On minimal free resolutions of projective varieties of degree = codimension + 2
The paper is a supplement to the joint work of the author, \textit{J. Stückrad} and \textit{W. Vogel} [see J. Pure Appl. Algebra 71, No. 2/3, 203-231 (1991; Zbl 0749.14032)] on the structure of non-degenerate, reduced and irreducible varieties \(X \subset \mathbb{P}^ n_ k\), \(k\) an algebraically closed field, of `almost minimal' degree. In general one has \(\deg X\geq\) codim \(X+1\), while `almost minimal' means \(\deg X = \text{codim} X + 2\). In the present paper there is a discussion of the minimal free resolution of \(X\) when it is of `almost minimal' degree. In the case \(X\) is arithmetically Cohen-Macaulay the coordinate ring is an extremal Gorenstein ring of size 2 [see the reviewer, J. Algebra 64, 93-101 (1980; Zbl 0449.13008)]. So it has a linear resolution and there are explicit formulas for the Betti numbers. In the general case the author uses the 2-regularity of \(X\) in order to show that the minimal generators of the syzygy modules are concentrated in at most two degrees. This implies several results about the Betti numbers of \(X\), in particular in large homological degrees. These statements become more explicit in small dimensions (for curves) or under additional assumptions.
minimal generators of the syzygy modules
scientific article; zbMATH DE number 427766