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A bound for the Castelnuovo-Mumford regularity of log canonical varieties. (English)
J. Pure Appl. Algebra 215, No. 9, 2180-2189 (2011).
The author studies the Castelnuovo-Mumford regularity of $S = R/I = k[x_0, \dots, x_n]/I$ where $I$ is a homogeneous ideal. Recently, as pointed out in the paper under review, there has been substantial interest in trying to bound the regularity of $S$ based upon the singularities of the Proj of $S$, see for example {\it M. Chardin} and {\it B. Ulrich}’s [“Liaison and Castelnuovo-Mumford regularity”, Am. J. Math. 124, No.~6, 1103‒1124 (2002; Zbl 1029.14016)] and also {\it T. de Fernex} and {\it L. Ein}’s [“A vanishing theorem for log canonical pairs”, Am. J. Math. 132, No. 5, 1205‒1221 (2010; Zbl 1205.14020)]. In this paper, the authors prove the following Theorem 1.1. With notation as above, suppose that $I = (f_1, \dots, f_t)$ where the $f_i$ are of degree $d_1 \geq d_2 \geq \dots \geq d_t \geq 1$ and also that $X = \mathrm{Proj} (R/I)$ is of codimension $r$ in $\mathbb{P}^n$. If $X$ is a local complete intersection with log canonical singularities and $\dim X \geq 1$, then $$\mathrm{reg} (R/I) \leq \frac{(\dim X + 2)!}{2}\left( \sum_{i = 1}^r d_i - r\right).$$ In the special case that $I$ is saturated, de Fernex and Ein obtained a somewhat better bound in the paper mentioned above. Chardin and Ulrich studied the question under the weaker hypothesis that $X$ has canonical singularities. The author points out that the main idea of this theorem is similar to the one given by Chardin and Ulrich mentioned above. However, in order to do this, the author first establishes certain fundamental results on (families of) varieties with local complete intersection log canonical singularities, which may be useful in other applications as well.
Reviewer: Karl Schwede (University Park)