Liaison Theory in Arbitrary Codimension

Abstract

In this chapter we introduce liaison (or linkage) theory for subschemes of arbitrary codimension in projective space. The largest part of the theory has been developed around the notion of “linking” by complete intersections, but another large part has also been developed around “linking” by Gorenstein ideals. From now on we will refer to these as “complete intersection liaison” and “Gorenstein liaison” respectively, and we will abbreviate them by CI-liaison and G-liaison. The most complete picture (in terms of theory and also applications) has been in the case of codimension two, where we saw in Example 4.1.11 that the Gorenstein ideals are precisely the complete intersections. Even in higher codimension, where much less is known, there is more work in the complete intersection setting than in the more general Gorenstein setting. For this reason, the first version of this book treated only the case of complete intersections. We will still treat the codimension two case in the next chapter.