On the regularities of arithmetically Buchsbaum curves (Q1374877)

Let \(k\) be an algebraically closed field of characteristic zero and \(\mathbb{P}^r\) the projective \(r\)-space over \(k\); let \(X\subset \mathbb{P}^r\) be a closed subscheme. Recall that \(X\) is said arithmetically Buchsbaum if its homogeneous coordinate ring is Buchsbaum. Buchsbaum rings have been intensely studied in recent years, as well as the notion of Castelnuovo-Mumford regularity \(\text{reg} X\) [see for example: \textit{J. Stückrad} and \textit{W. Vogel}, ``Buchsbaum rings and applications. An interaction between algebra, geometry, and topology'' (1986; Zbl 0606.13017) and Math. Ann. 276, 341-352 (1987; Zbl 0628.14037)]. In this paper the author, starting from some results of Stückrad and Vogel (see the above mentioned papers), studies the Castelnuovo-Mumford regularity of an arithmetically Buchsbaum curve; in particular he proves that a nondegenerate integral arithmetically Buchsbaum curve \(C \subset \mathbb{P}^r\) is contained in a surface with minimal degree, when \(\text{reg} C= \lceil (\deg C-1)/(r-1) \rceil +1\) and \(\deg C\gg r\) (where, for a rational number \(x\), one sets \(\lceil x\rceil: =\min \{n\in\mathbb{Z} \mid n\geq x\})\); moreover the author gives a criterion for curves contained in a smooth surface scroll to be arithmetically Buchsbaum.