Upper bounds for the degrees of the equations defining locally Cohen- Macaulay schemes
Following recent activities in finding degree bounds for the defining equations of projective varieties, this paper deals with locally Cohen- Macaulay schemes. In algebraic terms, one considers an equidimensional homogeneous algebra \(A=k[X_ 0,...,X_ n]/{\mathfrak a}\) over a field k such that \(A_ P\) is a Cohen-Macaulay ring for all homogeneous prime ideals \(P\neq M:=(X_ 0,...,X_ n)A\). For such an algebra, the local cohomology modules \(H\) \(i_ M(A)\) are of finite lengths, \(i=0,...,d-1\), where \(d:=\dim (A)\). Put \(\lambda_ i=\min \{n;\quad M\quad nH\quad i_ M(A)=0\},\) \(I(A)=\sum^{d-1}_{i=0}\left( \begin{matrix} d-1\\ i\end{matrix} \right)length h_ AH\) \(i_ M(A),\quad and\Lambda (A)=\sum^{d- 1}_{i=0}\left( \begin{matrix} d-1\\ i\end{matrix} \right)\lambda_ i.\quad Then\) the authors show that the degree of the equations of a minimal basis of \({\mathfrak a}\) is bounded above by both numbers \(e-n+d+I(A)\) and \(e+\Lambda (A)\), where e denotes the multiplicity of A. These two bounds are mutually independent. If A is an integral domain with [\({\mathfrak a}]_ 1=0\) and k is algebraically closed of characteristic zero, then A is defined by forms of degree at most \(t+\Lambda (A)\), where t is the least integer such that \(e\leq t(n+1-d)\). This result is a generalization of a result of \textit{R. Treger} on arithmetically Cohen-Macaulay varieties [Duke Math. J. 48, 35-47 (1981; Zbl 0474.14030)] and settles in the affirmative a question raised by \textit{P. Maroscia} and \textit{W. Vogel} on arithmetically Buchsbaum varieties [Math. Ann. 269, 183-189 (1984; Zbl 0533.14022)]. For a Buchsbaum domain A, the reviewer and \textit{G. Valla} have obtained the better bound \(t+\min \{2;I(A)\}\) [Acta Math. Vietnam. (to appear)]. This bound has been improved to \(t+1\) by \textit{J. Stückrad} and \textit{W. Vogel} [Math. Ann. 276, 341-352 (1987; Zbl 0628.14037)]. Finally, the authors also give an upper bound for the defining equations of a Cohen-Macaulay scheme in terms of the lower bound.
degree bounds for the defining equations
scientific article; zbMATH DE number 4031866