On the defining equations and syzygies of arithmetically Cohen-Macaulay varieties in arbitrary characteristic
Let \(V \subseteq \mathbb{P}^n_K\) denote an arithmetically Cohen-Macaulay nondegenerate absolutely integral scheme of degree \(d\) and of codimension \(c\) over an arbitrary field \(K\). Upper bounds for the degree of defining equations have been described by several authors, e.g., \textit{R. Treger} [Duke Math. J. 48, 35-47 (1981; Zbl 0474.14030)] and \textit{Ngo Viet Trung} and \textit{G. Valla} [Math. Ann. 281, 209-218 (1992; Zbl 0616.14039)], provided \(K\) is a field of characteristic zero. In the paper under review the author proves that some of these results hold regardless of the characteristic and can be extended to degree bounds of higher syzygies. In particular it is shown that the defining ideal \(I(V)\) of \(V\) is generated by forms of degree \(\leq \lceil d/c \rceil\). In the case of \(c \geq 5\), \(d > (c + 1)^2\) there is a characterization when \(I(V)\) has a minimal generator of degree \(\lceil d/c \rceil\). In the case of characteristic zero the main technical tool is the general position lemma, which may fail if \(\text{char} (K) > 0\) [see \textit{J. Rathmann}, Math. Ann. 276, 565-579 (1987; Zbl 0595.14041)]. Avoiding this difficulty the basic idea is to apply a vanishing result of \textit{D. Eisenbud} and \textit{J. Koh} [Adv. Math. 90, No. 1, 47-76 (1991; Zbl 0754.13012)] bounding the length of the linear part of a resolution of the canonical module. This provides also degree bounds on the minimal generators of the higher syzygy modules.
bounds for the degree of defining equations
scientific article; zbMATH DE number 783927