Chapter 1 The Intersection Pairing for One-Dimensional Schemes In SGA4 XVIII and [D], Deligne defines an intersection pairing which assigns to any pair of line bundles on a smooth projective family of curves a line bundle on the base scheme. In this context, he interpreted the Grothendieck-Riemann-Roch theorem not as an equality between certain elements of the Picard group of the base, but as a canonical isomorphism between the corresponding line bundles on the base. In Part I of this work I give a development of Deligne's intersection pairing and the associated Riemann-Roch isomorphism. I do so not only for smooth families of curves, but for a larger class of families of curves. I do not assume that the generic fiber is non-singular or even irreducible. My development of the intersection pairing was inspired by the approach of L. Moret-Bailly in [MB] where he developed the pairing for certain types of generically smooth families (of total dimension 2). His method was to define the intersection pairing in terms of the determinant of cohomology, and then to prove that this pairing has all the desired properties. The reader should also be aware of generalizations of Deligne's work by R. Elkik [El] and J. Prank [Fr]. I have divided the development of the intersection pairing into two stages: this chapter is concerned with the "absolute" case, i.e., a projective one-dimensional scheme over a field the next chapter is concerned with the "relative" case, i.e., a flat, projective family of one-dimensional schemes over an arbitrary integral base scheme. This seems to be a good division of labor: in the first chapter we don't have to worry about the more technical issues such as flatness criteria or relative cohomology and their determinants. The basic ideas can be presented more clearly. Then in the second chapter we extend the constructions to the case when the base scheme is any integral scheme. 1
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