ebook img

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

305 Pages·1990·4.6 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

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

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 Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved © 1990 Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 1990 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owners. 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

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.