Table Of ContentAlgebraic Geometry
and
Commutative Algebra
in Honor of Masayoshi NAG ATA
VOLUME II
Edited by
Hiroaki HIJIKATA
Heisuke HIRONAKA
Masaki MARUYAMA
Hideyuki MATSUMURA
Masayoshi MIYANISHI
Tadao ODA
Kenji UENO
1988
ACADEMIC PRESS
Harcourt Brace Jovanovich, Publiahera
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Copyright ©1988 by the Editorial Commitee for Algebraic Geometry and
Commutative Algebra in Honor of Masayashi Ν AG ATA. All rights reserved.
No part of this publication may be reproduced or transmitted in any form
by any means, electronic or mechanical, including photocopy, recording, or
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from the publisher.
Published in the United States by Academic Press, Inc., Orland, Florida
32887
Library of Congress Catalog Card Number 88-082929
ISBN 0-12-348032-9
Mathematics Subject Classification(1980) 13-06,14-06, 05-02,12A55,13C10,
13D03, 13E05, 13H05, 13H10, 13N05, 14C17, 14C20, 14D20, 14D22, 14H10,
14H45, 14H99, 14J15, 14J25, 14J30, 14J40, 14K99, 32G13, 32J25, 34A20.
ISBN
Table of Contents of Volume I
Shreeram S. ABHYANKAR — Determinantal Loci and Enumerative Com-
binatorics of Young Tableaux. 1
Yoichi AOYAMA and Shiro GOTO — A Conjecture of Sharp - The Case
of Local Rings with dimnonCM < 1 or dim < 5. 27
Michael ARTIN and Christel ROTTHAUS — A Structure Theorem for
Power Series Rings. 35
Wolf BARTH and Ross MOORE — On Rational Plane Sextics with Six
Tritangents. 45
Shizuo ENDO — On Rings of Invariants of Finite Linear Groups. 59
John FOGARTY — Invariant Differentials. 65
Takao FUJITA — Classification of Polarized Manifolds of Sectional Genus
Two. 73
Rajendra V. GURJAR and Masayoshi MIYANISHI — Affine Surfaces with
κ < 1. 99
Hiroaki HIJIKATA — On the Convolution Algebra of Distributions on
Totally Disconnected Locally Compact Groups. 125
Masa-Nori ISHIDA — The Local Cohomology Groups of an Affine Semi-
group Ring. 141
Christian U. JENSEN and Noriko Yui — Quaternion Extensions. 155
Toshiyuki KATSURA — On the Discriminants of the Intersection Form on
Néron-Severi Groups. 183
Joseph LIPMAN — On Complete Ideals in Regular Local Rings. 203
Masaki MARUYAMA — On a Compactification of a Moduli Space of Stable
Vector Bundles on a Rational Surface. 233
Hideyuki MATSUMURA — On the Dimension of Formal Fibres of a Local
Ring. 261
Tzuong-Tsieng MOH — On the Classification Problem of Embedded Lines
in Characteristic p. 267
N. MOHAN KUMAR, M. Pavaman MURTHY and Amit ROY — A Cancel-
lation Theorem for Projective Modules over Finitely Generated
Rings. 281
At sushi MORIWAKI — Semi-ampleness of the Numerically Effective Part
of Zariski Decomposition, II. 289
David R. MORRISON — On the Moduli of Todorov Surfaces. 313
Shigeru MUKAI — Curves, K3 Surfaces and Fano 3-folds of Genus < 10. 357
4
Iku NAKAMURA — Threefolds Homeomorphic to a Hyperquadric in P . 379
Algebraic Geometry and Commutative Algebra
in Honor of Masayoshi NAGATA
pp. 405-431(1987)
On Weierstrass Models
Noboru NAKAYAMA*
Introduction.
The theory of elliptic surfaces of Kodaira [14], [15], plays an important role
in the study of compact complex surfaces. In [14], he constructed a basic elliptic
surface, i.e., an elliptic surface with a section, canonically from functional and
homological invariants. Further it was shown that every elliptic surface can be
obtained from the basic elliptic surface which has the same functional and ho-
mological invariants by repatchings and logarithmic transformations. In higher
dimensional case, generalizing the argument of Kodaira, Kawai [8] and Ueno
[26] succeeded in constructing a basic fibration from functional and homological
invariants.
One of the main purposes of our paper is to construct a basic fibration as a
Weierstrass model. Even though our construction is different from theirs, it has
almost the same properties. The rest of this paper is devoted to the study of
elliptic threefolds with a section. Especially such threefolds with κ = 0, p = 1
g
are easilly classified.
The author expresses his thanks to Professors S. Iitaka and Y. Kawamata
for their advice and encouragement.
Notation.
(1) Let X be a complex variety. An open subset U of X is called a Zanski
open set if X \ U is a proper analytic subset of X. U is called big if U is Zariski
open and codim(X \ U) > 2.
(2) An elliptic fibration π : X —> S is defined to be a proper surjective
morphism with connected fibers between normal complex varieties X and S
whose general fibers are nonsingular elliptic curves.
(3) Let D be an effective Cartier divisor on a variety X. A defining equation
ν of D is a section ν G T(X,Ö(D)) such that div(t/) = D.
X
(4) The definition of canonical, terminal, and log-terminal singulanties are
the same as in [11], [13]. Note that if (Χ,Δ) is log-terminal, where X is a
* partially supported by the Fujukai foundation.
Received October 13, 1986.
Revised September 1, 1987.
406 Ν. ΝΑΚΑΥΑΜΑ
surface, then Χ has only quotient singularities. A good minimal model means
a projective variety with only canonical singularities whose canonical divisor is
semi-ample.
(5) Let £ be a reflexive sheaf of rank one on a normal complex variety S.
m
We denote the double dual of £® by £M. For any coherent sheaf F on 5, F"
denotes the double dual of F.
a
(6) For a Q-divisor D = Σ i A> we denote [D] — Σ[α»]Ζ^, where [o»] is the
integral part of a^.
§0. Variation of Hodge structures.
Let π : X —• S be a projective surjective morphism between complex man-
ifolds X and S. Assume that π is smooth over an open subset S° whose
complement is a divisor with only normal crossings. Then the local system
HQ := Α*π Ζχ| ο forms a variation of Hodge structures (V.H.S.) on S°. Let
Φ 5 UL ι ι
HQ := HQ®OSO. We denote by H (resp.Ίϊ) the upper (resp. lower) canonical
e u0 0
extension of Wj to 5, and let F ( «j) (resp. F^'WJ)) be the induced nitrations
from the Hodge filtrations of Ho. For the definition of canonical extensions, see
[16] or [19].
Moriwaki [19] proved the following theorem, which is also announced by
Saito. It is derived from his argument in [23].
Theorem (0.1). There exist isomorphisms
tiK+Ox^Gr^H*-*) for every i,
where d := dimX — dim S.
As a consequence of (0.1), we have the following:
Theorem (0.2). (The canonical bundle formula of elliptic fibrations)(cf.[6])
Let π : X —» S be a projective elliptic fibration between complex manifolds X
and S. Assume that π is smooth over an open subset S° of S whose complement
is a divisor with only normal crossings. Then
(1) C := ΐΡπ+Οχ is an invertible sheaf,
l
(2) ωχ = π*(ωs Θ C~) 0 Οχ(Ό) for an effective divisor D on X.
l
(3) D = 7Γ*(Σ·(1 - mJ)Fi) + E-G, where
Q
(a) the Fi 's are irreducible divisors on S, the m» denote the multiplicities of
the general fibers over F{,
(b) Ε is an effective Q-divisor on X such that the natural injection Οχ —>
Οχ([τηΕ]) induce isomorphisms π*Οχ = π+Οχ([πιΕ]) for any positive inte-
ger m,
(c) G is an effective Q-divisor on X such that coa\mw(G) > 2.
On Weierstrass Models 407
Proof. Since π is a projective morphism, by (0.1),
1 1
wx/s = ^ r w j) = (R'^Ox)- s ζ;- .
-1
Therefore we have an injection π * —• α>χ. Hence ωχ = ^(us^C )®
Οχ(Ό) for some effective divisor D on X. By the canonical bundle formula for
elliptic surfaces (see [15], [26]), D can be written as ττ*(Σ*(1 — m~l)Fi) + Ε
over a big open subset of S. Thus we obtain (c). q.e.d.
Remark (0.3). According to Kawamata [10, Theorem 20, (3)], we obtain
£-12 £ j r .0 () l005(£^ we h e r
1
(a) J : 5 —• P is the J-function of π,
(b)U^i = ^\^°,and
(c) the dj/12 are rational numbers in [0,1) corresponding to the types of the
general singular fibers over Dj. Here, if the singular fibers are multiple fibers
on Dj then a; = 0.
y }
Corollary (0.4). Let π : X —• S be an elliptic fibration. Suppose that X
has only canonical singulanties and that kKx = n*L for some positive integer
k and for some line bundle L on S. Then there exists an effective Q-divisor Δ
on S such that (S, Δ) is log-terminal and that Κχ =q 7r*(Ä"s -f Δ).
Proof. By flattening of π, we have the following commutative diagram:
Y X
where
(1) / : Y —• Γ satisfies the same condition as that of (0.2), hence
K = f*(K -C + ]T(1 - m-^Fi) + E-G,
Y q T
i
(2) λ and μ are proper bimeromorphic morphisms and
(3) codimA(G) > 2.
Thus we have
^O(mK) = μΛ*Ογ{τηΚ + mG)
x x γ
m
= μ.{Ο(τηΚ) ® C~ <g> Ο(Σί0- ~ Ό™*ί))
τ τ τ
= μ+(Ο(™Κ) <S> J*0(m/12) <g> 0(Σ; majDj)
τ τ Τ
(ZOTiEiil-rn^mFi)),
408 Ν. ΝΑΚΑΥΑΜΑ
where m is a positive common multiple of fc, 12, and all the πΐ{. Take a general
smooth divisor Η on Τ such that Ο(Η) ^ J*C*(l). Then
τ
π*Οχ{πιΚ) = μ*Ο(τη{Κ + 1/12 · Η + YaD + ^ (1 - m" 1)^)).
χ τ τ j J j
Since UK χ — π*£, π+Οχ(τηΚχ) is a line bundle. Put
1
Δ := μ,(1/12 · Η + ^ α,Ζλ, + ]Γ(1 - m" )^)-
j *
Then ϋίχ = π*(Κ + Δ). Furthermore
Q 3
tf + 1/12 - F + ^ α,£> + - m-^Fi > μ*{Κ + Δ).
T ; 3
3 i
Hence (5, Δ) is log-terminal. q.e.d.
(0.5). In the remaining of this section, we recall the construction of the basic
families of elliptic curves due to Kodaira [14].
Let Ho be a variation of polarized Hodge structures (V.P.H.S) of rank 2,
weight 1 on a complex manifold S°. Then we have the period map
f : S° —• Η
from a universal covering r : S° —• S° of S° to the upper half plane Η := {ζ G
C I Im ζ > 0}. We have the monodromy representation
Φ :πι(5°) —• S£(2,Z)
such that
f (/? - y) = (a^f (y) + b)/(cf(y) + cfc),
ß ß
where y , and
For β e 7Ti(5°) and for integers ni,ri2, an automorphism <7(/3;ni,ri2) of 5° χ C
defined by
(y, 0 «— (0 · y. + · (C + nif(y) + * )),
2
where y G 5°, ( G C. Then G := ni, 712)} forms a group and acts on
S° χ C properly discontinuously and freely. Hence we obtain a quotient manifold
B° := 5° x C/G and an induced smooth elliptic fibration p° : B° 5°. Note
On Weierstrass Models 409
that p° has a section σ : S° —> B° defined by σ(χ) = the class of (Î/,Ο), where
r(y) = x. We have also an isomorphism
as a V.H.S.
§1. Elementary properties.
Let S be a complex variety, £ a line bundle on 5, and α and b sections of
4 6 3 2
£~ and £~ , respectively, such that d := 4a + 276 is not identically zero on
2 3
S. Put Ρ := Ps(C>s0£ 0£ ) and let ρ : Ρ -> 5 be the natural projection and
(9p(l) the tautological line bundle. We denote by X, F and Ζ the sections of
-2 _3
(9p(l)<g>£ , (9ρ(1)®£ and Op(l) which correspond to the natural injections
2 2 3
£ —• 0 Θ£ θ£ ,
5
3 2 3
£ —> o<?0£ 0£ ,
and
2 3
C?s —> <9s6£ e£ ,
respectively.
Définition (1.1). By W(£, a, 6) we denote the divisor on Ρ defined by the
2 z 2 z
equation YZ — (X + aXZ + bZ) and call this the Weierstrass model over S
of type (£, a, 6).
(1.2). The following properties of W := W(£, a, 6) are easily shown.
(1) W is also a complex variety and ρ : W -» 5 is a proper flat surjective
2
morphism whose fibers are irreducible cubic curves in P .
(2) If S is normal, then W is also normal. If S is Gorenstein, then W is also
_ 1
Gorenstein and ω\γ = P*(^5 0 £ ).
2 3 3
(3) The natural projection O5 0 £ 0 £ -> £ gives a section σ : 5 -> Ρ
such that σ(£) C W and that σ(5) is a Cartier divisor on W. We denote a(S)
by Σ(£, α, 6) and call this the canonical section of W(C, a, b).
(4) Let s be a defining equation of Σ := Σ(£,α,6). Then there exist an
2
isomorphism φ : O(l) -> 0\γ(3Σ) and sections / G Γ(0ιν(2Σ) ®p*£~ ) and
w 3 3
$ G Γ(£ν(3Σ) <g>p*£- ) such that = s, V>P0 = /* and ψ(Υ) = 9.
(5) There exists an isomorphism ρ^Ο^Σ) = £ and using /, g, s in (4), we
can identify
p*O(Z) = Os-s,
w
2 2
p+O(2Y,) = Os'S ®C · f,
w
and
z 2 3
p+O(3V) = Ό s · s 0 £ · /* 0 £ · g,
w
410 Ν. ΝΑΚΑΥΑΜΑ
with the relation
2 3 4 6 6
g = / + α/5 + bs in p*O(6V) <g> C~.
w
(6) We have
1 j
Rp+O = £, Rp+O(mY,) = 0 for ; > 1 and m > 1.
w w
Let Τ C W be another irreducible Cartier divisor isomorphic to S by p.
Then, we have the following:
Lemma (1.3). (1) There is an automorphism j : W = W over S such that
;(Σ) = Γ, ;(Γ) = Σ, and j -j = id.
(2) There exists the translation L(T) : W —• W which induces χ \-+ χ +
T(p(:c)) on smooth fibers W ( ) under the group structure with Σ(ρ(χ)) = 0.
p x
From this we can prove that W^/S has a group structure over S with Σ
being zero, where := {x G W \ ρ is smooth at x}.
Lemma (1.4). Let η : W(£,a,6) —• W(£',a',6') 6e an isomorphism over
S such that η(Σ(£, a, 6)) = Σ(£',α', &'). T/ien Mere existe a nowhere vanishing
- 1 4 6
section ε G Γ(£, £' ® £ ) suc/i ί/ιαί a = α'ε, 6 = δ'ε, and that η is the
morphism
ζ
(X : Υ : Ζ) y—> (εΧ : Y : εΖ).
4 6
Especially i/(£, α,δ) = (£',α',δ'), then ε G C*, ε = 1 or ε = 1. Moreover, if
4 6
ε ^ 1, then 6 = 0, and i/ε φ 1, ί/ien a = 0.
Corollary (1.5). // λ : W —• W is an automorphism over S, then there
exists an irreducible Cartier divisor TonW isomorphic to S by ρ such that λ
is a composition of one of the automorphisms in (1.4) and L(T).
Definition (1.6). Let S be a complex manifold. A triplet (£,a, 6) is said
to be minimal if there is no prime divisor Δ on S such that div(a) > 4Δ
and div(6) > 6Δ. If (£,α,δ) is minimal, then W(£,a, b) is called a minimal
Weierstrass model.
Any Weierstrass model is obtained from a minimal Weierstrass model by
blowing ups and downs.
§2. Elliptic fibrations.
Theorem (2.1). Let π : X -* S be an elliptic fibration between com-
plex manifolds. Assume that π has a section σ : S —• X. Then there exist
a Weierstrass model W(£, a, 6) over S and a proper bimeromorphic morphism
μ : X -> W(C,a,b) over S such that a(S) = μ*Σ(£,α,6).
On Weierstrass Models 411
For a proof, according to Mumford and Suominen [20], we consider the
following long exact sequences
(* ) 0 —• π*Οχ((τη - l)T) —• π+Οχ{τηΤ) —. π*Ο{πιΤ) —.
m τ
—> tfîr.Ox((m - 1)T) —• Ä ^ C ^ m T) —- 0,
and isomorphisms
f
Ι?π*Οχ((τη - 1)T) ^ Ä V.(9x(mT) for m G Ζ and i > 2,
where Τ := σ(5) and 5 is a defining equation of Γ. By (Ai), it is easy to see
that
and the following sequence is exact:
! 1
0 —> π.Οχ(Γ) —• Α π.Οχ —* Ä ^ * (Γ) —• 0.
Hence π*Ο(Τ) = O · s.
χ s
Let us consider (Λ™) for m > 2. Our first question is the surjectivity of
π*Ο(τηΤ) π*Ο{πιΤ). We need the following:
χ τ
Lemma (2.2). //dimS = 1, Men π*Οχ(πιΤ) —• 7r*<9r(mT) *5 surjective
for m > 2.
1
Proof. If ϋΓχ is π-nef, then Ι^π.ϋ^Τ) = 0. Thus R n^O(mT) = 0
x
for m > 1 by (Am) for m > 2. Hence π*(9χ(τηΤ) -» π*(9:τ(7πΤ) is surjective
in this case. If K is not π-nef, then by contracting (-l)-curves contained in
x
a fiber, we obtain a minimal elliptic fibration g : Y —• S and a contraction
λ : X F. Therefore it is enough to prove that if π.(9χ(τηΤ) -+ π+Οτ{τηΤ)
is surjective for m > 2 and if ρ : Ζ —• X is the blowing up at a point ρ G Γ,
7 ;
then (π · p)+öz{mT') -> (π · p^Or^mT ) is surjective for ra > 2, where T is
-1
the strict transformation of T. Put Ε := ρ (ρ). Then p*T = T' + E. Then we
have
p*O(mT') = p+O{-mE) <g> O{mT) = xtÇ · O (mT),
z z x x
and
p+0>(mT') = p+0,(-mE) <g> 0{mT) = m™ · O (mT),
T T T r
where trip is the maximal ideal at p.
Since Τ is a section of π, π+(χη™>Ο(τηΤ)) = m™ 'π+Ο(τηΤ). The natural
τ (j>) τ
homomorphism 7r*(m™^) —• xrÇ induces the following commutative diagram:
m?yK.O{mT) —> m™ . π.ϋ(τηΤ)
{p x (p) τ
I II
mT
K*(K'°x( )) —> π.(τη™·ϋ (τηΤ)).
τ
