2 WAYNE AITKEN 1. Preliminaries (1.1) Fractional Notation for Line Bundles. Throughout this work the terms line bundle and invertible sheaf will be used synonymously. For L and M line bundles on a scheme X we will use the following notation: L H f —- = L g M v where M v is the dual line bundle to M. M Given a pair of isomorphisms of line bundles, A: L -^ U and fi: M ^ M'', define ^ : M ~* M7 to be the isomorphism A ® (/^v)_1 where /i v is the dual isomorphism from (M') v to M v induced by \i. Let L and M be line bundles on a scheme X with sections, I of L and m of M, on an open subscheme £/. Also assume that m is non-zero at every point of U. We use the notation l def , v = Z (8)771 m where rav is the homomorphism from Mlf/ to Ox It/ sending m to 1. So is a m L section of -— on U. M (1.2) Meromorphic Sections. We will now establish the terminology and ba- sic properties concerning meromorphic sections of line bundles and other related matters. This terminology will be used throughout this work. We will always assume that the schemes we deal with are Noetherian and sep- arated. Such a scheme X has a finite number of associated points, where, by def- inition, x G X is an associated point if and only if the maximal ideal of the local ring Ox,x is a n associated prime ideal of Ox,x- In other words, x G X is an associ- ated point if and only if every element of the maximal ideal of the local ring Ox,x is a zero divisor in this ring. On an affine scheme, X = Spec A, a function / G A is a zero divisor of A if and only if the zero set of / contains at least one associated point of X. A flat morphism between two schemes sends associated points to associated points. The closure of any associated point is called an associated component. In par- ticular, each component of X is an associated component. Those associated com- ponents which are not components of X are called embedded components and their generic points are called embedded points. In many common situations there are no embedded points. For example, if a scheme is reduced it has no embedded points. Likewise, if a scheme is Cohen-Macaulay it has no embedded points. Conversely, if a scheme is one-dimensional and contains no embedded points, then it is Cohen- Macaulay. Zero-dimensional schemes are, of course, always Cohen-Macaulay. Let L be a line bundle on a scheme X. Intuitively, a meromorphic section I of L is a section of L which is defined and nowhere vanishing on an open set U which
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