On projective Cohen-Macaulayness of a Del Pezzo surface embedded by a complete linear system

Abstract

Let k be an algebraically closed field. We understand by a Del Pezzo surface X over k a non-singular rational surface on which the anti-canonical sheaf ―wx is ample. We call the self-intersectionnumber d=a)x of wx the degree of X, then we get that 1^J^E9. It is well known that X is isomorphic to PlxP which has degree 8, or an image of P2 under a monoidal transformation with center the union of r―9―d points which satisfiesthe following conditions: (a) no three of them lie on a line; (b) no six of them lie on a conic; (c) there are no cubics which pass through seven of them and have a double point at the eighth point. Conversely any surface described above is a Del Pezzo surface of the corresponding degree ([8,in, Theorem 1]). It is also well known that ―o)x is very ample when d^3 and that ample divisors on X of degree 3, which is a cubic surface, are very ample too. In this paper we will get that ample divisorson X of degree d^3 are very ample and that ample divisors on X of degree 2 [resp. 1] other than ―o>x [resp. ―o)x nor ―2^x] ore very ample. A closed subscheme V in PN is said to be projectively Cohen-Macaulay if its affine cone is Cohen-Macaulay. It is equivalent to that H1(PN,Jv(m))=0 for every meZ and H\V, Ov(m))-0 for every meZ and 0<?<dim V. In this paper, we will get that (p\o\(X) is projectively Cohen-Macaulay for a very ample divisor D on X, where <p[D[ is the morphism from X to JPdiml£ defined by the complete linear system ID I of D. We also study the homogeneous ideal I(D)=Ker \sr(D) ― c I\nD) defining <pim(X). These results will be stated and proved in §3 and §5. The fourth section will be devoted to a study on ―ncox of a Del Pezzo surface X of degree 1 or 2. In §1 we will compute the dimension h\D) of the z-th cohomology group H\X,Ox{D)) of the invertible sheaf OxiP) corresponding to a divisor D.