Anmerkungen:
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Beschreibung:
This thesis attempts to investigate the Noether, Dedekind, and Kähler differents for a 0-dimensional scheme X in the projective n-space P^n_K over an arbitrary field K. In particular, we focus on studying the relations between the algebraic structure of these differents and geometric properties of the scheme X. In Chapter 1 we give an outline to the problems this thesis is concerned with, a brief literature review for each problem, and the main results regarding these problems. Chapter 2 contains background results that we will need in the subsequent chapters. We introduce the concept of maximal p_j-subschemes of a 0-dimensional scheme X and give some descriptions of them and their Hilbert functions. Furthermore, we generalize the notion of a separator of a subscheme of X of degree deg(X)-1 to a set of separators of a maximal p_j-subscheme of X. In Chapter 3 we explore the Noether, Dedekind, and Kähler differents for 0-dimensional schemes X. First we define these differents for X, and take a look at how to compute these differents and examine their relations. Then we give an answer to the question "What are the Hilbert functions of these differents?" in some cases. In Chapter 4 we use the differents to investigate the Cayley-Bacharach property of 0-dimensional schemes over an arbitrary field K. The principal results of this chapter are characterizations of CB-schemes and of arithmetically Gorenstein schemes in terms of their Dedekind differents and a criterion for a 0-dimensional smooth scheme to be a complete intersection. We also generalize some results such as Dedekind's formula and the characterization of the Cayley-Bacharach property by using Liaison theory. In addition, several propositions on the uniformities are proven. In Chapter 5 we are interested in studying the Noether, Dedekind, and Kähler differents for finite special classes of schemes and finding out some applications of these differents. First, we investigate these differents for reduced 0-dimensional almost complete intersections X in P^n_K ...