Lecture Notes ni Mathematics Edited yb .A Dold and .B Eckmann 6501 ciarbeglA yrtemoeG tserahcuB 1982 Proceedings of the International Conference held ni Bucharest, Romania, August 2-7, 1982 Edited yb .L B~descu and .D Popescu galreV-regnirpS nilreB Heidelberg New York oykoT 1984 Editors Lucian B~.descu Dorin Popescu The National Institute for Scientific and Technical Creation Bdul Pacii 220, ?9622 Bucharest, Romania AMS Subject Classification (1980): 14-XX ISBN 3-540-12930-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12930-8 Springer-Vertag New York Heidelberg Berlin Tokyo work This si to subject .thgirypoc llA rights era ,devreser the of part or whole the whether lairetam si ,denrecnoc of those specifically ,noitalsnart reprinting, esu-er of ,snoitartsulli ,gnitsacdaorb noitcudorper yb photocopying enihcam similar or ,snaem dna data in storage .sknab rednU § the of 54 namreG Copyright waL copies where era private than other for made ,esu is fee a elbayap to tfahcsllesegsgnutrewreV" ,"troW .hcinuM © yb galreV-regnirpS 1984 Heidelberg Berlin detnirP ni ynamreG Printing dna Beltz binding: ,kcurdtesffO .rtsgreB/hcabsmeH 012345-0413/6412 FOREWORD The "Week of the Algebraic Geometry" held in Buchar~st bet- ween August 2-7, 1982, was organized by the Department of Mathema- tics of INCREST Bucharest, with the hospitality of the University of Bucharest. This volume contains a part of the lectures and talks given during the Conference, as well as two other papers (by L.B~descu and A.Constantinescu), which were not presented because of lack of time. We are grateful to all the participants for their contribu- tion to the success of the Conference, and in particular, to the contributors of this volume. We also thank INCREST Bucharest for the generous financial support and Springer-Verlag for accepting to publish these Procee- dings. The Editors P A R T I C I P A N T S • L BADESCU (Bucharest) C .BANICA (Bucharest) S.BARCA~[ESCU (B~chare st ) S .BASARAB (Bu chare st) M.BECHEANU (Bucharest) C. BORCEA (B~/chare st ) • A BREZULEANU (B~ c hat e st ) V. BRINZANESCU (Bucharest) A. BU'fUM (Buchare st ) • N BURUIANA (Bucharest) F. CATANESE (pisa) G. CHIRIACESCU (Bucharest) A .CONSTANT!NESCU (Bucharest) A .DIMCA (Buchare st) M. FEUSTE L (Berlin) • M FI ORENTINI (Fe rrara ) • G GA LBURA (Bucharest) G. -M. OREUEL (E:aiserslsutern) I • P ONESCU (Bucharest) S .KLEIMAN (M.I.T.) H. KURKE (Berlin) • A LASCU (Montreal) N. MA NOLA C HE (B~ch are st ) E. HA LAN Y A (Bucharest) B.MARTIN (Berlin) S .MUKAI (Nagoya) G.PFISTER (Berlin) D .POPESCU (Bucharest) N. RADU c~XB( hare st ) M .ROCZEN (Berlin) P.RUSSEIL (Montreal) M -STOIA (B~chare st) D .STEFANESCU (Bucharest) R .URSIANU colB( har • st ) T .ZINK (Berlin) CONTENTS I. B~descu, .L - Hyperplane sections and deformations .......... I .2 BrTnz~nescu, V. Topologically trivial algebraic and M.Stoia - 2-vector bundles on ruled surfaces. II ......... 34 3. Buium, A. - On surfaces of degree at most 2n+I in pn . . . . . . . . . . . . . . . . . . . . . . . . 47 4. Catanese, .F Commutative algebra methods and equations of regular surfaces . . . . . . . . . . . . 68 5. Constantinescu, A. - On the algebraization of some complex schemes .................... 112 6. Fiorentini, M. Two theorems of G.Gherardelli on and Lascu, A. curves simple intersection of three surfaces . . . . . . . . . . . . . . . . . . . . . . . . 132 7. Ionescu, P. Embedded projective varieties of small invariants . . . . . . . . . . . . . . . . . . . 142 8. Kurke, H. and Some examples of vector bundles Theel, .H on the flag variety F(1,2) .............. 187 9. Martin, .B and Distinguished deformations of Pfister, .G isolated singularities of plane curves . . . . . . . . . . . . . . . . . . . . . . . . 255 10. Popescu, .D On Zariski's uniformization theorem ......... 264 11. Roczen, M. Some properties of the canonical resolutions of the 3-dimensional singularities n n n A ,D , E over a field of characteristic ~2 ............... 297 12. Russell, P. Factoring the Frobenius morphism of an algebraic surface ............... 366 HYPERPLANE SECTIONS AND DEFORMATIONS Lucian Badescu Introducti on This paper is a continuation of ES ~ and E43- Here we especially determine all normal projective varieties X containing a certain given projective variety Y as an ample Cartier divisor. In many c~ses we shall be dealing with, the va- riety X turns out to be a cone over Y, provided that X is assumed to be singu- lar. Some situations of this kind were already encountered in ~SJ and ~4S. The paper is divided in four sections. The first one deals with the cases in which Y is either an elliptic curve (Theorem 1), or a smooth curve of genus >/2 (The- orem 2). Since in CS S and E4J we classified all smooth projective 3-folds con- taining & geometrically ruled surface as an ample divisor, it is natural also to see what is going on in the singular case. And indeed, section 2 takes up this problem, giving a complete answer if Y is the surface F (Theorems 3 and e 4), and a partial one if Y is a pl-bundle over a smooth non-rational curve (Theorem .)6 The higher dimensional case .i( .e when Y is a pn-bundle over a smooth curve, with n~2) is much easier to handle. In section 3 we improve a result of Fujita concerning the Grassmann variety (Theorem 7). Finally, the last section relates the results obtained previously with the deformation theory. After this paper was written we got a paper of Fujita (Rational retractions onto ample divisors, Sci. Papers Coll. Arts and Sci. 33 (1983) 33-39), in which, using our idea of proving Theorem 7 below (essentially contained in the simple and rather formal lemma 3 below), he gives a nice criterion concerning the exis- tence of a rational retraction ~ of the inclusion YCX (in the notations of lemma 3). This criterion applies in many cases for which dim(Y)~3. A part of the results of this paper were found when the author spent the academic year 1981-1982 at the Institute for Advanced Study (Princeton) with the NSF Grant MCS 77-18723 A04. He wants to thank this institution for the hos- pitality and excellent conditions extended to him. He also thanks P. Ionescu for some stimulating discussions. Some of these results were announced (with- out proofs) in a note with the same title come out in "Recent Trends in Math:', Reinhardsbrunn 1982, Teubner-Texte zur Math. Band 50, Leipzig 1983. Terminology and notations. Throughout this paper we shall fix an algebrai- cally closed base field k. In general the terminology and notations are standard, with th~ following precisations. Unless otherwise stated, all schemes we shall be dealing with will be alge- braic schemes over .k The term "algebraic variety" means an irreducible and re- duced algebraic scheme over .k By a polarized variety we understand a pair (Y,L) consisting of a projective variety Y and an ample line bundle L over .Y The graded k-algebra S - S(Y,L) as- sociated to the polarized variety (Y,L) is the algebra ~ H°(Y,L n) graded in the natural way. Consider the polynomial algebra S~zJ over S in one variable z graded by the condition that deg(sz n) - deg(s) + n for every homogeneous ele- ment s~S. Then the variety C(Y,L) ~ Proj(S~zS) will be referred to as the pro- jective cone associated to the polarized variety (Y,L). Let Y be an effective Cartier divisor on the variety X. We shall denote by 0x(Y ) the invertible sheaf (or line bundle) associated to the divisor Y, and by Ny, X ~ 0x(Y)~ 0y tks normal bundle of Y in X. A global equation of the divi- sor Y on X is a section ~ E R°(X,Ox(Y)) whose associated divisor is .Y If Z is an arbitrary algebraic scheme over k, we shall denote by % the Grothendieck dualizing sheaf of Z. If D is an effective Cartier divisor on Z, then one has the adjunction formula Df(6 ~ %@0z(D)@0 .D This gives in par- ticular the genus'formula of a curve om a smooth projective surface. If DI,... ,D d are Cartier divisors on a proper d-dimensional scheme Z over k, then DI°D2..°D d will denote the intersection number of DI,°.°,D .d In parti- cular, if DI m D for every i m l,...,d, then D1. D2...D d will be also denoted by D "d , or simply by D . d If F is a coherent sheaf on the scheme Z and D a Cartier divisor on Z, then we shall denote by F(D) the sheaf F®Oz(D). If moreover Z is proper over k we shall denote by hi(Z,F) the dimension over k of the vector space Hi(Z,F). If E is a vector bumdle over Z, E stands ~or the dual of E. If E is of rank one, we shall also write E -1 instead of E. If S is a graded k-algebra and r a natural number, then (r) S is the graded k-algebra such that (S ~rj) = S , where S denotes the homogeneous part of n m degree m of .S ~. Surfaces eontainin 6 a 6iven curve as an ample Cartier divisor Let us begin by recalling two well-known results: 1 Theorem A. Let X be a normal pro~ective surface containin6 Y ~ P as an _smple Cartier divisor. Then (up to an isomorphism) one has one of the follo- wing three possibilities: a) X is p2 and y is either a straight line or a conic in p2 b) X is the ~eometrisall~ ruled surface Fe = P(Op~Opd(-e)) (e~o) and 1 Y is a section of the canonical projection p:F ~P ; . . . . . . pl S c) X is the pro~ective cone over with respect to the s-fold Verones ~ 1 s embeddin~g v :P P • (s~2) and y is the intersection of X with the h~er- S plane at infinity of pS+l. Theorem A is classical. A modern reference for it is ~8~. Theorem B (Char(k) ~ o). Let X be a smooth projective surface containi.n~ the elliptic curve Y as an ample divisor. Then (up to isomorphism) one has one of the followin~ two possibilities: a) X is a Del Pezzo surface anal -y is a canonical divisor on X; b) X is a ~eometricall~ rule~ surface p:P(E) ~ Y over Y and the inclu- sion YCX is a section of p (with E a rank two vector bundle over Y). Theorem B is also classical. A modern reference for it is [~. In connec- tion with theorem B, it is natural to classify also all normal (singular) pro- jective surfaces containing a given elliptic curve as an ample Cartier divisor. As far as we know such a classification is not explicitly containeal in any pa- per, although it turns out to be closely related to the classification of all d d surfaces of degree d in P which are not contained in any hyperplane of P (see [SD)- The result is the following: Theorem i (Char(k) i 01. Let X be a normal (sit~ular) projective surface containin~ the elliptic curve y as an ample Cartier divisor. Then one has one of the followin~ two possibilities: a) X is a surface with only rational double points as singularities and -Y is a canonical divisor on X. These surfaces are classified in E~], E~J~ ~ (see Theorem C below). b) X is the projective cone over the polarized curve (Y,Ny,x) , and y is embed~e~ in X as the infinite section (i.e. X is &n elliptic cone over y). Proof. Let f:~ ~X be the minimal desing~alarization of X, i.e. the ex- ceptional fibres of f do not contain any exceptional curve of the first kind. Since Y does not meet the singular locus of X, Y is also contained in X and and the normal bundles of Y in X and of Y in X are the same. In particular, y2)g ( >o and Y.E~o for every integral curve E on X. The exact sequence yields the exact sequence of cohomology By duality and the Kodaira-Ramanujam vanishing theorem, Bl(oJW~-~Of(Y))~ = ,o which implies that q = hl(o~) = .l._~)~Ycc(lh If q = o the map 6 is surjective, and therefore there is a section sCH (X,~O~(Y)) 0 ~ whose restriction to Y is ;l in other words, there is a canonical divisor K on ~ of the form K = D-Y, with D~o and Supp(D)~Y = .~ Since Y is ample on X, the support of D (if D ~ o) is contained in the excep- tional fibres of .f On the other hand, since f is minimal, o~K.E = D.E - Y.E D.E for every irreducible component E of the exceptional fibres of .f Then a standard argument (see p)'QpoS. E~57, ~) shows that D~<O. Recalling that D~/o, ~e ~et D ~ ,o and in particular, ~/f-Y ~ %/~-Y. This ~ impiies that i~ in- vertible and isomorphic to Ox(-Y), and that f~((~X) = £cY~. By [Z] X has only rational double points as singularities. Therefore q = o leads to case a). Now assume q = .1 First we show that X is ruled. Claim. Assuming that X is not ruled, then ~.Z>~O for every integral cur- ve Z onX such that Z2> o and pa(Z)~l. Proof of the claim. We shall proceed by induction on the number n of qua- dratic transformations in order to reach from its (unique) minimal model (see ~). If n = ,o i.e. if ~ is itself the minimal model, then the classifi- cation of surfaces (loc. cit.) ~hows that ~l~.Z~/o for every curve Z on ~. Thus we can assume n>o, and let E he an exceptional curve of the first kind on ~. Let ~ 6 :~ ~ X be the morphism contracting E to a smooth point xEX, and set Z' = 5"(Z). We have Z 2' 2 = Z 2, + m where m>/o is the multiplicity of the point x on 'Z (m = o if x~Supp(Z')). Therefore Z'2~/Z2> .o Moreover, pa(Z' )>~pa(Z)>/l. Using the inductive hypothesis we infer that 60~.Z'~o. But o.<c~.z, -- ~(~). , ) G~(z ~ ~(~).z ÷ m~(~).E ~ e'(C~).Z. Therefore ['(O~).Z>/o. Since ~ = 6~'(O~)~O[(E), we have O.Y~.Z = ¢~*(6t~).Z + E.Z >/ >/E.Z>/o, the last inequality coming from the fact that E and Z are different integral curves. The claim is proved. Returning to the proof of theorem 1 and using the claim (in the assumption that ~ is not ruled), one gets dLr~.Y>/o. Then the genus formula yields the desired contradiction: I = pa(y) = i/2.y 2 + i/2.6~Y~.y + i /> 1/2.¥ 2 + i>i. This proves that X is ruled if q = .i Now we can apply corollary 2.4 of f~ in order to deduce that the inclusion YCX is equivalent to a section of a geometrically ruled surface p:P(E) > .Y The point is to show that the inclusion YCX actually coincides to a sec- tion of a geometrically ruled surface X = P(E) ~Y. But this is standard as following. Let g:~ ~¥ be the fulling morphism of X. We have to check that all the fibres of g are irreducible, knowing that Y.F = 1 for a general fibre F of .g Since ~ is the minimal desingularization of X, there are no exceptio- nal curves of the first kind not meeting Y. Consider a commutative diagram Y where ~ is a birational morphism (such a diagram always exists because the 1 minimal models of surfaces birationaily equivalent to P ~ Y are the geometri- cally ruled surfaces g':P(E)>Y over Y). Since Y is a section of ,g ~(Y) is also a section of g', and in particular, ~(Y) is smooth. Let m be the ma- ximal number of irreducible components of fibres of ,g and let F be a fibre of g having exactly m components. An easy induction on m shows that if m>l, there is an exceptional curve ~ E of the first kind contained in F such that Y]/'E = .~ In this way, the assumption that m~>l contradicts the minimality of the desingularizaticn f:X > X. Therefore m = ,i or else ~ is geometrically ruled over y and the inclusion YCX is a section of g:~ > .Y Write ~ = P(E), with E a normalized vector bundle of rank 2 over Y (see ~, page 373). Set e = -deg(E), and let C be the minimal section of P(E). o Then 2 C -e and Op(E)(Co) = Op(E)(l .) Assume that the vector bundle E is in- s = decomposable. Then by loc. cit., Theorem 2.15 (page 377) and Propositions