CLASSIFICATION OF RIGID ANALYTIC SURFACESKENTARO MITSUI

Contents
1. Introduction
2. Preliminaries
2.1. Intersection Theory
2.2. Coherent Modules and Flat Base Changes
2.3. Resolutions of Singularities
2.4. Fields of Meromorphic Functions
3. Elliptic Fibrations and Jacobian Fibrations
3.1. Definitions and Fundamental Results
3.2. Canonical Bundle Formula3.3. Jacobian Fibrations
3.4. Noether’s Formula
4. Classifications
4.1. Proper Regular Surfaces of Algebraic Dimension Two
4.2. Proper Regular Surfaces of Algebraic Dimension One
4.3. Proper Elliptic Fibrations
References

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1. Introduction
The main purpose of the present paper is to give classification of proper
regular rigid analytic surfaces. The classification is similar to Kodaira’sclassification of complex analytic surfaces. Some analogous theorems holdand certain positive characteristic phenomena occur.
We treat proper regular rigid analytic surfaces of algebraic dimension
one or two, especially elliptic surfaces. Bombieri and Mumford studied
Enriques’ classification of algebraic surfaces in arbitrary characteristic andgave a list of possible invariants and combinations of multiple fibers forcertain elliptic surfaces in [1]. We give a similar list for rigid analytic surfaces. In the rigid analytic case, non-algebraic elliptic surfaces appear withThis work was supported by Grant-in-Aid for JSPS Fellows (21-1111).
1

2

KENTARO MITSUI

the invariants and combinations of multiple fibers that can never appear in
the algebraic case, nor in the complex analytic case.
Ueno first studied rigid analytic surfaces in [29]. We gave the detailed
proofs of the study in [23] and [21] partially. Using these results, we study
proper regular rigid analytic surfaces. In the following sketch of the results,
for simplicity, we assume that the base field is algebraically closed.
First, we review and develop the fundamental theories of rigid analytic
surfaces. We treat the intersection theory for non-proper normal rigid analytic surfaces, flat base change theorems, resolutions of singularities, andfields of meromorphic functions.
Second, we study rigid analytic elliptic fibrations. We give a relative duality theorem, a canonical bundle formula, and a relative Noether’s formula.
In the algebraic case, these formulas correspond to special cases of theGrothendieck duality theorem, the canonical bundle formula of Bombieriand Mumford [1], and the relative Riemann-Roch theorem of Deligne [12]respectively. As for the relative Noether’s formula, we have to modify
the relative Riemann-Roch theorem by the study of reductions of curvesof genus one [19] since the fibrations are not always smoothly fibered.
Third, we study proper smooth rigid analytic surfaces of algebraic dimension one or two. We prove that proper smooth rigid analytic surfaces ofalgebraic dimension two are projective rigid analytic surfaces and those ofalgebraic dimension one are rigid analytic elliptic surfaces.
Finally, we study proper rigid analytic elliptic fibrations. The canonicalbundle formula and Noether’s formula enable us to classify minimal proper
rigid analytic elliptic fibrations that are not properly elliptic. We give a listof possible invariants and combinations of multiple fibers for these surfaces.
Let us be more precise on the first two parts of the story. In the first part,
we algebraize the normalizations and blowing-ups of rigid analytic surfaces
locally in order to apply Lipman’s desingularizations [17]. Then we obtainregular models of quasi-compact rigid analytic surfaces. Using this result,
we construct algebraic reduction morphisms.
In the second part, we reduce the discussion on rigid analytic elliptic
fibrations to the corresponding algebraic arguments by using Conrad’s étale
coverings [9] and the Zariski’s main theorem of de Jong and van der Put[11]. Since the base changes of rigid analytic elliptic fibrations via the étale
coverings are projective, by Köpf’s GAGA theorems [15], we may apply
algebraic results to the covering elements. Finally, to conclude the proof ofthe desired theorems, we use the faithfully flat descent theory of Bosch andGörtz [3].
Notations and Conventions. We fix a complete non-Archimedean valuation field K with a non-trivial valuation and assume that rigid analytic

CLASSIFICATION OF RIGID ANALYTIC SURFACES

3

spaces are defined over K. We mainly use the terminologies and notationsof [4], [23], and [21].
Acknowledgment. The author thanks Kenji Ueno for support and suggestions. Ueno gave him unpublished literatures [26] and [24]. He also thanks
Fumiharu Kato for support and warm encouragement.
2. Preliminaries
2.1. Intersection Theory. In this subsection, we assume that X is a normal
surface. We explain the intersection theory on X. For a divisor D on X, let
us denote the line bundle OX (D) by [D]. For an effective divisor D on X, let
us denote a coherent sheaf OX /OX (−D) on X by OD .
First, we summarize the results on the intersection theory on proper normal surfaces in [21, 5]. Assume that X is proper over K. For two line
bundles L1 and L2 on X, we define the intersection number of L1 and L2byχ(OX ) − χ(L∨1 ) − χ(L∨2 ) + χ(L∨1 ⊗ L∨2 ).
Let us denote this integer by L1 ·L2 . For a line bundle L and for two divisors
D and E on X, we abbreviate L · [D], [D] · L, and [D] · [E] to L · D, D · L,and D · E respectively.
For any two line bundles L1 and L2 , the following equalities hold:(1) L1 · OX = OX · L1 = 0;
(2) L1 · L2 = L2 · L1 .
Let L1 , L2 and L3 be line bundles on X. Assume that one of these line
bundles is isomorphic to [D] where D is a divisor. Then the following
equality holds:
L1 · L2 ⊗ L3 = L1 · L2 + L1 · L3 .
PP
In particular, for any two divisors i∈I ai Di and j∈J b j D j , the following
equality holds:
XXX
a i Di ·b jD j =ai b j Di · D j .
i∈I

j∈J

i∈I, j∈JP

Let D be a divisor i∈I ai Di on X. Let πi : Dni → Di be the normalizationof a prime divisor Di . Then, for any line bundle L on X, the following
equality holds:X(∗)
OX (D) · L =ai degDni π∗i L.
i∈I

The local intersection number of two distinct prime divisors D1 and D2
at a point p on D1 ∩ D2 is the integer dimK (OX /OX (−D1 ) + OX (−D2 )) p . Let

4

KENTARO MITSUIus denote this number by I(p, D1 , D2 ). For any two distinct prime divisors
D1 and D2 on X, the following equality holds:
X
I(p, D1 , D2 ).
D1 · D2 =p∈D1 ∩D2

We have the Riemann-Roch theorem for proper smooth surfaces. For any
line bundle L on a proper smooth surface X, the following equality holds:
L · KX∨ − L · L∨
+ χ(OX ).
2
For a general normal surface X, we define the intersection number of aproper divisor D and a line bundle L by the equality (∗). Then the otherresults hold.
We define a virtual genus π(D) of a divisor D on a normal surface by theequality:
D · D + D · KX
π(D) :=+ 1.
2
When the surface X is smooth and the divisor D is proper and smooth, thevirtual genus π(D) is equal to the arithmetic genus of D. Using a local
tubular neighborhood covering of D (Theorem 2.12 in [21]), we can show
this fact.
χ(L) =

2.2. Coherent Modules and Flat Base Changes. In this subsection, we
introduce sheaves of morphisms between coherent modules. Then we state
the flat base change theorems of coherent sheaves.
Let F and G be coherent OX -modules on a rigid analytic space X. For anadmissible affinoid open subset U, put
HomOX (F , G)(U) := HomOX (U) (F (U), G(U)).
For two admissible affinoid open subsets U and V of X, we assume that U
contains V. Since any affinoid algebra is Noetherian, by (3.E) in [20], there
exists a canonical isomorphism:
HomOX (V) B(F (V), G(V))  HomOX (U) (F (U), G(U)) ⊗OX (U) OX (V).
Thus, by [4, 9.4.2], the above definition gives a sheaf on each admissible
affinoid open subset. Then we obtain an OX -module. Since the right-handside of the above definition is a finite A-module, the OX -module is coherent.
Let us denote this coherent OX -module by HomOX (F , G). For simplicity,
we putF ∨ := HomOX (F , OX ).
This notation is compatible with the same one for line bundles.

CLASSIFICATION OF RIGID ANALYTIC SURFACES

5

A morphism f : X → Y of rigid analytic spaces is said to be flat if themorphism f satisfies the following condition, Let Sp A and Sp B be admissible affinoid open subsets of X and Y respectively. Assume that thepreimage f −1 (Sp B) contains Sp A. Then the corresponding K-algebra homomorphism B → A is flat. This condition is equivalent to the following
local one. For any point x in X, the associated local ring homomorphism
OY, f (x) → OX,x is flat. We show this fact in the following way. First, notethat any stalk of the structure sheaf of any rigid analytic space is Noetherian(Proposition 7 in [4, 7.3.2]). Let m and n be the maximal ideals of A and B
corresponding to the points x and f (x) respectively. Then the diagram
Bn/ OY, f (x)/ OX,xAm/Ob

Y, f (x)


/O
b

X,x

bX,x and O
bY, f (x) are the completions of OX,x and OY, f (x)is commutative where O
with respect to the maximal ideals respectively. By Proposition 3 in [4,7.3.2] and (24.A) Theorem 56 in [20], the horizontal homomorphisms in
the above diagram are faithfully flat. Then Proposition 3 in [4, 7.3.2] and(20.G) Application 3 in [20] shows that the morphism Bn → Am is flat if
and only if the morphism OY, f (x) → OX,x is flat. This proves the fact.
By (3.E) in [20], we obtain the following proposition.
Proposition 2.1. Let f : X → Y be a flat morphism of rigid analytic spaces.
Then, for two coherent OY -modules F and G, there exists a canonical isomorphism:f ∗ HomOY (F , G)  HomOX ( f ∗ F , f ∗ G).
In particular, there exists a canonical isomorphism:f ∗ (F ∨ )  ( f ∗ F )∨ .
The following proposition shows that base change preserves flatness.
Proposition 2.2. Any base change of any flat morphism of rigid analytic
spaces is flat.
Proof. Let f : X → Y and u : Z → Y be morphisms of rigid analytic spaces.
Assume that the morphism f is flat. We show that the base change of f via u
is flat. We may assume that the rigid analytic spaces X, Y, and Z are affinoid
spaces. Thus, it suffices to show that, for two homomorphisms B → A andB → C between affinoid algebras, if the homomorphism B → A is flat, thenthe base change C → A b⊗B C is flat. Further, we may assume that C is finite
over B or isomorphic to the free affinoid algebra BhXi in one indeterminate
over B.

6

KENTARO MITSUIThe first case follows from the isomorphism A b⊗B C  A ⊗B C (Proposition 6 in [4, 3.7.3]). Then we consider the second case. We take a maximal
ideal m′ of the free affinoid algebra AhXi over A. Let m, n, and n′ be theprime ideals of A[X], B[X], and BhXi that are the pull-backs of m under
the natural homomorphisms respectively. Then the prime ideals are maximal since Corollary 3 in [4, 6.1.2] implies that the quotient rings A[X]/m,
B[X]/n, and BhXi/n′ are finite extension fields of the base field K.
The same corollary gives an isomorphism A[X]/mn  AhXi/m′n for any
d m of A[X]m with respect to m
positive integer n. Thus, the completion A[X][is isomorphic to the completion AhXim′ of AhXim′ with respect to m′ . By
dm
(24.A) Theorem 56 in [20], the natural homomorphisms A[X]m → A[X][ m′ are faithfully flat. Therefore, the natural homomorand AhXim′ → AhXi
phism A[X]m → AhXim′ is flat. In the same way, the natural homomorphism
B[X]n → BhXin′ is flat. Since the ring A is flat over B, the localization A[X]m
is flat over B[X]n . Therefore, by (20.G) Application 3 in [20], the natural
homomorphism BhXin′ → AhXim′ is flat. This proves the proposition.

The following proposition shows that cohomology commutes with flatbase change under certain assumption.
Proposition 2.3. For a proper morphism f : X → Y of rigid analytic spaces
and a flat morphism u : Z → Y of rigid analytic spaces, let g and v be themorphism of rigid analytic spaces in the following Cartesian diagram:
X ×Y Z

v

/X

g



f
u



/ Y.
Z
Assume that the preimage of any point on Y is of dimension zero. Then fora coherent OX -module F , there exists a canonical isomorphismu∗ Rq f∗ F  Rq g∗ v∗ F
for all q ≥ 0.
Proof. First, note that Lemma 1 in [4, 9.6.2] shows that the base change g
is proper. We may assume that the rigid analytic spaces Y and Z are affinoid
spaces. We write Y = Sp A and Z = Sp B. Since the sheaves appearing
in the above isomorphism are all coherent modules, it suffices to show that
there exists a canonical isomorphism:
H q (X, F ) ⊗A B  H q (X ×Sp A Sp B, v∗ F ).
By the same method as in the proof of Satz 3.5, we may assume thatthe affinoid algebra A is an Artin ring. Since the affinoid algebra B is also
an Artin ring by assumption, Corollary 3 in [4, 6.1.2] shows that B is a

CLASSIFICATION OF RIGID ANALYTIC SURFACES

7

finite A-algebra. Thus, Proposition 6 [4, 3.7.3] shows that, for any affinoid
algebra C over A, the natural homomorphism C ⊗A B → C b⊗A B is an
isomorphism. Therefore, since the affinoid algebra C is a Noetherian ring,⊗A B
for any finite C-module M, the natural homomorphism M ⊗A B → M b
is an isomorphism.
Since the morphism f is proper, there exists a finite admissible affinoid
covering U of X. Since the morphism f is separated, the covering U isacyclic for F . Then the pull-back of the covering U under v is a finite admissible affinoid covering of X ×Sp A Sp B that is acyclic for v∗ F . Therefore,
the proposition follows from the flatness of u.
A rigid analytic space is said to be geometrically connected (resp. geometrically reduced) if any base field extension of the rigid analytic spaceis connected (resp. reduced). If the base field is perfect, any rigid analytic
space is geometrically reduced (Lemma 3.3.1 1 in [8])Proposition 2.4. If a proper reduced connected rigid analytic space X isgeometrically connected (resp. geometrically connected and geometricallyreduced), then the finite K-algebra OX (X) is a purely inseparable extension
field of K (resp. isomorphic to K).
Proof. First note that cohomology commutes with base change of the basefield (Theorem A.1.2 in [10]). Using the completion of the separable closure, and algebraic closure of K and Corollary 2 in [4, 9.3.3], we can prove
this proposition by the same method as in the proof of Corollary 3.3.21 in[18].

2.3. Resolutions of Singularities. Schoutens gave the resolutions of singularities of rigid analytic spaces over an algebraically closed base field of
characteristic zero in [28]. Here, we give the resolutions of singularities ofrigid analytic spaces of dimension two over an arbitrary base field.
Theorem 2.5. For any quasi-compact reduced rigid analytic space X ofdimension two, there exists a quasi-compact reduced regular rigid analytic
space Y and a bimeromorphic morphism Y → X.
Proof. First, note that any affinoid algebra is excellent ([8, 1]). Therefore,
the regular locus of any reduced affinoid space is a non-empty Zariski open
subset and any normal affinoid space of pure dimension two has at mostfinitely many singular points. We normalize X (see [8, 2.1]) and blow up
all the singular points of the resulting surface (see Definition 4.1.1 in [10]).
We have only to show that we can obtain a regular surface in a finite succession of this procedure. To show this, we may assume that X is an affinoid
space. Then the analytic blowing-up of X coincides with the analytification

8

KENTARO MITSUIof the algebraic one. Thus, the termination follows from the remark B in
the introduction of [17].

2.4. Fields of Meromorphic Functions. In this subsection, we assume
that X is a proper irreducible surface. We review some facts on the fieldof meromorphic functions M(X) on X. Then we construct the algebraic
reduction morphism.
The algebraic dimension of X is the transcendental degree of M(X) over
K. By a(X) we denote this integer. The following theorem on M(X) is a
special case of Theorem 5.1 (ii) in [2].
Theorem 2.6. The field of meromorphic functions M(X) is an algebraic
function field over K. Moreover, the algebraic dimension a(X) is at mosttwo.
Lemma 2.7. Assume that X is normal. Let Y be a projective regular modelof the algebraic function field M(X). Then there exist a proper regular surface Z, a bimeromorphic morphism π : Z → X, and a proper morphismρ : Z → Y such that the associated morphism OY → f∗ OZ is an isomorphism. Moreover, the morphism ρ is surjective and has connected fibers.
Proof. Let f0 be the constant function 1X on X. We choose a generating
family { f1 , . . . , fn } of M(X) over K such that the map ( f0 : · · · : fn ) gives
the embedding of Y to a projective space. We define the analytic subset
P Γ of
X × Y in the following way. For each i = 0, . . . , n, we write ( fi ) = D aiD D.
Put bD := min
Pi=0,...,n aiD . Let {Sp A j } j∈J be a finite locally principal covering
for a divisor D bD D. For each j ∈ J, let g j be a local defining function ofthis divisor on Sp A j . Put
U j := Projan A j [X0 , . . . , Xn ]/(Xi − ( fi /gi )X0 )i=1,...,nwhere the functor Projan is defined in [10]. Let V j be the intersection of two
analytic subspaces U j and Sp A j × Y of PnSp A j . Then each V j is proper overSp A j . Gluing all V j , we obtain the proper rigid analytic space Γ, the proper
modification Γ → X, and the proper morphism Γ → Y. We resolve the singularities of the reduction of Γ (Theorem 2.5). Let Z be the resulting proper
regular surface. We also obtain the bimeromorphic morphism π : Z → X
and the proper morphism ρ : Z → Y.
We have the Stein factorization β ◦ α : Z → S → Y (Proposition 5 in [4,9.6.3]). Since the morphism β is finite, the rigid analytic space S is projective (Theorem 5.4 in [23]). Since the rigid analytic space Z is reduced andirreducible, the rigid analytic space S has the same property. Therefore, therigid analytic space S is the analytification of a projective integral scheme
over K (Theorem 5.3 in [23]). On the other hand, by Theorem 4.3 in [2],∼ M(Z). Thus, the K-algebra
we have the K-algebra isomorphism M(X) →CLASSIFICATION OF RIGID ANALYTIC SURFACES

9homomorphism M(Y) → M(Z) is an isomorphism. Since the isomorphism
factors M(S ), the K-algebra homomorphism M(Y) → M(S ) is an isomorphism. Therefore, the morphism β is an isomorphism. This proves that the
triple Z, π, and ρ is the desired one.
The last statement follows from Lemma 4 in [4, 9.6.3].

3. Elliptic Fibrations and Jacobian Fibrations
3.1. Definitions and Fundamental Results. The following lemma reduces
the study of rigid analytic fibered surfaces to the study of algebraic fiberedsurfaces.
Lemma 3.1. For any proper flat surjective morphism ρ : Y → T to a regular curve with fibers of dimension at most one, there exist two morphisms
α1 : T 1 → T and α12 : T 2 → T 1 satisfying the following conditions.
(1) The morphism α1 is an admissible affinoid covering.
(2) The morphism α12 is a finite étale Galois covering.
(3) The base change ρ ×T T 2 is locally projective.
Proof. By Theorem 2.1.4 in [9], there exist two morphisms α3 : T 3 → T
and α34 : T 4 → T 3 satisfying the following conditions.
(1) The morphism α3 is an admissible affinoid covering.
(2) The morphism α34 is étale, quasi-compact, and surjective.
(3) The base change ρ ×T T 4 is locally projective.
By Proposition 3.1.4 in [11] and Lemma 4.4 in [23], there exist three morphisms α35 : T 5 → T 3 , α56 : T 6 → T 5 and α64 : T 4 → T 6 satisfying thefollowing conditions.
(1) The composite α35 ◦ α56 ◦ α64 is equal to the morphism α34 .
(2) The morphism α35 is an admissible affinoid covering.
(3) The morphism α56 is étale, finite, and surjective.
(4) The morphism α64 is an open immersion.
We define the morphism α1 : T 1 → T by the collection of all the connectedcomponents of T 5 . We choose a connected component Sp A of T 5 and aconnected component Sp B of α−1
56 (Sp A). Since the morphism Sp B → Sp A
is finite, it is the analytification of Spec B → Spec A. The corresponding K-algebra homomorphism A → B induces a K-algebra homomorphism
M(A) → M(B). Since the field extension M(B)/M(A) is finite and separated, we take a normal closure of M(B) over M(A). Let C be the integral
closure of A in M(B). Since any affinoid algebra is excellent ([8, 1]), thealgebra C is a finite A-affinoid algebra.
Applying Lemma 2.4 in [23] to each image of ramification points of
Sp C → Sp A, we may assume that the morphism Sp C → Sp A is étale.

10

KENTARO MITSUIWe define the morphism α12 : T 2 → T 1 by the collection of the morphisms
Sp C → Sp A. Then the desired conditions are fulfilled.
An elliptic fibration is a fibered surface with fibers of arithmetic genus
one satisfying the following condition. There exists a nowhere dense analytic subset S 0 of S such that, for any p ∈ S − S 0 , the fiber π−1 (p) issmooth and geometrically connected over K(p). A proper regular surface
X is called an elliptic surface if X admits a structure of an elliptic fibration.
The analytification of an algebraic elliptic fibration over an affinoid algebra
A is an elliptic fibration over Sp A.
We fix a regular curve S . Let C be the category of regular curves withétale quasi-compact morphisms to S . We sometimes abbreviate an object
f : T → S of the category C to T . A morphism from f1 : T 1 → S to
f2 : T 2 → S in C is a morphism α : T 1 → T 2 of rigid analytic spaces satisfying the following conditions.
(1) The morphism α is étale and quasi-compact.
(2) The composite f2 ◦ α is equal to f1 .
Remark. The étaleness of α follows from that of f1 and f2 . Lemma 4.4 in[23] shows that the image of any morphism α : T 1 → T 2 in the category C
is always an admissible open subset of T 2 .
The category C admits fiber products.
Let F be the category of elliptic fibrations over C. An object of thecategory F is a quadruple (Y, T, ρ, f ) satisfying the following conditions.
(1) The element f : T → S is an object of the category C.
(2) The triple (Y, T, ρ) is an elliptic fibration.
We sometimes abbreviate this quadruple to Y. A morphism from (Y1 , T 1 , ρ1 , f1 )to (Y2 , T 2 , ρ2 , f2 ) in F is a pair (α, β) satisfying the following conditions.
(1) The morphism α : T 1 → T 2 is a morphism in the category C(2) The morphism β : Y1 → Y2 is a morphism of rigid analytic spaces
such that the diagram
Y1

β

/ Y2

ρ1



T1

ρ2
α


/ T2

is Cartesian.
The category F admits fiber products.
Let F s be the category of elliptic fibrations over C with a section, which
is a subcategory of F . An object of F s is a quintuple (Y, T, ρ, f, s) satisfying
the following conditions.

CLASSIFICATION OF RIGID ANALYTIC SURFACES

11(1) The element s : T → Y is a section of the fibration ρ.
(2) The quadruple (Y, T, ρ, f ) is an object of the category F .
We sometimes abbreviate this quintuple to Y. A morphism from (Y1 , T 1 , ρ1 , f1 , s1 )to (Y2 , T 2 , ρ2 , f2 , s2 ) in F is a morphism (α, β) of the category F such thatβ ◦ s1 = s2 ◦ α. We sometimes abbreviate this morphism to β. The category
F s admits fiber products.
We fix an elliptic fibration π : X → S . Let FX be a full subcategory of
F whose object (Y, T, ρ, f ) satisfies that the fibration ρ : Y → T is given by
the base change X ×S T → T of the fibration π : X → S via a morphism
f : T → S of C. We also define a full subcategory FX,s of F s in the sameway.
To study the category FX , we introduce a more manageable subcategory
F1 . Let C1 be the full subcategory of C whose object f : T → S satisfies
that the quasi-compact regular curve T is an affinoid space. The category
C admits fiber products and contains fiber products over S . Let F1 be thefull subcategory of FX whose object (Y, T, ρ, f ) satisfies that the morphism
f is an object of C1 and the morphism ρ is projective. In this case, we canalgebraize ρ (see [23, 5.1]), i.e., the elliptic fibration ρ : Y → Sp A is theanalytification of an algebraic elliptic fibration p : Y → Spec A.
A coherent module on C1 is the following data.
(1) Coherent OT -module GT for each object T of C1 .
(2) Isomorphism γ12 : α∗21 GT1 → GT2 for each two objects T 1 and T 2
and each morphism α21 : T 1 → T 2 of C1 . The isomorphisms satisfy
the equality γ12 ◦ α∗21 γ23 = γ13 when the equality α32 ◦ α21 = α31
holds.
Using coherent modules on total spaces, we define coherent modules on F1
in the same way.
A coherent module on the base spaces of F1 is the following data.
(1) Coherent OT -module GT for each object (Y, T, ρ, f ) of F1 .
(2) Isomorphism γ12 : α∗21 GT1 → GT2 for each two objects (Y1 , T 1 , ρ1 , f1 )and (Y2 , T 2 , ρ2 , f2 ) and each morphism β21 : Y1 → Y2 of F1 . The isomorphisms satisfy the equality γ12 ◦ α∗21 γ23 = γ13 when the equality
β32 ◦ β21 = β31 holds.
We define morphisms between coherent modules in the natural way.
We define coherent modules on C and FX , and coherent modules on thebase spaces of FX in the same way. Then these are completely determinedby the coherent modules on S and X respectively. Thus, we have the following proposition.

12

KENTARO MITSUI

Proposition 3.2. The inclusion functors C1 → C and F1 → FX induces theequivalences of the categories of coherent modules respectively. In particular, the categories of coherent modules on C1 and F1 are equivalent to thecategories of coherent modules on S and X respectively.
The inclusion functor F1 → FX also induces the equivalence of categories of coherent modules on the base spaces. In particular, the categoryof coherent modules on the base spaces on F1 is equivalent to the categoryof coherent modules on S .
Proof. This proposition follows from Lemma 3.1 and the faithfully flat descent theory (Theorem 3.1 in [3], Theorem 4.2.8 in [10]).
Lemma 3.3. If a fibered surface (X, S , π) admits a section and the basespace S is quasi-compact, the projection π is projective. In particular, any
fibered surface over a proper regular curve with a section is projective overK.
Proof. Since the section of π defines an S -ample line bundle (see Definition
3.2.2 in [10]), gluing a finite number of the projective morphisms that isgiven in Theorem 3.2.7 in [10], we obtain the desired result.

3.2. Canonical Bundle Formula. We prove the canonical bundle formulafor minimal elliptic fibrations.
Let p : Y → T be an algebraic fibered surface over an algebraic regularcurve. Assume that the base space T is given by the spectrum Spec A of anaffinoid algebra A. In this case, the morphism p is automatically projective(Theorem 2.6 in [16]). Thus, we have a relative dualizing sheaf ωY/T for
the fibration p (see Definition 6.4.7 in [18] and Theorem 6.4.32 in [18]).
Lemma 3.4. We keep the above notations. Then, for any quasi-coherent
OY -module F , there exists a canonical isomorphism:∨p∗ HomOY (F , ωY/T )  R1 p∗ F .
The isomorphism commutes with étale base change of affinoid algebras.
Proof. The isomorphism follows from the definition of the relative dualizing sheaf. The base change property follows from Theorem 6.4.9 (b) in[18].
Lemma 3.5. We keep the above notations. Assume that the arithmetic genusof the fibers is equal to one. Suppose that the fibration
P Y is minimal. Let
C be the fiber at a closed point of T . We write C = i∈I biCi . Then eachintersection number ωY/T · Ci is equal to zero.

CLASSIFICATION OF RIGID ANALYTIC SURFACES

13

Proof. The intersection number ωY/T · C is equal to zero by the adjunction
formula (Theorem 9.1.37 in [18]). Since the fibration is minimal, Proposition 9.3.10 (b) in [18] shows that each intersection number ωY/T · Ci is
non-negative. Therefore, each integer ωY/T · Ci is equal to zero.
For a positive integer n, let us denote the relative dualizing sheaf on nDt
by ωnDt .
Lemma 3.6. We keep the above notations. Assume that the fibration p isa minimal elliptic fibration. Let T0 be the finite set of closed points where
the fiber is not smooth. For each t ∈ T0 , let mt be a multiplicity of the fiber
p−1 (t). We write the fiber mt Dt . Then there exists a canonical isomorphism
ωY/T  p∗ p∗ ωY/T ⊗OY OY (D)where the divisor D is given by the following equality:XD=
a t Dt .
t∈T0Here, each integer at satisfies the inequalities 0 ≤ at < mt . The isomorphism
commutes with étale base change of affinoid algebras.
In particular, if the total space Y and the base space T are smooth, thenthere exists a canonical isomorphism
KY  p∗ p∗ ωY/T ⊗OT KT ⊗OY OY (D)where KY and KT are the canonical bundles on Y and T respectively. In
this case, the relative dualizing sheaf ωY/T is isomorphic to KY ⊗OY p∗ KT∨ .
Proof. Since the generic fibers are genus one curves, the direct image sheafp∗ ωY/T is free of rank one. Thus, the natural morphism p∗ p∗ ωY/T → ωY/T
is injective. Since the inclusion morphism is an isomorphism on the genericfibers, we obtain the isomorphism ωY/T  p∗ p∗ ωY/T ⊗OY OY (D) where D
is the zero divisor of the inclusion morphism, which is vertical with respect
to p.
We determine this effective
P divisor D. Let C be the fiber at a closed
point of T . We write C = i biCi . Then, by Lemma 3.5, the integer D ·Ci is Pequal to zero. Therefore, by Theorem 9.1.23 in [18], we may write
D = t∈T at Dt . Since the two line bundles p∗ p∗ ωY/T and ωY/T admit thesame global sections over any admissible open subset of T , the inequalities
follow. This proves the first assertion.
We prove the second assertion. Since the dualizing sheaf ωY/T is given
by KY ⊗OY p∗ KT∨ in this case, the second assertion follows from the first
assertion.
14

KENTARO MITSUI

We fix a fibered surface π : X → S . The analytifications of the isomorphisms in the above lemmas give an isomorphism in the category of coherent modules on the base spaces of F1 and the category of coherent modules
on Fi . Proposition 3.2 gives the corresponding isomorphism in that of FX .
In particular, we obtain the following theorems. Since the canonical bundle
commutes with étale base change (Theorem 6.4.9 in [18]), the relative dualizing sheaf ωX/S corresponds to the coherent module on F1 given by the
relative dualizing sheaves on the total spaces of the fibrations.
Theorem 3.7 (relative duality theorem). We keep the above notations. Thenfor any coherent OX -module F , there exists a canonical isomorphism:∨π∗ HomOX (F , ωX/S )  R1 π∗ F .
In particular, there exists a canonical isomorphism:∨π∗ ωX/S  R1 π∗ OX .
Theorem 3.8 (canonical bundle formula). We keep the above notations.
Assume that the fibration π is a minimal elliptic fibration. Let S 0 be thefinite set of closed points where the fiber is not smooth. For each s ∈ S 0 , letm s be a multiplicity of the fiber p−1 (s). We write the fiber m s D s . Then there
exists a canonical isomorphism
ωX/S  p∗ p∗ ωX/S ⊗OX OX (D)where the divisor D is given by the following equality:
XD=
as Ds.
s∈S 0Here, each integer a s satisfies the inequalities 0 ≤ a s < m s .
In particular, if the total space Y and the base space T are smooth, thenthere exists a canonical isomorphism
KX  π∗ π∗ ωX/S ⊗OS KS ⊗OX OX (D)where KX and KS are the canonical bundles on X and S respectively. In
this case, the relative dualizing sheaf ωX/S is isomorphic to KX ⊗OX p∗ KS∨ .
From now on, we assume that the fibration π is a minimal elliptic fibration. To study multiple fibers, we refer to the arguments in [1]. Let TX be
the torsion submodule of R1 π∗ OX . A multiple fiber m s D s is said to be tamely
ramified if TX,s is equal to zero. Otherwise, it is said to be wildly ramified.
By ν s we denote the order of the normal bundle NDs /X of D s .
Lemma 3.9. We keep the above notations. Then, for any positive integer n,
the equality χ(OnDs ) = 0 holds. In particular, the inequality h1 (OnDs ) > 0
hold.

CLASSIFICATION OF RIGID ANALYTIC SURFACES

15

Proof. By Lemma 2.4 in [23], it suffices to show the corresponding lemmain the algebraic case. The adjunction formula (Theorem 9.1.37 in [18])gives the relative dualizing sheaf ωnDs of nD s :ωnDs  (OX (nD s ) ⊗ ωX/S )|nDs .
By Corollary 7.3.31 in [18], the equality 2χ(OnDs ) = deg ωnDs holds. Thus,
the adjunction formula on the generic fibers yields the equality χ(OnDs ) = 0.
Therefore, since the dimension h0 (OnDs ) is not equal to zero, the dimension
h1 (OnDs ) is also not equal to zero.
Proposition 3.10. We keep the above notations. Assume that the fibration X
is minimal. Then the order ν s divides both m s and a s + 1. If the base field K
is separably closed, then the quotient m s /ν s is a power of the characteristicof K.
Proof. Since the normal bundle NDs /X is given by OX (D)|D , the order ν sdivides m.
We show that the order ν s divides a s +1. Lemma 3.9 and the Grothendieckduality theorem show that the relative dualizing sheaf ωDs admits a non-zero
section h. Let D1 be the maximal effective divisor such that D1 ≤ m s D s andh|D1 = 0. Put D2 := D s − D1 . Then we define the coherent OD2 -module S
by the following sheaf exact sequence:h0 −→ OD2 −→ OD2 (−D1 ) ⊗ ωDs −→ S −→ 0.
The intersection number of the divisor D s and any prime divisor appearing
in D s is equal to zero. Thus, by Lemma 3.5, the degree of the restriction of
the line bundle ωDs to any prime divisor appearing in D s is equal to zero.
Therefore, by the Riemann-Roch theorem (Theorem 7.3.26 in [18]) for D2 ,the equality
χ(OD2 (−D1 ) ⊗ ωDs ) = χ(OD2 (−D1 ))holds. Finally, we obtain the equalities:−D1 · D2 = χ(OD2 (−D1 ) ⊗ ωDs ) − χ(OD2 ) = χ(S).
Since the support of S is a finite number of points, the right-hand side ofthe equalities is at least zero. Therefore, by Theorem 9.1.23 in [18], we
conclude that the divisor D1 is equal to zero. In particular, the line bundle
ωDs is trivial. Thus, the order ν s divides a s + 1.
The last statement follows from Proposition 6.3.5 in [25].
Lemma 3.11. We keep the above notations. Then the function h0 (OnDs ) ofn is non-decreasing. Moreover, the inequality h0 (O(νs +1)Ds ) > 1 holds.

16

KENTARO MITSUIProof. The function h1 (OnDs ) of n is non-decreasing since the morphism
On1 Ds → On2 Ds is surjective for any positive integers n1 and n2 such thatn1 > n2 . Thus, the first assertion follows from Lemma 3.9. Since the linesbundle ND⊗νs /X
is trivial, we have the sheaf exact sequence:φ

0 −→ ODs −→ O(νs +1)Ds −→ Oνs Ds −→ 0.
Then the second assertion follows from the induced long exact sequencesince the homomorphism H 0 (φ) maps constants on (ν s + 1)D s into constantson ν s D s .
Proposition 3.12. We keep the above notations. Assume that the fibration
X is minimal and proper, and the total space X is smooth and geometricallyconnected. If the dimension h1 (KX ) is at most one, then the multiplicity m s
is equal to either a s + 1 or ν s + a s + 1.
Proof. The above lemma and the Grothendieck duality theorem give theinequality h0 (ω(νs +1)Ds ) > 1. The long exact sequence induced by the sheaf
exact sequence
0 −→ KX −→ KX ⊗OX OX ((ν s + 1)D s ) −→ ω(νs +1)Ds −→ 0
gives the inequality:
h0 (KX ⊗OX OX ((ν s + 1)D s )) > h0 (KX ).
Therefore, the inequality ν s + a s + 1 ≥ m s holds. The previous proposition
gives the inequality 1 + (a s + 1)/ν s ≥ m s /ν s of integers. This proves the
proposition.

3.3. Jacobian Fibrations. To obtain Noether’s formula for elliptic surfaces, we construct the Jacobian fibrations of elliptic fibrations. The Jacobian fibration is defined for all complex analytic elliptic surfaces with
reduced fibers. Here, we define the Jacobian fibration for all rigid analytic
elliptic surfaces. In the algebraic case, we may use minimal regular modelsof the Jacobian varieties of the generic fibers to analyze elliptic fibrations([19], [26], [24]). We use this idea to construct the Jacobian fibrations over(étale) affinoid open subsets. Then we glue these fibrations to obtain thedesired Jacobian fibrations.
We define the algebraic Jacobian fibration (J, T , q) of an algebraic elliptic fibration (Y, T , p) over an excellent regular ring of pure dimension
one by a minimal regular model of the Jacobian varieties of the genericfibers of p.
Remark. The existence of the minimal regular model of the Jacobian variety follows from the desingularization ([17]) and the existence theorem

CLASSIFICATION OF RIGID ANALYTIC SURFACES

17

of minimal regular model (Theorem 4.4 in [16], Theorem 1.2 in [7], Theorem 9.3.21 in [18]). The construction of algebraic Jacobian fibration commutes with étale base change and localization (Proposition 9.3.28 in [18]and Corollary 9.3.30 in [18]). When the projection p is smooth, the algebraic Jacobian fibration is canonically isomorphic to the dual of the identity
component of the Picard scheme (Théorème 3.1 in [13]) of the fibration p(see the proof of Proposition 4 in [5, 9.4]).
We fix an elliptic fibration π : X → S . The Jacobian functor over C is afunctor J from FX to FX,s such that the restriction J1 of J to F1 satisfies thefollowing conditions.
(1) For an object (Y, Sp A, ρ, f ) of F1 , let p : Y → Spec A be an algebraization of ρ (see [23, 5.1]). Then the object J1 (Y) is given by aquintuple (J1 (Y), Sp A, σ, f, s) where the fibration σ : J1 (Y) → Sp A
with the section s is the analytification of the algebraic Jacobianfibration p. Note that any affinoid algebra is excellent ([8, 1]).
(2) Let (α, β) be a morphism from (Y1 , T 1 , ρ1 , f1 , s1 ) to (Y2 , T 2 , ρ2 , f2 , s2 )in FX,s . Then the morphism J1 (β) is equal to the composite of the∼ J (Y ) × T (seeunique canonical isomorphism lY2 T2 T1 : J1 (Y1 ) →1 2
T2 1
Lemma 4.3 in [23] and Proposition 9.3.18 in [18]) and the projection
J1 (Y2 ) ×T2 T 1 → J1 (Y2 ). Note that the composite is equal to the∼composite of the unique canonical isomorphism rT1 T2 Y1 : J1 (Y1 ) →T 1 ×T2 J1 (Y2 ) and the projection T 1 ×T2 J1 (Y2 ) → J1 (Y2 ).
We show that there exists the Jacobian functor J up to canonical isomorphism. Then we define the Jacobian fibration of the elliptic fibration
π : X → S by J(X) → S .
Assume that the Jacobian functor J exists. Let (Y, T, ρ, f ) be an object of
FX . For each i = 1, 2, let αi : T i → T be an object of C. Put Yi := Y ×T T i .
The Cartesian diagram
Y1

/Y




/T

T1
gives the Cartesian diagram:
J(Y1 )/ J(Y)




/ T.

T1

18

KENTARO MITSUIThen the above diagram and the base change of J(Y1 ) → T 1 via T 1 ×T T 2 →T 1 give the commutative diagram consisting of Cartesian squares:
J(Y1 ) ×T T 2/ J(Y1 )/ J(Y)




/ T1


/ T.

T 1 ×T T 2

We may exchange 1 by 2 in the above diagram. Therefore, we obtain the∼ T × J(Y ).
canonical isomorphism ψY1 T Y2 : J(Y1 ) ×T T 2 →1 T
2
We prove the following two lemmas, which enable us to construct theJacobian functor J by gluing the functor J1 on the subcategory F1 of FX .
Lemma 3.13. Assume that the Jacobian functor J exists. We use the above
notations. Let φ : Y1 ×T T 2 → T 1 ×T Y2 be the canonical isomorphism. Thenthe diagram
J(Y1 ×T T 2 )J(φ)/ J(T 1 ×T J2 )rT 1 T Y2

lY1 T T 2

J(Y1 ) ×T T 2

ψY1 T Y2


/ T 1 ×T J(Y2 )is commutative.
Remark. It is important that by the above commutative diagram, we canconstruct the morphism ψY1 T Y2 from the morphisms lY1 T T2 , rT1 T Y2 , and φ,which derive from the base changes of the fibrations Y → T , J(Y1 ) → T 1 ,and J(Y2 ) → T 2 . In other words, we may construct the morphism ψY1 T Y2
without the fibration J(Y) → T .
Proof. The commutative diagram consisting of Cartesian squares
Y1 ×T T 2

/ Y1

/Y




/ T1


/T

T 1 ×T T 2

gives the commutative diagram consisting of Cartesian squares:
J(Y1 ×T T 2 )/ J(Y1 )/ J(Y)




/ T1


/ T.

T 1 ×T T 2

We may exchange 1 and 2 in the above diagram. The two resulting diagrams
prove the lemma.
CLASSIFICATION OF RIGID ANALYTIC SURFACES

19

Lemma 3.14. Assume that the Jacobian functor J exists. For each i =1, 2, 3, let Yi and Yi′ be objects in FX . Assume that the two squares in thecommutative diagram
Y2 o

Y1

/ Y3





β3

β2



Y2′ o

p2

p3

Y1′

/ Y′

3

∗

are Cartesian. Then the equality J(p2 ) J(β2 ) = J(p3 )∗ J(β3 ) holds.
Proof. The above diagram gives the commutative diagram consisting ofCartesian squares:
J(Y1 )

J(Y2 ) o/ J(Y3 )J(β2 )J(Y2′ ) oJ(p2 )J(β3 )J(Y1′ )J(p3 )/ J(Y ′ ).
3

The lemma follows from this diagram.



Using the Galois descent of projective varieties over the generic pointsof Dedekind rings, by Lemma 3.1, Lemma 3.3, Lemma 3.13, and Lemma
3.14, we obtain the Jacobian functor.
Theorem 3.15. The Jacobian functor exists up to canonical isomorphism.
3.4. Noether’s Formula. In this subsection, we show Noether’s formulafor minimal elliptic fibrations over the perfect base field K. Note that, when
the base field K is perfect, regularity is equivalent to smoothness over K in
both the rigid analytic case and the algebraic case.
First, for the Jacobian fibration (J, S , σ, s) for a minimal elliptic fibration(X, S , π), we construct the canonical OS -module homomorphism:
τX : R1 π∗ OX → R1 σ∗ O J .
Let T be the spectrum Spec A of an excellent regular ring A of dimension
at most one. For the algebraic Jacobian fibration (J, S, q) of a smoothly
fibered algebraic elliptic fibration (Y, T , p), we construct the canonical Aisomorphism
τY : H 1 (Y, OY ) → H 1 (J, OJ )in the following way. By Theorem 1 in [5, 8.4], we have the isomorphisms∼ H 1 (Y, O ) and Lie(Pic ) →∼ H 1 (J, O ). By the proof ofLie(PicY/T ) →Y
J/TJ
Proposition 4 in [5, 9.4], we have the isomorphism from J to the identity
component of PicJ/T . These isomorphisms give the desired isomorphism.
The A-module isomorphism τY commutes with flat base change.

20

KENTARO MITSUILet T be the spectrum Spec A of a finite direct sum A of excellent discrete valuation rings. For the algebraic Jacobian fibration (J, S, q) of analgebraic elliptic fibration (Y, T , p), by Theorem 3.8 in [19], there exists a
canonical A-module homomorphism
τY : H 1 (Y, OY ) → H 1 (J, OJ )such that the following diagram
H 1 (Y, OY )

H 1 (Yη , OYη )

τY

τYη

/ H 1 (J, O )J/ H 1 (Jη , OJ )η

is commutative where Yη is the generic fibers over the generic points Jη andthe vertical arrows are the localizations induced by the open immersions.
The A-module homomorphism τY commutes with étale base change.
Let T be the spectrum Spec A of an excellent rings A of pure dimension
one. For the algebraic Jacobian fibration (J, S, q) of an algebraic elliptic
fibration (Y, T , p), we construct the canonical A-module homomorphism
τY : H 1 (Y, OY ) → H 1 (J, OJ )in the following way. Let T0 be the finite number of closed points where
the fiber is not smooth. We write the generic fibers Yη over the genericpoints Tη . Let T1 be the open subscheme T − T0 of T . Let T2 be the unionof all localizations at points in T0 . For i = 1, 2, put Yi := Y ×T Ti andJi := J ×T Ti . Since the A-module H 1 (J, OJ ) is free of rank one, we may
paste τY1 and τY2 to construct τY such that the diagramτY

H 1 (Y, OYP)

PPP
PPP
PPP
PP'

H 1 (Y2 , OY2 )τY1

H 1 (Y1 , OY1 )/ H 1 (J, O )
J
PPP
PPP
PPP
PP'
τY2
/ H 1 (J , O2

/ H 1 (J , O
1

OOO
OOO
OOO
OOO
'


H 1 (Yη , OYη )

τYη

J2 )

J1 )

OOO
OOO
OOO
OO'

/ H 1 (Jη , OJ )η

is commutative where the non-horizontal arrows are the localizations. The
A-module homomorphism τY commutes with étale base change.

CLASSIFICATION OF RIGID ANALYTIC SURFACES

21

Lemma 3.16. Using the above notations, the kernel of τY is equal to thetorsion submodule of the A-module H 1 (Y, OY ). Moreover, the kernel andcokernel of τY have the same length.
Proof. This lemma follows from the result on τY2 in Theorem 3.8 in [19]and Theorem 3.1 in [19].

By Proposition 3.2, we obtain the desired OS -module homomorphism
τX : R1 π∗ OX → R1 σ∗ O J
for a minimal elliptic fibration (X, S , π) with its Jacobian fibration (J, S , σ).
Let TX be the torsion submodule of R1 f∗ OX . Then the quotient R1 f∗ OX /TX
is a line bundle LX on S . The above lemma gives the following corresponding theorem.
Theorem 3.17. Using the above notations, the kernel of τX is equal to TX .
Moreover, the kernel and cokernel of τX have the same length.
Next, for an algebraic elliptic fibration (Y, T , p) without multiple fibers,
we construct the canonical injective OY -module homomorphism∆Y/T : OY → (p∗ ωY/T )⊗12
where ωY/T is a relative dualizing sheaf for the fibration p. Note that thedirect image sheaf p∗ ωY/T is a line bundle on T . Since the fibration p
has no multiple fibers, the higher direct image sheaf R1 p∗ OY is also a line
bundle on T .
First, we assume that the base space is an affine scheme Spec A. We
define the canonical injective A-module homomorphism∆Y/T : A → H 0 (Y, ωY/T )⊗12 ,which commute with étale base change, in the following way. For each i =η, 1, 2, we define Ti and Yi in the same way as in the above discussion. For
each i = 1, 2, put Ai := H 0 (Ti , OTi ). Note that the A-module H 0 (Y, ωY/T )and the Ai -module H 0 (Yi , ωYi /Ti ) for each i = η, 1, 2 are free of rank one.
We take the injective A1 -module homomorphism ∆Y1 /T1 and the injective
Q(A)-module homomorphism ∆Yη /Tη in [19, 5] such that the diagram
A1


Q(A)

∆Y1 /T1/ H 0 (Y , ω
1∆Yη /Tη

Y1 /T1 )⊗12/ H 0 (Yη , ωY /T )⊗12ηηis commutative where the vertical A1 -module homomorphisms are localizations (see Théorèm 9.9 in [12], Construction 7.5 in [12], and Proposition

22

KENTARO MITSUI

5.5 in [19]). We also define the injective A2 -module homomorphism ∆Y2 /T2such that the diagram∆Y2 /T2

A2/ H 0 (Y , ω
2



Q(A)

∆Yη /Tη

Y2 /T2 )⊗12/ H 0 (Yη , ωY /T )⊗12ηηis commutative where the vertical arrows are localizations.
Finally, we define the injective A-module homomorphism ∆Y/T such thatthe diagram
∆Y/T

A CC

CC
CC
CC
C!

∆Y2 /T2

A2


A1 B

/ H 0 (Y, ω

Y/T )

⊗12

SSS
SSS
SSS
SSS
SS)
/ H 0 (Y , ω
2∆Y1 /T1

/ H 0 (Y , ω
1

BB
BB
BB
BB
! 

Q(A)

Y1 /T1 )Y2 /T2 )

⊗12

⊗12

SSS
SSS
SSS
SSS
SS)

∆Yη /Yη/ H 0 (Yη , ωY /T )⊗12ηηis commutative.
The discriminant of the fibration p : Y → T at a point t on T is givenby the length of the cokernel of ∆Y/T at t (see [19, 5]). By disct (Y/T ) we
denote this number.
From now on, we consider elliptic surfaces (X, S , π). We define deg R1 π∗ OX
by the integer deg LX + length TX .
Proposition 3.18. We keep the above notations. Then the equality
χ(OX ) = − deg R1 π∗ OXholds.
Proof. By Leray’s spectral sequence for π and OX , we obtain the two equalities
h0 (OX ) = h0 (π∗ OX )andh2 (OX ) = h1 (R1 π∗ OX )and the exact sequence:
0 −→ H 1 (π∗ OX ) −→ H 1 (OX ) −→ H 0 (R1 π∗ OX ) −→ 0.

CLASSIFICATION OF RIGID ANALYTIC SURFACES

23

These give the equalities:
χ(OX ) = χ(π∗ OX ) − χ(R1 π∗ OX ) = χ(OS ) − χ(LX ) − length TX .
Then the proposition follows from the Riemann-Roch theorem for S .

Theorem 3.17 and the above proposition give the following theorem.
Theorem 3.19. For any minimal proper elliptic fibration X with its Jacobian fibration J, the equality
χ(OX ) = χ(O J )holds.
By Lemma 3.3, the Jacobian fibration over a proper base space is projective over K. By Chow’s theorem (Theorem 5.3 in [23]), we may algebraize
this fibration.
Theorem 3.20 (Noether’s formula). For any minimal proper elliptic fibration X over S with its Jacobian fibration J, the equality
X
disc s (J/S )12χ(OX ) =s∈Sholds.
Remark. Using Lemma 2.4 in [23] and Theorem 5.9 in [19], we can calculate the discriminants without constructing the Jacobian fibration.
Proof. The sheaf exact sequence∆ J/S

0 −→ OS −→ (σ∗ ω J/S )⊗12 −→ Coker(∆ J/S ) −→ 0
gives the equality:X
disc s (J/S ) = χ (σ∗ ω J/S )⊗12 − χ(OS ).

s∈S

Since the Grothendieck duality theorem gives the isomorphismσ∗ ω J/S  (R1 σ∗ O J )∨ ,by the above proposition and the Riemann-Roch theorem for S , the equality
X
disc s (J/S ) = 12χ(O J )s∈Sholds. Therefore, by the above theorem, we obtain the desired equality.

24

KENTARO MITSUI

4. Classifications
4.1. Proper Regular Surfaces of Algebraic Dimension Two. In this subsection, we prove that any proper regular surface of algebraic dimension
two is a projective surface. We also give a criterion for algebraicity.
We have shown that we can blow down exceptional curves of the firstkind on proper smooth surfaces (Theorem 4.1) in [21]. Here, we give more
precise result in the following lemma when the surfaces are projective.
Lemma 4.1. If we blow down an exceptional curve of the first kind on aregular projective surface, then the resulting surface is also regular andprojective.
Proof. By the same method as in the proof of Theorem 5.7 in [14], we canshow that the blowing-downs of a projective regular surface is a projective
regular surface.

Theorem 4.2. Any proper regular surface of algebraic dimension two is a
projective surface.
Proof. Assume that the algebraic dimension a(X) of X is equal to two. Wetake Y, Z, π, and ρ in Lemma 2.7. By Hopf’s theorem (Theorem 2.2 in[23]), the morphism ρ is a finite succession of blowing-ups. Thus, therigid analytic space Z is projective. The theorem follows from the previous lemma.
We give a criterion for algebraicity.
Theorem 4.3. If a proper regular surface admits a divisor whose selfintersection number is positive, then the surface is projective.
Proof. Assume that a divisor D on a proper regular surface X has a positive
self-intersection number. We may assume that the divisor D is effective. We
take Y, Z, π, and ρ in Lemma 2.7. We have only to show that the algebraic
dimension a(Z) is equal to two. Let E be the pull-back of the divisor D
under π. Then the effective divisor E have the same self-intersection number
as D’s by Proposition 3.1 in [23]. The long exact sequence induced by thesheaf exact sequence
0 −→ [(n − 1)E] −→ [nE] −→ OE ⊗OX [nE] −→ 0
shows that, for all sufficiently large integers n, the dimension of the kernel
Ker(H 1 (X, [(n − 1)E]) → H 1 (X, [nE]))is equal to zero. Therefore, the same long exact sequence gives the inequality
E·E 2
n + an + b(∗)
h0 ([nE]) >2

CLASSIFICATION OF RIGID ANALYTIC SURFACES

25

where a and b are suitable constant integers. In particular, the algebraic
dimension a(Z) is positive.
Suppose that the algebraic dimension a(Z) was equal to one. In this case,the rigid analytic space Y is a proper regular curve. We write E = E1 + E2
where the image of any prime divisor appearing in E1 under ρ is a point on
Y and that of any prime divisor appearing in E2 is equal to Y. Since theK-algebra homomorphism M(Y) → M(Z) is an isomorphism, the equality
h0 ([nE]) = h0 ([nE1 ])holds. We choose an effective divisor F on Y such that E1 ≤ ρ∗ F. Since theinequality
h0 ([nE1 ]) ≤ h0 ([nF])
holds, by the Riemann-Roch theorem and the above equality, the inequality
h0 ([nE]) ≤ (deg F) n + cholds where c is a suitable constant integer. The above inequality contradicts the inequality (∗). Thus, the algebraic dimension a(Z) is equal to two.
Therefore, the theorem follows from the previous theorem.

4.2. Proper Regular Surfaces of Algebraic Dimension One. In this subsection, we prove that any proper regular surface of algebraic dimension
one is a fibered surface with fibers of arithmetic genus one. In particular,when the base field K is perfect, we prove that any proper smooth surface
is an elliptic surface.
Assume that the algebraic dimension of a proper regular surface X isequal to one. We take Y, Z, π, and ρ in Lemma 2.7.
Lemma 4.4. We keep the above notations. Then the image of any divisor
on Z under π is a finite number of points.
Proof. Suppose the image of a prime divisor D on Z under π was Y. Choose
be a point p on Y. Then the preimage E of p under π is a divisor on Z. Since
the self-intersection E · E is equal to zero and the intersection number D · E
is positive, there exists an integer n such that the self-intersection of thedivisor D + nE is positive. However, this contradicts Theorem 4.3. This
proves the claim.
Lemma 4.5. We keep the above notations. There exists a proper morphism
σ : X → Y such that the composite π ◦ σ is equal to ρ.
Proof. By Hopf’s theorem (Theorem 2.2 in [23]), we have a finite sequenceof blowing-ups at a single pointZn ←− Zn−1 ←− · · · ←− Z1 ←− Z0 .

26

KENTARO MITSUI

where Z0 and Zn are equal to Z and X respectively. The extension theorem(Theorem 2.1 in [23]) shows that if the morphism ρ factors Zi , then themorphism also factors Zi+1 . Thus, we obtain the morphism σ : X → Y suchthat the composite π ◦ σ is equal to ρ. By Proposition 4 in [4, 9.6.2], themorphism σ is proper.
Lemma 4.6. Any proper regular surface of algebraic dimension one is afibered surface with fibers of arithmetic genus one.
Proof. We show that the morphism σ in the above lemma gives a structureof a desired fibration. Choose a point p on Y. Let D be the pull-back of
the divisor p under σ. By Lemma 2.4 in [23] and the adjunction formula(Theorem 9.1.37 in [18]), we have only to show that the intersection number
D · ωX/Y is equal to zero. Suppose that the integer D · ωX/Y was not equal tozero. The Grothendieck duality theorem gives the equality:
h1 (OD ⊗OX [mD] ⊗OX ωnX/Y ) = h0 (OD ⊗OX [−mD] ⊗OX ω(1−n)
X/Y ).
We may take the integer n such that the above dimensions vanishes and theabsolute value |n| is greater than one. Then the long exact sequence that is
induced by the sheaf exact sequence⊗n⊗n0 −→ [(m−1)D]⊗OX ω⊗n
X/Y −→ [mD]⊗OX ωX/Y −→ OD ⊗OX [mD]⊗OX ωX/Y −→ 0

shows that, for all sufficiently large integers |n|, the dimension of the kernel
1⊗nKer(H 1 (X, [(m − 1)E] ⊗OX ω⊗nX/Y ) → H (X, [mE] ⊗OX ωX/Y ))is equal to zero. Therefore, the long exact sequence gives the inequality(∗)

h0 ([mE] ⊗OX ω⊗nX/Y ) > |mnD · ωX/Y | + c

for all positive integers m where c is a suitable constant integer. In particular, for a sufficiently big integer m0 , the line bundle [m0 D] ⊗ KX⊗n admits a non-zero section s. Suppose that the line bundle [(m0 + 1)D] ⊗ KX⊗nadmits a section t that is not a section of the line bundle [m0 D] ⊗ KX⊗n .
Then the quotient t/s is a section of [D]. Thus, the K-algebra isomorphism∼ M(X) gives the inequality:
M(Y) →
h0 [(m0 + 1)D] ⊗ KX⊗n ≤ h0 [m0 D] ⊗ KX⊗n + 1.
However, this contradicts the inequality (∗). Therefore, the intersection
number D · KX is equal to zero.
Theorem 4.7. Assume that the base field K is perfect. Then any propersmooth surface of algebraic dimension one is an elliptic surface.
Proof. We show that the morphism σ in the above lemma gives a structureof an elliptic fibration. We have only to show that all fibers of σ except
for finitely many points are prover smooth curves. By the Jacobi criterion

CLASSIFICATION OF RIGID ANALYTIC SURFACES

27(Proposition 2.5 in [6]) and Theorem 2.6 in [27], the singular locus of σ isan analytic subset that is not equal to X. The Lemma 4.4 shows that theimage of the singular locus is a finite number of points. Then the theorem
follows from Proposition 2.9 in [6].

4.3. Proper Elliptic Fibrations. In this subsection, we classify minimal
proper elliptic fibrations that are not properly elliptic when the base field isalgebraically closed.
For a proper irreducible surface X, put
N(X) := {m ∈ Z | H 0 (X, KX⊗m ) , 0, m ≥ 0}.
For any integer m ∈ N(X) and any two sections f and g of KXm with g , 0,
the quotient f /g defines a meromorphic function on X. Then, for eachm ∈ N(X), the quotients { f /g | f, g ∈ H 0 (X, KX⊗m ), g , 0} form a subfield ofM(X). Let us denote this field by Km (X). We define the Kodaira dimension
κ(X) of X by the equality:
max tr.degK Km (X), if N(X) , ∅,m∈N(X)
κ(X) := −∞,otherwise.
By definition, the Kodaira dimension of X is at most the algebraic dimension of X.
The relative duality theorem (Theorem 3.7), the canonical bundle formula(Theorem 3.8), and Proposition 3.18 show the following theorem.

Theorem 4.8. Assume that the base field is perfect. Let π : X → S be a minimal proper elliptic fibration. Then there exists a canonical isomorphism
KX  π∗ L ⊗OX OX (D)where the line bundle L on S and the divisor D on X satisfy the following
conditions:(1) Let TX be the torsion submodule of R1 π∗ OX . Then the degree of L
is equal to χ(OX ) + length TX − 2χ(OS ).
(2) Let S 0 be the subset of points on S where the fiber is not smooth.
We write the multiplicity m s and the fiber m s D s at s ∈ S 0 . Then thedivisor D is given by the equality:
XD=
as Ds.
s∈S 0Here, each integer a s satisfies the inequalities 0 ≤ a s < m s .
An elliptic surface X is said to be properly elliptic if the Kodaira dimension of X is equal to one. By the Serre duality theorem (Theorem in [30,5.1]), the equality h1 (KX ) = h0 (OX ) holds for a proper smooth surface X.
Proposition 3.10 and Proposition 3.12 give the following classification.

28

KENTARO MITSUI

Type κ(X) χ(OX ) g(S ) length(TX ) multiple fibers
char K
(i)
−∞
0
0
0
none
(m − 1/m)(m1 − 1/m1 , m2 − 1/m2 )(1/2, 1/2, m − 1/m)(ii) −∞
001(a/m∗ )
,0∗(a/m1 , m2 − 1/m2 ),0(iii) −∞
100none
(m − 1/m)
(iv)
0
0
0
0
(1/2, 2/3, 5/6)
(1/2, 3/4, 3/4)
(2/3, 2/3, 2/3)
(1/2, 1/2, 1/2, 1/2)
(v)
0
0
0
1
(0/2∗ , 1/2, 1/2)
2∗
(1/2 , 1/2)
2
(1/3∗ , 2/3)
3∗
(1/4 , 3/4)
2
(2/4∗ , 1/2)
2∗
(2/6 , 2/3)
2
(3/6∗ , 1/2)
3
a∗
(vi)
0
0
0
2
(0/p )
p,0(0/pa∗ , 0/pb∗ )
p,0
(vii)
0
0
1
0
none
(viii) 01
00(1/2, 1/2)
(ix)
0
1
0
1
(0/2∗ )
2
(x)
0
2
0
0
none
Table 1. The possible invariants of minimal proper elliptic
fibrations π : X → S that are not properly elliptic. The integer χ(OX ) is the Euler characteristic of the structure sheaf
OX . The integer g(S ) is the genus of the base curve S . The
OX -module TX is the torsion submodule of the higher direct
image sheaf R1 π∗ OX . The notation (a1 /m1 , . . . , an /mn ) describes the multiple fibers of the fibration π. The integer miis the multiplicity of the i-th multiple fiber. The integer ai isthe coefficient of the divisor in the canonical bundle formula.
The symbol ∗ means that the multiple fiber is wildly ramified.

CLASSIFICATION OF RIGID ANALYTIC SURFACES

29

Theorem 4.9. Assume that the base field K is algebraically closed. Then
Table 1 gives the possible invariants of minimal proper elliptic fibrations
that are not properly elliptic.
Remark. For example, in positive characteristic p, there exists an elliptic
fibration of Type (ii) with the combination (pn − n − 1/pn∗ ) for any positive integer n. If the integer n is greater than one, then the surface is notalgebraic. We discuss details in the forthcoming paper [22].
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Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502,
Japan
E-mail address: mitsui@math.kyoto-u.ac.jp