 1 1

x x

X X

@ @ A A

K V + f x=w K + D D V : =

i i

i min

F F

j j

V 2V T

j = =

j 1 1

f f x + > f x+

i i

The i are strictly increasing ( 1 2) and we can apply Lemma 5.3

i I = f ;::: ;rg

m2N

to obtain a sequence m 0 of elements in : 1 with

m m

= f x f x m 2 N ; i

i min for all

m i

i 0

i2I

x = f 6 k 6 m j i = ig m >

where # 0 1 k . Now we deﬁne for 0

x < F ; f ; m

if i 0

m m +

x 1

F = +

m 1 :

V : 0 ; otherwise

I contend that this sequence has the required properties.

m 0

D = K + D n = V V

F i2I i

For abbreviation, set m : and .Let be

j i =

j 1 0

V V

the unique element between 0 and i . It is not difﬁcult to see that the following

statements hold true:

F = V m > n +

(i) m 0 if and only if 0 1.

m 6 n i = ;::: ;r; x = f 6 j 6 m j F > V g i (ii) For all and 1 we have # 1 j .

0 i

i = ;::: ;r V = fm 2 N j F > V g

m i

(iii) For all 1 we have i # 0 .

0 i 0

m 6 n D V > w D i > D V >

m m

(iv) If 1, we have m for all and 0 i

0 i

w D i

m for at least one .

0 i

m > n i 6 D V 6 w D m (v) If then for all we have 0 m .

0 i

Using these statements, properties (1)–(4) for the sequence m are easy to

derive. 2

F H X; ! i

With the help of the we can now construct a basis of W=R .

F V T

PROPOSITION 5.5. Let i 1 be a sequence in satisfying the properties

: i 2 N b 2 fa ;::: ;a g

g +

listed in Lemma 5 4. For every let i 1 2 2 be an element

F 2V R=m t 2 m

representing the class n .Let denote a prime element

6 i 6 g

of R and set for 0 1

0 1

i Y

d x e

i 0

@ A

! = t x b 2 H X ; ;

i j K =K

: X

y =

j 1

e = w K + D w : F

where i : and is deﬁned as before Lemma 5 4.Then

j = j 1

comp4166.tex; 8/08/1998; 8:53; v.7; p.19

! ;::: ;! R H X; !

g =R

(1) 0 1 is an -basis of X

g 1

X X

X 1 1 2

e = V V + + V ;

(2) i 1

2 2

V>V V>V =

i 0 0 0

V 'V

' even odd

V :

where is deﬁned as in Lemma 5 4.

! > ! 2 i Proof. (1) By Lemmas 5.1 and 5.2 we have div i 0 and therefore

0 0

X; ! i = ;::: ;g ! K H X ; H

i K =R

X for 0 1. Since the form a -Basis of

1 0

H X; ! =R

, they are linearly independent. We show that they span X .For

X =K

= t hx x=y hx 2 R [x] d 6 g this let ! d (where is primitive of degree 1)

H X; ! =R

be an arbitrary element of X . By Lemma 5.1 and by construction of

F w K + h 6 e ! > e 6 e

i d

the ,wehave divvert d .Sincediv 0, it follows and

c 2 R ! c ! = h x x=y

therefore there exists an element , such that d 1 d for a

x 2 R [x] d d

polynomial h of degree at most 1. We proceed by induction on . F

(2) By property 3 of the i ,wehave

e = K F D F :

i i+ F i+

1 i 1 =

j 1

F = V i > V = g K V = D V = F Since furthermore i for and 0, it

0 0 0 j 0

follows

1 i 1

X X X X

e = K F D F :

i F i

i ( )

i= j = > i> 0 i 1 1 1

For the ﬁrst term of this equation, we have

X X

K F = V 'V :

i 2

2V T

1 V

V 2 N

n 2

K F D F =

F j

This follows from the deﬁnition of and property 2 of the i .Since

D F = fV 2 V T j nV 2 N ;V 6 F ; V 6 F g

F i j

2 # 2 i ,wehaveforthe j

second term

i 1

X X X

D F = ; V V i

F 1

i> j =

2V T

1 1 V

nV 2

2N F

where we have again made use of property 2 of the i . Plugging in these expres-

2 sions into , we get the required formula.

comp4166.tex; 8/08/1998; 8:53; v.7; p.20

AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 57

g + g +

4 4 4 2

D = ! ;::: ;! A a a

g i j

6=j Let : 2 i and 0 1 as in the above

proposition. Then obviously

g e +

8g 4

= D t ! ^ ^ ! ;

g =R

X 0 1

g 1

t 2 R e = e

where is a prime element and i . Therefore we have =

i 0

g 1

= g v D g + e :

s i =R

ord X 8 4 (1) = i 0

It is easily seen that

X X

D = V + V + + V + V :

v 2 2 2 1 2 1 2 V>V

V>V0 0

V 'V ' even odd

Plugging in this and the formula of Proposition 5.5.2 into (1), we get

= V + g V

s =R

ord X 2 1

V>V0

' even

V g V : +2 (2)

V>V0

' odd

V T

There is an obvious one-to-one correspondence between vertices > 0 of

V 2 V n > V

and its edges (to n 1 there corresponds the edge that connects

0 0

V V V < V ;V 2 V

and ,where is characterized by n 1), and therefore also a

V 7! p V T fV g Y V

bijection from 0 to the set of double points of k .Weset

'p = 'V p = V Z g +

V V

: and : .Let k be the 2 2 -pointed prestable

Y V C V =

genus-0 curve obtained from k by contracting all components with 1

Y r Y ! Z

k k

of k (cf. end of Section 4) and let : be the contraction map. Observe

p p Y r p =r p 'p ='p

that for two double points 1, 2 of k with 1 2 ,wehave 1 2 ,

' Z

so that and are deﬁned on the set of double points of k . Observe furthermore

q Z Y q k

that for a double point of k the number of double points of lying above is

q 'q

one, if ' is even and it is two, if is odd. So we can reformulate (2) as

= q + g q

ord X 2 1

q 2Z

k double point

' even

+ q g q :

4 (3)

q 2Z

k double point

q ' odd

comp4166.tex; 8/08/1998; 8:53; v.7; p.21

58 IVAN KAUSZ

q Z Z ;Z Z k

Let be a double point of k and let 1 2 be the closed subsets of

Z Z fq g k

k whose union is and which intersect precisely in . Then one of these

'q Z

two subsets carries exactly marked points of k , while the other carries

g + 'q X ! Z k

2 2 of these. The admissible double covering k is ramiﬁed

over q if and only if is odd. It follows from Lemma 1.6 that in this case the

X q l = f q ;g q g l

double points of k lying above are of type ,where min ,

'q l = f q ;g q g

whereas if is even, they are of type l ,where min 1 . After these considerations it is clear that (3) implies the formula of the theorem.

Remark. From the above proof it is easy to see that there are hyperelliptic

g 1

X=K x=y ^ ^ x x=y <

curves , such that ords d d 0 for any choice of

= f x

equation y . This is in opposition to the case of elliptic curves, where

x=y =

ords d 0 if and only if the corresponding equation is minimal. Still one

y = f x f

could think of calling an equation minimal,ifords is minimal –

g 1

x=y ^^x x=y

or, what is the same, if ords d d is maximal (cf. [Lock], where

such an attempt has been made). But unlike in the case of elliptic curves, we will

> =g =f

s s =K in general have ord min 1 ord X ,where min for a minimal

= f x X=K equation y for the hyperelliptic curve .

6. Good reduction and characteristic two There should be an analogue of Theorem 3.1 in characteristic two. In this section,

we prove this analogue only in the case of good reduction and generic characteristic

= R; K; k ; v K 6=

6 2. More precisely, let be a discrete valuation ring where char

X=R R g

2andk is perfect and let be a smooth proper -curve of genus with

g +

g 0 1 8 4

X 2 M =^ H X ;

K K

hyperelliptic generic ﬁbre K .Let : be

X =K

g +

g 0 1 8 4

=^ H X;

deﬁned as in Section 2. As in Section 3, M : deﬁnes

X=R M

an integral structure on K and it makes sense to speak of the order of vanishing

s 2 R

ords of at the closed point Spec . Our objective is to prove that regardless

= k

of char we have ords 0, thereby justifying the power of 2 involved in the

R =R

deﬁnition of . It is clear that it is enough to show this after a base extension ,

=R R where R is a discrete valuation ring dominating . So the statement will follow from Lemma 6.1 and Propositions 6.2 and 6.3 below.

LEMMA 6.1. After some base-change with a discrete valuation ring dominating

U X R , there exists an open afﬁne subset which is isomorphic to the Spec of

B = A[y ]=y + ay + b;

2 A = R [x] f x=ax bx 2 K [x]

for some a; b : such that the polynomial 4

+ ax 6 g + bx 6 g +

is separable and of degree 2 g 2 and such that deg 1, deg 2 2.

x; b x 2 k [x] a x = g +

For the reduced polynomials a we have deg 1 or

x > g + deg b 2 1.

comp4166.tex; 8/08/1998; 8:53; v.7; p.22

Proof. By [L-K] (cf. proof of their Proposition 5.14 and Theorem 4.12), there

= X ! Y exists a faithfully ﬂat ﬁnite R morphism of degree two onto a twisted

1 0

R R =R P over Spec . Passing, if necessary, to an etale´ surjective extension ,we

1 P may assume that Y is isomorphic to ([L-K] Corollary 3.4). We may choose this

1 R

X ! P P

1;K

isomorphism such that K is unramiﬁed over the point at inﬁnity of K

1 0 1 1

P U = P P = R [x] P P P

1 1;K

.Let : 1 Spec ,where is the closure of in .

K R R 0

= U = B R [x] B

Then U : Spec for a ﬁnite ﬂat -algebra of degree two. By

R [x] [Sesh], B is free as an -module and it follows

B A[y ]=y + ay + b;

= ( )

2 A = R [x] a

for some a; b : . We chose a representation such that has minimal

a 6 g + b 6 g + degree. Then it is easy to see that deg 1 and deg 2 2. Regarding

y k X y > f a; bg p 1 as an element of k ,wehavediv min deg 2 deg ,where

p Y = P n 6 g 1

is the place at inﬁnity of k . Since, by Riemann–Roch, for any the

k ' 2 k X ' > n p 1

-vectorspace of functions k with div is contained in the

g 2

span of 1 ;::: ;x , the last statement of the lemma follows.

x= y + a

PROPOSITION 6.2. In the situation of Lemma 6 :1, the differential d 2 is

x x= y + ai = ;::: ;g nowhere vanishing on U and the differentials d 2 0 1) 1

extend to regular global sections of .

X=R

' =B x B y =F x +

Proof. It is easy to see that the morphism : d d x d

B=R

F y ! B ' x = F ' y = F B

y x y d ,deﬁnedby d , d , is an isomorphism of -

modules This proves the ﬁrst part of the proposition. As to the second part, it sufﬁces

x= y + a to check that the restrictions of the differentials x d 2 on the generic ﬁbre

U 2

K extend to regular global sections of . But this is well-known.

X =K

g +

4 4

x 2 R [x] : D =

PROPOSITION 6.3. Let f as in Lemma 6 1 and set : 2

f f f D R

,where is the discriminant of (cf. Section 2 .Then is a unit in .

k 6= k =

Proof.Ifchar 2, this is clear. So assume char 2. In what follows,

n i m i

C P = u T Q = v T C [T ] i

if is any ring and i , are two polynomials in ,

= i=

i 0 0

n;m

P; Q 2 C P Q

we denote by R the resultant of and (c.f. [vdW] Sections 34,

T ;

2 2;1 2 1

F; F F; F F x; y = y + axy + bx Q = R P = R y

35). Let : , : x , : and

y y

2 R P

A the leading coefﬁcient of .Thenwehave

+ ; g +

2g 2 4 2 2

R : P; Q=A D x

We can read this equation as a formal identity between polynomials in the coef-

+ ; g +

2 2g 2 4 2

x bx D 2 R A j R P; Q

ﬁcients of a and and conclude that and .

[x] ! k [x] h 7! h A 6= Let R , be the residue homomorphism. If 0, then we have

2 2

+ ; g +

2g 2 4 2

= g + D = A R P; Q A =

deg P 2 2and . If to the contrary 0, then

6 g Q = g it is easy to see that we may assume deg P 2 and deg 4 and that we have

2 2 g

2g;4

D = R P; Q D 6= X 2

. Thus in any case 0 by smoothness of k . x

comp4166.tex; 8/08/1998; 8:53; v.7; p.23

7. Asymptotic of metrics X=R

Let R be the ring of integers in a number ﬁeld and let beasemistablecurve

> X=R

with smooth hyperelliptic generic ﬁbre of genus g 2. Assume that is the R

minimal regular model of its generic ﬁbre. Let s be a closed point of Spec . Denote

X=R X=R s

by s the number of singular points in the geometric ﬁbre of over .

A ;A ;B j

By an elementary estimate of the coefﬁcients of 0 i in the expression for

=R ord X in 3.1, we deduce the inequality

X=R : 6 g

s s

=R ord X (1)

The purpose of this section is to globalize this inequality in the sense of Arakelov- geometry (Theorem 7.7) and to deduce from this an upper bound for the self-

intersection of the relative dualizing sheaf on X=R (Corollary 7.8). To this end, we will establish an analogue of (1) at the inﬁnite place (Theorem 7.1). We start with

some deﬁnitions and notations.

> X ! S

Let g 2 be an integer. Let : be an analytic stable curve of genus

g ! ! X=S

. Then we denote by X (or by ) the relative dualizing sheaf of .

X ! S =S

If : is generically smooth and hyperelliptic, we denote by X the

g +

8 4

canonical section of det X , deﬁned as in Section 2.

jj jj !

A C -metric Mod on is a rule that to each analytic stable curve

X =M

g g

g S jj jj

X=S of genus over a complex manifold associates a continuous metric

on the line bundle X , such that the rule is compatible with any basechange and

C X=S

such that jj jj is ,if happens to be the universal local deformation of a

1 !

stable curve. It is easy to see that C -metrics on exist(cf.[Bost3],4.2.4,

X =M

g g p. 249). It would be desirable to have a canonical choice of such a metric, but I do not know of any such construction. Because of its bad behavior in the neighborhood

of degenerate ﬁbres (cf. [Jor], [Wen]), the Arakelov-metric (cf. [Ara2], p. 1178 or

1 !

[Sou], p. 338) unfortunately cannot be extended to give a C -metric on .

X =M

g g

C g C Let C= be a smooth proper curve of genus over . The hermitian scalar

R 0 1

hs ;s i = i= ^ s jj jj H X;

product 2 s . induces a metric on ,

1 2 C 1 2 0

C=C which we will call the natural metric. We will call natural metric and will denote

0 1

jj jj = H X; C

by the same symbol also the metric induced by 0 on C= det ,

C=C and on tensor powers of this vectorspace.

L C jj jj jj jj L

Let be a line bundle on and let and L be metrics on and ,

C=C ;

1 0 1 2

s ;s L L L respectively. For two global C -sections of (of )their scalar

1 2 C product is deﬁned by

1 Z

s ;s i = hs x;s xi x: h 2 : d

1 2 L 1 2

C jj jj

Here the scalar product under the integral is induced by the metric L (by

jj jj jj jj C

the metrics L and ), and d is the volume form on locally deﬁned by

comp4166.tex; 8/08/1998; 8:53; v.7; p.24

2 1

= i= ^ =jj jj

d 2 ,where is an invertible local section of .The

C=C

C; L

vectorspace H can be identiﬁed with the kernel of the Dolbeault-complex

1 1

0;1

C C; L !C C; L

1 ?

C; L= = =

and we have canonically H coker im ker . Restriction of

1 1

2 0;1 0

C C; L C C; L H C; L the L -norms of and gives metrics on and

1 0 1 1

C; L L = H X; L H X; L

H andbythiswegetanormon : det det ,

jj jj = jj jj

jj jj 2

L;jj jj;jjkj

which we denote by the symbol .TheQuillen-metric Q;

L;L

on is deﬁned as

0 =

1 2

jjjj = ; jj jj 2

Q : det

L;L

1 2

where det is the regularized determinant of the Laplace-Operator

0 (cf. [S-A-B-K], Ch. V, for the deﬁnition of det ).

Let jj jj still denote a smooth hermitian metric on the line bundle .The

C=C jj jj

delta invariant of C with respect to the metric is deﬁned as the real number

jj jj

; C = : 12 log

jj jj0

jj jj jj jj

C C=C where 0 is the natural metric on C= and is the Quillen-metric on

associated to jj jj. Now we have all necessary deﬁnitions at hand, to be able to formulate the

analogue of inequality (1) ‘at inﬁnity’.

> C jj jj !

THEOREM 7.1. Let g 2 be an integer and ﬁx a -metric Mod on .

X =M b

g g

> c 2 R

Let " 0. There exists a constant such that for all smooth hyperelliptic C curves C= we have the inequality

jj jj 6 g + " C +c; C

log C= 0

C C jj jj where is the delta-invariant of with respect to the metric induced by Mod

on C .

The proof of the theorem will be given after Propositions 7.3–7.6 below.

C g +

DEFINITION 7.2. (a) Let Y= be a 2 2 -pointed stable genus-zero curve and

2 Y Y let p be a singular point. The partial normalization of is the disjoint sum of

two connected pointed curves. Let k be the number of marked points on one of its

= k; g + k

components and put m : min 2 2 . The following cases are possible

= i + i 2f ;::: ;[g = ]g (1) m 2 2forsome 0 1 2 .

comp4166.tex; 8/08/1998; 8:53; v.7; p.25

62 IVAN KAUSZ

Then we call to be of type i .

= j + j 2f ;::: ;[g= ]g

(2) m 2 1forsome 1 2 .

Then we call to be of type j .

C g > p 2 X

(b) Let X= be a stable curve of genus 2andlet be a singular point.

X p

Let X be the partial normalization of at . The following cases are possible

(1) X is connected. Then we call to be of type 0.

j 2f ;::: ;[g= ]g

(2) X is the disjoint union of two prestable curves. Let 1 2 be the

minimum of their respective geni. Then we call to be of type j .

g > C

Let X0 be a hyperelliptic stable curve of genus 2 over . In what follows,

!U X

we review the construction of the universal local deformation Xhsc hsc of 0 0 as a hyperelliptic stable curve (cf. [H-M], Section 4, [C-H], Section 4). Let X0

be the modiﬁcation of X0 in its direction-reversing double points. By deﬁnition, Y

there exists an n-pointed prestable genus-zero curve 0 and an admissible double

! Y n = g +

covering X0 0.ByLemma1.6wehave 2 2, and it is easy to see that

Y Y !U

stability of X0 implies that 0 is in fact a stable pointed curve. Let spc spc be the

U universal local deformation of Y0 as a stable pointed curve. The locus spc spc

of singular curves is a normal-crossing divisor whose branches correspond to the

2g 1

U C

singular points of Y0. We may assume that spc is an open neighborhood of

t ::: t = t ;::: ;t

and that 1 d 0 is an equation of spc,where 1 2 1 are the standard

2g 1

C d Y I

coordinates on and is the number of double points of 0.Let i (resp.

J k 2 f ;::: ;dg ft = g j

) be the subset of indices 1 , such that k 0 is the branch

Y j

of spc, which corresponds to a double point of type i (resp. of type )of 0

= ;::: ;[g = ] j = ;::: ;[g= ] U

(i 0 1 2 , 1 2 ). Choose some small neighborhood adc

2g 1

C z ;::: ;z U

of the origin of and let 1 2g 1 be the standard coordinates on adc. U Consider the morphism from Uadc to spc, given by the equations

t = z i 2 J;

i for

t = z : i

i else

!U Y U !U

Let Yadc adc be the pull-back of spc by adc spc. There exists a admissible

!Y X

double covering Xadc adc which is unique up to isomorphism. The ﬁbre of adc

X X !U

over the center of Uadc is isomorphic to 0.Thecurve adc adc is semistable.

! U

It is the blow-up of a hyperelliptic stable curve X adc along the closed

00 00

X X ! U

subvariety D of direction-reversing double points. adc is unique

00 X

up to isomorphism. The special ﬁbre X0 is isomorphic to 0. Now consider the

2g 1 C morphism of Uadc to deﬁned by

u = z i 2 I ;

; u = z i

i else

2g 1

u ;::: ;u C X ! U

1 2g 1 being the standard coordinates on .Thecurve adc

X !U U descends to a hyperelliptic stable curve : hsc hsc over a neighborhood hsc

comp4166.tex; 8/08/1998; 8:53; v.7; p.26

AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 63

2g 1

X !U X of the origin of C and : hsc hsc is the universal local deformation of 0 as a hyperelliptic stable curve. In the following diagram, which gives an overview of the construction, the two extreme squares are cartesian

blow-up :

00 admiss double

- -

Y Y X X

Xhsc adc covering adc spc

? ? ? ?

? ramiﬁed ramiﬁed

U U ======U ======U

hsc covering adc adc adc covering Uspc

2 f ;::: ;[g = ]g j 2 f ;::: ;[g= ]g U

For i 0 1 2 (resp. for 1 2 ), let hsc hsc

(resp. hsc hsc) be the locus of curves carrying a double point of type i (resp.

u =

2I k

of type ). It is the normal-crossing divisor given by the equation k 0

u =

2J k

(resp. by k 0).

PROPOSITION 7.3. Let be the canonical section of the line bundle

8g 4

! U

det . Then we have the following equality of divisors on hsc

g = ] [g= ]

[ 1 2 2

X X

0 i

=g + i + g i + j g j :

div hsc 2 1 hsc 4 hsc

= j =

i 1 1 C

Proof.Let D be the unit disc in . It sufﬁces to prove that the stated equality !U holds for the pull-back of the divisors under any morphism D hsc. But this can be shown analogously to the proof of Theorem 3.1. Alternatively, one may derive the equality from Proposition 4.7 of [C-H] and the fact that over the moduli space

of smooth hyperelliptic curves, our canonical section provides a trivialization of

g + 2

the 8 4 th power of the Hodge-bundle.

X ! U X

Now let : sc sc be the universal deformation of 0 as a stable curve (cf.

= ;::: ;[g= ] U

[Bi-Bo] pp. 19–21). For j 0 2 ,thelocus sc sc of curves carrying

a double point of type j is a normal crossing divisor. We may identify sc with

[g= ]

2 j

3g 3

C =

an open neighborhood of 0 2 such that the divisor : sc is =

sc j 0

f = f = t t ;::: t

given by 0, where : j (the functions being the standard =

i 1 1 3 3

3g 3

d X

coordinates on C and the number of singular points of 0). By the universal

! U

property of Xsc sc, we have a canonical morphism from (a neighborhood

U X X

of 0 in) Uhsc to sc, such that hsc is the pull-back of sc via this morphism. The !U following proposition follows from the above construction of Xsc sc.

PROPOSITION 7.4. We have the following equalities between divisors on Uhsc

comp4166.tex; 8/08/1998; 8:53; v.7; p.27

64 IVAN KAUSZ

g 1 2

0 0 i

= + ;

sc hsc 2 hsc =

i 1

j j

= j = ;::: ;[g= ]: sc hsc for 1 2

A special case of Theorem 4.1 in [Fa-Wu],¨ p. 17 is the following

PROPOSITION 7.5. Let be an invertible section of det . Then there exist

;C > U 2U U constants C1 2 0, and a neighborhood of 0 sc such that on sc we

have the inequalities

C C 1 2 1 2

6 jj jj 6 C :

C1 log 0 1 log

jf j jf j

Remark. Using results of Y. Namikawa, we have shown in [Ka] the following

! h U

more precise statement: Let be an invertible section of det and let : sc

R ! be deﬁned by sc +

: =

h : max log

i= ;:::;d

t j

1 j

0 00

Then there are constants 0

on U sc we have the inequalities

0 2

h 6 jj jj 6 C h ; C 0

where is the rank of the ﬁrst homology group of the graph associated to the stable

curve X0. In what follows, however, we only need the statement of Proposition 7.5 above. One of the hardest ingredients of our proof of Theorem 7.1 is the following

consequence of a result of Bismut and Bost ([Bi-Bo], Theorem 2.2) that describes

the asymptotic of the Quillen-metric (observe that the bundle denoted by in

[Bi-Bo] is inverse to our determinant bundle det ).

PROPOSITION 7.6. Let be an invertible section of det . There exists a

U U

continuous function h1 on sc, such that on sc sc we have the equality

jj jj = jt j + h :

Q i

12 log U log 1 =

i 1 I

Proof of Theorem 7.1.Let g be the (coarse) moduli space of smooth hyperel-

g C M

liptic curves of genus over . It is a closed subset of the moduli space g of all

I I M

g g smooth curves. Let g be the Zariski-closure of in the moduli space of stable

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AN UPPER BOUND FOR ! FOR HYPERELLIPTIC ARITHMETIC SURFACES 65

[C ] 7! jj jj [C ] 7! C I R

g C

curves. The maps C= and from to are continuous.

I I I

g g

Since g is compact, it sufﬁces to show that for each point of there exists

V I jj jj < g + " C V \I

g g C

a neighborhood such that log C= over .

X !U Such a point is represented by a hyperelliptic stable curve X0.Let : hsc hsc

be the universal local deformation of X0 as a hyperelliptic stable curve. Adopting

the above notation for the various divisors on Uhsc,let

a 2 U ; O ; i = ;::: ;[g = ]; U

i hsc hsc an equation for hsc 0 1 2

b 2 U ; O ; j = ;::: ;[g= ]; j

hsc Uhsc an equation for hsc 1 2

; ! : 2 U hsc det an invertible section

By deﬁnition of the delta invariant, Proposition 7.4 and Proposition 7.6, there exists U

a continuous function '1 on hsc, such that

g = ] [g= ]

[ 1 2 2

X X

X =U = ja j ja j jb j jj jj + ' : j

hsc hsc log 0 2 log i log 12 log 0 1

= j = i 1 1

On the other hand, from Proposition 7.3 we have

g = ]

[ 1 2

jj jj = g ja j + i + g i ja j

log Xhsc hsc 0 log 0 2 1 log =

i 1

g= ]

[ 2

j g j jb j + g + jj jj + ' ; +

4 log j 8 4 log 0 2 =

j 1 U

for some continuous function '2 on hsc. From Proposition 7.5 it follows that in

= + jj jj j

the neighborhood of hsc : i hsc hsc the function log 0 is negligible

ja j jb j j

compared with log i and log . Therefore an elementary comparison of the

ja j jb j X =U j

coefﬁcients of log i and log in the above expressions for hsc hsc and

jj jj U U

X U

log hsc = hsc 0 proves the existence of an open neighborhood hsc of hsc

\ U such that over U hsc hsc the inequality

jj jj < g + " X =U

X U

log hsc = hsc 0 hsc hsc

U ! I [X ] 2 I g

holds. The canonical map hsc g identiﬁes an open neighborhood of 0

with the quotient of Uhsc by the natural action of the ﬁnite group Aut 0 .In

U 2 particular it is open, so we may take V as the image of under this morphism.

In what follows, we assume that the reader is familiar with the basic notions of

(low-dimensional) Arakelov-theory, as presented for instance in Soule’s´ Bourbaki

K S = O X=S

talk [Sou]. Thus, in particular, if be a number ﬁeld, : Spec K and

comp4166.tex; 8/08/1998; 8:53; v.7; p.29

an arithmetic surface, we have the notion of the degree of metrized line bundles on X

S and the intersection pairing between metrized line bundles on .

jj jj ! "> K

THEOREM 7.7. Fix a C -metric Mod on and an 0.Let be a

X =M

g g

X ! S = O

number ﬁeld. Let : Spec K be a minimal regular arithmetic surface >

with smooth hyperelliptic generic ﬁbre of genus g 2. Assume good reduction at

jj jj

all places dividing 2 and semistable reduction at all other places. Let 0 be the

! K ,! C X

natural metric on det X . For each embedding : let be the

X jj jj

delta-invariant of the Riemann-surface with respect to the metric Mod.Then

jj jj ;" jj jj there exists a constant c Mod , depending only on the metric Mod and on

", such that the following inequality holds

! X

2 X

g + "

! ; jj jj 6 N p + X

p =S

deg det X 0 log +

8 g 4

+ cjj jj [K Q ]: Mod ;" : Proof. This is an immediate consequence of inequality (1), Theorem 6 and

Theorem 7.1. 2

COROLLARY 7.8. Assume the conditions of the above theorem. Then there exists

jj jj ;" jj jj " a constant c Mod , depending only on Mod and , such that the following

inequality holds

0 1

2 X

g + "

@ A

X ! :! 6 N p +

=S X=S

X 3 1 log +

2 g 1

K,!C

2jS j

p :

c jj jj ;" [K Q ];

+ Mod :

! :! X

=S X=S

where the intersection number X and the delta-invariants are

jj jj ! =S understood with respect to the metric induced by Mod on X . Proof. By a theorem of Deligne ([Del], Theorem 11.4, cf. also [Sou], Theor´ eme` 4),

we have

! :! = ! ; jj jj N p [K Q ] ag ;

p Q

=S X=S X=S

X 12 deg det log :

p2S

g for some constant a , which depends only on the genus. With this, the corollary

follows immediately from Theorem 7.7. 2

It was ﬁrst observed by A. N. Parshin that a certain upper bound for the self- 2

intersection ! of the relative dualizing sheaf on arithmetic surfaces would have

interesting number-theoretical consequences as, for example, the abc-conjecture (cf. [Par1], [Par2], [Par3]).

comp4166.tex; 8/08/1998; 8:53; v.7; p.30

The shape of the upper bound as proposed by Parshin was suggested by the analogous geometric situation of a surface ﬁbred over a curve, where it follows from the so-called Bogomolov–Miyaoka–Yauinequality between the Chern classes of the surface. In the sequel, L. Moret-Bailly formulated as a hypothesis (cf. [MB] 2

‘Hypothese` BM’ (3.1.2)) a more general shape of an upper bound of ! which

he showed would still imply e.g. the abc-conjecture. Observe that Corollary 7.8 gives an upper bound which has the form required in [MB]. One might object 2

that in Moret-Bailly’s hypothesis the self-intersection ! and the delta-invariant

are deﬁned with respect to the Arakelov-metric. But it is easy to see that

2 1

! X C

in the hypothesis one could take and with respect to any -metric on

! , and draw the same conclusions. Only the constants which appear in the

X =M

g g estimates, would depend on the choice of the metric. Nevertheless our result does not sufﬁce to draw arithmetic consequences at least not along the lines of [MB]. In fact, recall the crucial argument in [MB]

which shows that Hypothese` BM implies a version of Mordell’s conjecture: Let

K K B be a smooth proper curve over a number ﬁeld . After replacing by some

ﬁnite extension and B by a ﬁnite etale´ covering, we may assume, by Kodaira’s

construction, that over the base B there exists a non-isotrivial smooth family of

! B curves V . One gets an upper bound on the height of rational points of

B by applying Hypothese` BM to models of the ﬁbres of the family. In view of Corollary 7.8, it is therefore natural to ask whether there exists a proper smooth curve which parametrizes a non-isotrivial family of smooth hyperelliptic curves.

Unfortunately, the answer to this question is negative. This fact seems to be well-

V ! B

known, but for sake of completeness we give a short proof: Let : be a

smooth family of hyperelliptic curves. By Theorem 3.1, the degree of det V=B

vanishes. As has been shown by Arakelov, this implies the isotriviality of V=B ([Ara1], Lemma 1.4 and Corollary 1 to Theorem 1.1).

Acknowledgements

Part of this paper was written during a stay at the Institut des Hautes Etudes´ Scientiﬁques, whose hospitality is gratefully acknowledged.Finally, I want to thank Uwe Jannsen warmly for constant encouragement and many helpful discussions.

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