Projections of singular varieties and Castelnuovo-Mumford regularity
Let \(X\subset {\mathbb P}_{\mathbb C}^r\) be a smooth variety of dimension \(n\) and degree \(d\). The Castelnuovo-Mumford regularity is \(\text{reg}\,(X):=\{t\in{\mathbb Z}\mid H^i({\mathcal I}_X(t-i))=0 \;\text{for}\;i>0\}\). There is a well-known conjecture concerning \(\text{reg}\,(X)\), which says that \(\text{reg}\,(X)\leq d-r+n+1\). This conjecture is true for integral curves [\textit{L. Gruson, R. Lazarsfeld, C. Peskine}, Invent. Math. 72, 491--506 (1983; Zbl 0565.14014)], and for smooth surfaces [\textit{R. Lazarsfeld}, Duke Math. J. 55, 423--429 (1987; Zbl 0646.14005)]. In higher dimension the problem is still open. For singular varieties only the case of curves has been dealt with, in the already quoted [Zbl 0565.14014]. For higher dimensional singular varieties the previous bound is easily seen to be false; hence it is meaningful to look for a bound in terms of the singularities, and this is the aim of this paper. In particular, if the singular locus is zero-dimensional and \(X\) has a good projection with respect to \(X_{\text{reg}}\).
scientific article; zbMATH DE number 5015901