On varieties of almost minimal degree in small codimension
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The aim of the present exposition is to investigate varieties of almost minimal degree and of low codimension, in particular their Betti diagrams. Here minimal degree is defined as $°X = \codim X + 2.$ We describe the structure of the minimal free resolution of a variety $X$ of almost minimal degree of $\codim X \leq 4$ by listing the possible Betti diagrams.
The most surprising fact is, that the non-arithmetically Cohen-Macaulay case of varieties of almost minimal degree can occur only in small dimensions (cf. Section 2 for the precise statements).
Our main technical tool is a result shown by the authors (cf. \cite{BS}), which says that besides of an exceptional case, (that is the generic projection of the Veronese surface in $\mathbb P^5_K$) any non-arithmetically normal (and in particular non-arithmetically Cohen-Macaulay) variety of almost minimal degree $X \subset \mathbb P^r_K$ (which is not a cone) is contained in a variety of minimal degree $Y \subset \mathbb P^r_K$ such that $\codim(X,Y) = 1.
12 pages. J. Algebra, to appear
12 pages. J. Algebra, to appear