Monomial Buchsbaum ideals in \({\mathbb{P}}^ r\)
Let N denote the nonnegative integers, let \(\{n_ 1,...,n_ r\}\subseteq N\) with \(g.c.d.(n_ 1,...,n_ r)=1. S_ 1=<n_ 1,...,n_ r>\) denotes the numerical semigroup \(\{v| v=\sum^{r}_{i=1}z_ in_ i,z_ i\in N\cup\{0\}\}, S_ 0=<n_ r- n_{r-1},...,n_ r-n_ 1,n_ r>.\) The set \(A(n_ r,S_ 1)=\{w(0)=0<w(1)<...<w(n_ r-1)\}\) of smallest elements of \(S_ 1\) in respective congruence classes mod n\({}_ r\) defines the Apery sequence of \(S_ 1\) mod n\({}_ r\). If \(v=\sum^{r}_{i=1}z_ in_ i\in S_ 1, \sum^{r}_{i=1}z_ i\) minimal, then \(\partial (v)=\sum^{r}_{i=1}z_ i\) is called the degree of v. \(P(n_ 1,...,n_ r)=P\) denotes the prime ideal in the polynomial ring \(k[x_ 0,...x_ r]=R\) over an arbitrary field k with generic zero \((t_ 0^{n_ r},t_ 0^{n_ r-n_ 1}t_ 1^{n_ 1},...,t_ 0^{n_ r-n_{r-1}}t_ 1^{n_{r-1}},t_ 1^{n_ r}).\) A local ring A is a Buchsbaum ring (B-ring) if for every parameter ideal (\b{X}) \(\ell (A/(\underline X)A)-e_ 0((\underline X),A)\) is independent of (\b{X}). Here ''\(\ell ''\) denotes length and \(''e_ 0''\) multplicity. \(P(n_ 1,...,n_ r)\) is a Buchsbaum ideal (B-ideal) if \((R/P)_{(x_ 0,...,x_ r)}\) is a B-ring. If P is a B-ideal then the invariant \(i(P)\geq 0\) is the difference between length and multiplicity. As a special case one obtains \(i(P)=0\) iff P is Cohen-Macaulay. - It is the purpose of this paper to characterize B-ideals \(P(n_ 1,...n_ r)\) explicitly by \(A(n_ r,S_ 0)\) and \(A(n_ r,S_ 1)\). This is done in terms of the degrees of the elements in the Apery sequences. From this the B-property is completely determined if either \(1\in S_ i\), \(i\in\{0,1\}\) or \(S_ 0=S_ 1=<n_ 1,n_ 2,...,n_ r>, n_ 1,...,n_{r-1}\) in arithmetic progression. Some other consequences derived are 1. i(P)\(=0\) if \(n_ r>\max_{i}\{\partial (s_ i')n_{r- 1}-s_ i'\}, 0\neq s_ i'\in A(n_{r-1},<n_ 1,...,n_{r-1}>).\) 2. A generalization of theorem 2.1 in ''Projections of one-dimensional Veronese varieties'', Math. Nachrichten 118, 47-67 (1984) by \textit{Ngô Viêt Trung}. 3. An extensive collection of examples, which includes the numerical semigroups \(S_ 1=<1,2,...,k,2k+1,3k+1>,\) \(k\geq 1\), with \(i(P(1,...,k,2k+1,3k+1))=1\) as a generalization of Macaulay's curve \(S_ 1=<1,3,4>\).
scientific article; zbMATH DE number 3871523