Asymptotic linear bounds for the Castelnuovo-Mumford regularity (Q2781384)

Continuing previous works by \textit{S. D. Cutkosky}, \textit{J. Herzog} and \textit{Nao Viet Trung} [Compos. Math. 118, No. 3, 243-261 (1999; Zbl 0974.13015)], and by \textit{S. D. Cutkosky}, \textit{L. Ein} and \textit{R. Lazarsfeld}, Math. Ann. 321, No. 2, 213-234 (2001; Zbl 1029.14022)], in the present paper the authors prove asymptotic linear bounds for Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not to be Noetherian.NEWLINENEWLINENEWLINEIn section 1 the authors prove that \(\text{reg} (G(R/I^n))\) is bounded by a linear function of \(n\) if \(\dim(R/I)\leq 1\), where \(R\) is a local ring, \(I\) an ideal of \(R\) and \(G(R/I^n)\) denotes the associated graded ring. In section 2, they prove an analogous boundedness result for \(\text{reg}(I^{(n)})\), where \(I\) is an ideal in a polynomial ring \(R\) and \(I^{(n)}\) denotes the \(n\)-th generalized symbolic power of \(I\), which is defined by taking those primary components of \(I^n\) which contain a given ideal. This result is proved under one of the following hypotheses: \(\dim(R/I) \leq 2\), or the singular locus of \(R/I\) has dimension \(\leq 1\), or \(I\) is a monomial ideal. Turning to the initial ideal in\((I)\) of a homogeneous ideal \(I\), in section 3, the authors prove that if \(\dim(R/I)\leq 1\), then \(\text{reg(in}(I^n))\leq\text{reg(in}(I))n\) for any \(n\geq 0\), and that \(\lim_{n\to+ \infty} {\text{reg(in}(I^n)) \over n}\) exists.