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On Projective Surfaces Arising from an Adjunction Process Us to Determine All the Pairs Japan. J. Math. Vol. 17, No.2, 1991 On projective surfaces arising from an adjunction process By M. ANDREATTA and A. LANTERI (Received May 21, 1990) Introduction We will consider the set S={(S, L)|S is a smooth projective surface and L E Pic(S) is very ample}. An adjunction process on the pair (S, L) in S is the construction of a new pair (S, H) in the following way: first contract all the -1 rational curves C on S with L•EC=1 in a new surface S, it: SS (this is called reduction); then consider on S the line bundle H=KS®L, where L=rr*L. The pair (S, L) is called the reduction of (S, L). If (S, L) is not in a particular subset E of S then (S, H) is again in S; this theorem, with a complete description of the pairs in E, is proved in [SV] and in [Se1] (see also [Io2]). Therefore there is a map RA: S-ES (where RA stands for reduction + adjunction) which associates to a pair (S, L) the pair (S, H)=(S, KS ® L). The map RA is not a surjection: since L is ample, by the Kodaira vanishing theorem, if (S, H) E ImRA, it must be that hi(H)=0, for i=1, 2. The natural question is then to characterize the pairs in S which are in the image of RA. In this paper we consider some particular subsets of S and we determine which pairs in these subsets are "adjunction pairs" in the above sense. We give a satisfac tory answer for the subset T={(S, H)} C S consisting of: a) surfaces in P4, i.e. h0(H)_??_5, (section 1), b) surfaces of degree less than or equal to 9, i.e. H2_??_9 (section 3), c) surfaces for which K71 is nef (section 2). In case b) our results are complete for H2_??_7 but there remain some doubtful cases for H2=8, 9. Our techniques heavily rely on the adjunction theory and the related classi fi cation results of surfaces of small sectional genus. In many instances they allow us to determine all the pairs (S, L) from which a pair (S, H) E ImRA comes via adjunction process. We are indebted to the referee for his useful observations. 286 M. ANDREATTA and A. LANTERI §0. Notation and preliminaries 0.1. We consider in this paper projective surfaces S, which means that S is a smooth, irreducible, projective scheme of dimension 2, defined over the field of complex numbers. We will use the standard symbols in algebraic geometry. In particular, if D is a divisor on a surface S we let: - OS(D) =: the invertible sheaf associated to D- hi(D)=hi(OS(D)) =: dim Hi(OS(D)) - D2 =: D•ED the self intersection of D- [D] =: the line bundle associated to an effective divisor D - q(S) =: h1(Os), the irregularity of S- g(D) =: 1/2 (D2+D•EKS)+1, the arithmetic genus of D- pg(S) =: dim H0(OS(KS)), the geometric genus of S - KS =: a canonical divisor on S the Kodaira dimension of S. We do not distinguish rotationally between a divisor and its associated line bundle. 0.2. We briefly introduce some well known classes of surfaces which are frequently used in the paper. We denote by Pn the n-dimensional projective space and by Opn(r) the r th-power of the hyperplane bundle. A surface S is called a geometrically ruled surface if S is a holomorphic P1 bundle, p: SR, over a non singular curve R. Equivalently, p: S=P(E)R for some rank 2 locally free sheaf on R. We can choose E in such a way that there exists a section, o, of p such that: Num(S) Z[o] Z[f], where f is a fibre of p, and f2=0, f•EƒÐ=1 and u2=-e=deg E. (~ denotes numerical equivalence). For r_??_0 we denote as Fr the geometrically ruled surface P(OP1®Opr(-r)). Fr is the unique holomorphic P1-bundle over P1 with a section u satisfying or2=-r. For r_??_1, o is unique. A surface S is called ruled if it is birational to a geometrically ruled one. A polarized surface (S, L) is called a scroll (respectively a conic bundle) if S is a ruled surface and O(L)f=OP1(1) for a general fibre f (respectively if O(L)f=OP1(2)). 0.3. Given a surface S we denote by B(S, p1, ..., pn) the surface obtained by blowing up the points pi on S. Let rr: B(S, p1, ..., pn)S be the blowing-up. Let L be a line bundle on S. We denote by |ar*L-t2p' the linear system Projective surfaces arising from an adjunction process 287 on B(S, p1, •c, pn) whose elements are in a one to one correspondence with the elements of the linear system |L| which pass through ti points of multiplicity ki. Of course fL tip i will denote the line bundle associated to the above linear system. Frequently the points p1, •c, pn will be assumed to satisfy some general po sition assumption in order to insure the very ampleness of 7r*L tip1. In this case we let Bn(S)=B(S, p1, •c, pn) for shortness. In case S=Fr or P2 for the ex plicit condition expressing the general position assumption with respect to a linear system see [Li, sec. 0]. 0.4. As discussed in the introduction our aim is to study the image of the "ad junction process" in some particular case: namely given (S, H) in T C S we ask whether H=KS®L for some (ample) line bundle L on S such that (S, L) is the reduction of a pair (S, L) E S; we will say that (S, H) is in ImRA. In particular if L itself is very ample, i.e. (S, L) E S then we will say that (S, H) is in ImA. Apart from some special cases [So,(2.4)] it is not yet known whether L is very ample and even if it is spanned by its global sections. So in our notation, it is not known if ImRA=ImA. REMARK0.4.1. We have two obvious necessary condition for (S, H) to be in ImRA. The first one is that i) L=H-KS is ample. The second one is that, since the reduction is a birational map, ii) there exists a pair (S, L) E s such that g(L)=g(L) and S is birational to S. Note that i) implies hi(H)=0, for i=1, 2. In the above hypothesis we will shorten our notation letting x=x(OS), pg=pg(S), q=q(S), g=g(L), d=L2,... and using the 'for the corresponding numerical invariants related to the line bundle H(g(H)=g', H2=d',...). 0.4.2. We recall the following relations between the characters of L and of H, which can be proved by using the adjunction formula, the Riemann-Roch theorem and the Kodaira vanishing theorem: i) d'-g'+2=g, ii) n+1=: h0(H)=pg+g-q, iii) d'+d=KS2+4(g-1). 0.5. We will frequently use the followingwell known theorem. CASTELNUOVO'SLEMMA. (see [G-H]). If C is an irreducible curve embedded in PC-1 and C belongsto no linear hyperplane PC-2, then, with d' the degree of C and g' the genus of C: 288 M. ANDREATTA and A. LANTERI where [ ] is the greatest integer function. §1. Some examples and surfaces in P4 1.1. Let (S, H) be a polarized pair in S. The fact that (1.1.1) hi(H)=0 for i=1, 2 is a necessary but not sufficient condition for the pair to be in the image of RA as the following example shows. EXAMPLE A. Let (S, H) be the minimal elliptic surface of degree 7 of P4 (e. g. see [La, §6] or [Io1, Prop. 8.7]). Since S is regular and |KS| consists of a linear pencil of plane cubics with respect to H, we have pg=2, q=0 and KS•EH=3. Therefore x(H)=x+H•E(H-KS)/2=3+(7-3)/2=5. Since h0(H)=5, this means that h1(H)=h2(H). On the other hand, by Serre duality, h2(H)=h0(KS-H)=0, since (KS-H)•EH<0. So (1.1.1) is satisfied. However (S, H) ImRA. Actually, let L=H-KS; then we have L2=(H- KS)2=1. Assume that (S, H) E ImRA; since 1_??_L2_??_L2=1 we have that (S, L)=(S, L), hence L is very ample and so (S, L)=(P2, Op2(1)), contradicting the fact that S is an elliptic surface. EXAMPLE B. Let S be a geometrically ruled surface polarized by a very ample line bundle H with q=0, 1 or 2. If q=0, by [Ha, p. 380], we have that 1.2. The line bundle H [ao+bf] on Fe is very ample if a>0 and b>ae; so (Fe, H) E ImRA iff it is in ImA, iff a>0 and b>ae+(e-2). In particular this shows that the quadric surface of P3 and the rational cubic scroll of P4 are in ImRA. When q=1 and 2 we restrict ourselves to the case of scrolls.
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