Algebraic Geometry: Proceedings of the Conference at Berlin 9–15 March 1988

305 Pages·1990·4.6 MB·English

by  F. A. Bogomolov, A. N. Landia (auth.), H. Kurke, J. H. M. Steenbrink (eds.)

ALGEBRAIC GEOMETRY
ALGEBRAIC GEOMETRY
Proceedings of the Conference at Berlin
9-15 March 1988
Edited by
H.KDRKE
Sektion Mathematik, Berlin, Germany
and
J. H. M. STEENBRINK
Mathematical Institute, Nijmegen, The Netherlands
Reprinted from
COMPOSITIO MATHEMAT ICA
Volume 76, Nos 1 & 2, 1990
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
ISBN-13:978-94-010-6793-5 e-ISBN-13:978-94-009-0685-3
001: 10.1007/978-94-009-0685-3
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COMPOSITIO MATHEMATICA
Volume 76, Nos 1 & 2, October 1990
Special Issue
ALGEBRAIC GEOMETRY
Proceedings of the Conference at Berlin, 9-15 March 1988
Preface vii
List of participants viii
F.A Bogomolov and AN. Landia: 2-Cocyc1es and Azumaya algebras under
birational transformations of algebraic schemes 1
W. Decker: Monads and cohomology modules of rank 2 vector bundles 7
A Dimca: On the Milnor fibrations of weighted homogeneous polynomials 19
AH. Durfee and M. Saito: Mixed Hodge structures on the intersection
cohomology oflinks 49
H. Esnault and E. Viehweg: Effective bounds for semipositive sheaves and
for the height of points on curves over complex function fields 69
J. Feustel: Eine Klassenzahlformel fUr singuUire Moduln der Picardschen
Modulgruppen 87
J. Franke: Chow categories 101
M. Furushima: Complex analytic compactifications of (:3 163
H. Hauser and G. Milller: Analytic curves in power series rings 197
B. Hunt: A Siegel Modular 3-fold that is a Picard Modular 3-fold 203
J. Jurkiewicz: Linearizing some Z/21L actions on affine space 243
I. Luengo and G. Pfister: Normal forms and moduli spaces of curve
singularities with semigroup <2p, 2q, 2pq + d> 247
U. Nagel: On Castelnuovo's regularity and Hilbert functions 265
D. Popescu and M. Roczen: Indecomposable Cohen-Macaulay modules and
irreducible maps 277
M. Szurek and J.A Wisniewski: Fano bundles of rank 2 on surfaces 295
Compositio Mathematica 76: vii, 1990.
Preface
The Conference on Algebraic Geometry, held in Berlin 9-15 March 1988, was
organised by the Sektion Mathematik of the Humboldt-Universitat. The organising
committee consisted of H. Kurke, W. Kleinert, G. Pfister and M. Roczen. The
Conference is one in a series organised by the Humboldt-Universitat at regular
intervals of two or three years, with the purpose of providing a meeting place for
mathematicians from eastern and western countries.
The present volume contains elaborations of part of the lectures presented at the
Conference and some articles on related subjects. All papers were subject to the
regular refereeing procedure of Compositio Mathematica, and H. Kurke acted as a
guest editor of this journal.
The papers focus on actual themes in algebraic geometry and singularity theory,
such as vector bundles, arithmetical algebraic geometry, intersection theory, moduli
and Hodge theory.
We are grateful to all those who, by their hospitality, their presence at the Con-
ference, their support or their written contributions, have made this Conference to a
success.
The editors
Compositio Mathematica 76: viii, 1990.
List of participants
R. Achilles, Halle G. Kempf, Baltimore
K. Altmann, Berlin W. Kleinert, Berlin
Y. Alwadi, Berlin S. Kloth, Kothen
S. Aouira, Berlin B. KreuBler, Berlin
A. Aramova, Sofia H. Kurke, Berlin
L. Badescu, Bucharest A. Landia, Tbilisi
C. Banica, Bucharest H. Lange, Erlangen
F. Bardelli, Pavia O. Laudal, Oslo
G. Barthel, Konstanz A. Lipkowski, Beograd
K. Behnke, Hamburg V. Lomadse, Tbilisi
A. Campillo, Valladolid N. Manolache, Bucharest
J. Castellanos, Madrid F. Marko, Bratislava
J. Coanda, Bucharest B. Martin, Berlin
W. Decker, Kaiserslautern H. Meltzer, Berlin
F. Delgado, Valladolid T. Muhammed, Halle
B. Dgheim, Berlin G. MUller, Mainz
A. Dimca, Bucharest W. MUller, Berlin
K. Drechsler, Halle U. Nagel, Halle
A. Durfee, Bonn N. Nakayama, Bonn
D. Eisenbud, Waltham Y. Namikawa, Bonn
H. Esnault, Bonn A. Nemethi, Bucharest
J. EBer, Bonn V. Palamodov, Moscow
J. Feustel, Berlin A. Parisinski, Gdansk
Th. Fiedler, Berlin G. Pfister, Berlin
T. Fimmel, Berlin D. Popescu, Bucharest
H. Flenner, Gottingen P. Pragacz, Torun
J. Franke, Berlin H. Reimann, Berlin
M. Furushima, Bonn M. Roczen, Berlin
F. Gaeta, Madrid H. Roloff, Erfurt
H. Grabe, Erfurt D. Rothe, Berlin
G.-M. Greuel, Kaiserslautern P. Schenzel, Halle
K. Haberland, lena H. Schonemann, Berlin
Z. Hajto, Krakow O. SchrOder, Halle
H. Hauser, Innsbruck T. Shiota, Princeton
F. Herrlich, Bochum Th. Siebert, Berlin
L. Hille, Berlin O. Song, Berlin
L. Hoa, Halle J. Spies, Berlin
Th. HOfer, Bonn J. Steenbrink, Nijmegen
R. Holzapfel, Berlin M. Szurek, Warsaw
B. Hunt, W. Lafayette J. Tschinkel, Berlin
D. Huybrecht, Berlin N. Tu Cuong, Hanoi
F. Ischebeck, M iinster E. Viehweg, Bonn
J. Jurkiewicz, Warschau W. Vogel, Halle
C. Kahn, Waltham U. Vollmer, lena
G. Janelidze, Tbilisi E. Zink, Berlin
Compositio Mathematica 76: 1-5, 1990.
© 1990 Kluwer Academic Publishers.
2-Cocycles and Azumaya algebras under birational
transformations of algebraic schemes
F.A. BOGOMOLOV1 & A.N. LANDIA 2
'Steklov Mathematical Institute of the Academy of Sciences of USSR, Vavilov Street 42, Moscow
117333, USSR; 2Mathematical Institute of the Academy of Sciences of Georgian SSR, Z.
Rukhadze Street 1, Tbilisi 380093, USSR
Received 17 November 1988; accepted in revised form 23 November 1989
The basic question whether the injection Br(X) -+ H2(X, (91)tors is an iso-
morphism arose at the very definition of the Brauer group of an algebraic scheme
X. Positive answers are known in the following cases:
1. the topological Brauer group Br(Xtop ) ~ H2(X, (9~p)tors ~ H3(X, Z)tors (J.-P.
Serre); in the etale (algebraic) case the isomorphism is proved for
2. smooth projective surfaces (A. Grothendieck);
3. abelian varieties;
4. the union of two affine schemes (R. Hoobler, O. Gabber).
The first author has formulated a birational variant of the basic question, while
considering the unramified Brauer group in [1]. The group Brv(K(X)) =
nBr(Av) S Br(K(X)) (intersection taken over all discrete valuation subrings Av of
the rational function field K(X)) is isomorphic to H2(X, (9*), where X is
a nonsingular projective model of X, i.e. a nonsingular projective variety
birationally equivalent to X.
QUESTION. Given a cocycle class )I E H2(X, (9*), is it possible to find a non-
singular projective model X such that )I is represented by a [pln-bundle (i.e. by an
Azumaya algebra) on X?
The case where X is a nonsingular projective model of V;'G, with G a )I-minimal
group and Va faithful representation of G, was considered in [2]. O. Gabber in
his letter to Bogomolov (12.1.1988) has given an affirmative answer to the
question in the case of general algebraic spaces. In this paper we give a simple
version of his proof for algebraic schemes.
Let X be a scheme, )I E H2(X, (9*), {V i} an affine cover of X. Then the restriction
of)l to each Vi is represented by an Azumaya algebra Ai' If we would have
isomorphisms Ai1u,nUj ~ AjlU,nup we could glue the sheaves {AJ and get an
2 F.A. Bogomolov and A.N. Landia
Azumaya algebra on X, representing y. But we have isomorphisms AilU,nUj ®
End(Eij) ~ AjlU,nUj ® End(Eji) for certain vector bundles Eij, Eji on Ui n Uj .
THEOREM. Let X be a noetherian scheme, y E H2(X, (9;). There exists a proper
birational morphism IX: X -+ X such that IX*(Y) is represented by an Azumaya
algebra on X.
Proof. It is enough to consider X which are connected. Suppose that {Ui} is an
affine open cover of X and that y is non-trivial on at least one Ui' We will
construct an Azumaya algebra on a birational model of X by an inductive process
which involves adjoining one by one proper preimages of the subsets Ui and, by
an appropriate birational change of the scheme and Azumaya algebra obtained,
extending the new algebra to the union. We start with some affine open subset U 0
and an Azumaya algebra Ao on it.
Now suppose by induction that we already have an Azumaya algebra Ak on the
scheme X k , a Zariski-open subset of the scheme Xk , equipped with a proper
birational map ak: Xk -+ X such that X k = akl(Uo u ... U Uk)' Let Uk+l inter-
sect Uo U ... U Uk and Uk+l = a- 1(Uk+d. Suppose that on Uk+1> y is re-
presented by the Azumaya algebra Ak+l' In the same vein as above we have an
isomorphism
and we need to extend Ek,k+l to X k and Ek+l.k to Uk+l from their intersection.
After this we will change Ak and a:'(Ak+ d by the other representatives
Ak ® End(Ek,k+d, a:'(Ak+d ®End(Ek+l.k) ofthe same Brauer classes and glue
these Azumaya algebras, hence the proof.
First, extend both sheaves E as coherent sheaves. This can be done by the
following
LEMMA. Let X be a noetherian scheme, U £: X a Zariski-open subset, E a co-
herent sheaf on U. Then there exists a coherent sheaf E' on X such that E'lu ~ E.
This is Ex. 11.5.15 in [4].
Note that we can assume that in our inductive process we add neighborhoods
U k+ 1 of no more than one irreducible component (or an intersection of
irreducible components) of X, different from those contained in X k• Thus we
assume X k n Uk + 1 to be connected and the rank of E to be constant on
X k n U k+ 1> hence E' will be locally generated by n elements, where n is the rank
of E.
LEMMA (see [3], Lemma 3.5). Let X be a noetherian scheme, E a coherent sheaf
on X, locally free outside a Zariski closed subset Z on X. Then there exists
a coherent sheaf I ofi deals onX such that the support of{9x/I is Z with thefollowing
2-Cocycles and Azumaya algebras 3
property. Let IX: X -+ X be the blowing up of X with center I, then the sheaf
ri(E) := the quotient of IX*(E) by the subsheaf of sections with support in IX - l(Z), is
locally free on X.
Proof. The proof consists of two parts. First: to reduce the number of local
generators to get this number constant on the connected components of X (the
minima are the values ofthe (local) rank function of E). Second, to force the kernel
of the (local) presentations (!)'P -+ Elv -+ 0 to vanish for all neighborhoods from
some cover {V}. Both parts are proved by indicating the suitable coherent
sheaves of ideals and blowing up X with respect to these sheaves. Let
(!)'P L Elv -0 be a local presentation of E. Then Ker(f) is generated by all
relations L7'= 1C iai = 0 where {ai} stand for the free basis of (!)'P. The coherent
sheaf of ideals in the first case is the sheaf defined locally as the ideal I v in (!)v
generated by all Ci such that L7'= 1C iai E Ker(f) and in the second case as J x =
Ann(Ker(f)). As the number of generators is constant in the case we are interested
in, we give the details only for the second part of the proof and refer to [3] for the
first.
Let IX: X' -+ X be the blowing up of X with respect to J x and let ri(E) be as in the
statement of the Lemma. Let
o -+ (Ker(J))lv' -+ (!)'P, L ri(E)lv' -+ 0
be the local presentation of ri(E). We have 1X-1(Ann(f)) 5; Ann(Ker(J)), Let
P E Z', V' = Spec(A') an affine neighborhood of p in X' and let Li'!: 1C iaiE
Ker(J) Iv ' map to a nonzero element in Ker(J)p' Denote by y a generator of the
invertible sheaf 1X-1(Ann(f)) on V" = Spec(A") 5; V' for suitable A". It is clear
that there exists for given p and V" a finite sequence of open affine neighborhoods
V1, ... , V; such that X'\Z' = V1, V" = V; and Vj n Vj+1 #- (j) for j = 1, .. "
s - 1. So suppose V' n (X'\Z') #- (j) and q E V" n (X\Z). Then (c;)q = 0 for
i = 1, ... , m and q E Spec(A~) hence ykCi = 0 for i = 1, ... , m for some k. Since y is
not a zero divisor, we conclude that Ci = 0 for i = 1" .. ,m. Thus (maybe after
considering a finite sequence of points q 1, ... , qs) we prove that (Ker(f))p is trivial
for every p EX'. 0
In this way we glue the two sheaves Ak and Ak+ 1 and get an Azumaya algebra on
Xk U ak + 1. As the scheme X is quasi-compact, we obtain an Azumaya algebra on
X after a finite number of such steps.
Now we have to show that this process can be done in such a way that the class
[A] of the Azumaya algebra A constructed in this way is equal to ri*(y). Again this
goes by induction on k. We have X k+ 1 = U U V with U = rik-t\(UO u .. · u Uk)
and V = rik-)-dUk+d. We have the exact sequence
4 F.A. Bogomolov and A.N. Landia
and by induction hypothesis, at+ 1t V) - [Ak+ 1] maps to zero in H2(U, (9*) E9
H 2(V,(9*) so it comes from peHi(U () V, (9*). By blowing up Xk+i we may
assume that p is represented by a line bundle which extends to U. Then p maps to
zero in H2(Xk+1> (9*), hence at+dy) - [Ak+i] = o. 0
Note that we need not bother about the compatibility of isomorphisms, because
at each step we choose a new isomorphism between the Azumaya algebra A on
U i u ... U Uj from the preceding step and Ak on Uk, modulo End(E), End(Ek)'
COROLLARY 1. Let G be afinite group, V afaithju[complex representation ofG.
Then there exists a nonsingular projective model X of V/G such that Br(X) =
H2(X, (9*).
Proof. The group H2(X, (9*) is a birational invariant of nonsi n gular projective
varieties and is isomorphic to H2(G, Ql/l::) if X is a model of V/G (see [1]). It
remains to recall that the group H2(G, Ql/l::) is finite. 0
COROLLARY 2. Let X be a noetherian scheme over C, Z a closed subscheme of
X and y e HHx, (9*). Then there exists a proper morphism oc: X' --t X which is an
isomorphism above X\Z and maps y to zero in H;-,(Z)(X', (9*).
Proof First, let's have IX(Y) map to zero in H2(X, (9*). To do this, desingularize
X by X' --t X. Then in the following exact sequence (in etale cohomology), p will
be injective:
Hi (X'\Z' ,(9*) --t Hl'(X', (9*) --t H 2(X', (9*) ~ H2(X'\Z' , (9*)
i
Hl(X, (9*)
The injectivity is due to the injectivity of H 2 (X', (9*) --t H 2 (K(X'), (9*) for
a nonsingular irreducible scheme X'.
Now y comes from y' e Hi (X'\Z', (9*) = Pic(X'\Z'). It is obvious that Picard
elements lift to Picard elements by the blowing ups from the theorem. Thus
from the diagram
Hi (X" , (9*) --t Hi (X"\Z" , (9*) --t Hl'(X" , (9*)
i i
Hi (X'\Z', (9*) --t Hl'(X', (9*)
we conclude that y becomes trivial on Z" by X" --t X' which extends y' to X".
o
Now let us return to the problem of an isomorphism Br(X) --t H2(X, (9*) for