Castelnuovo-Mumford regularity and reduction number of smooth monomial curves in \(\mathbb P^5\) (Q2845718)

Let \(K\) be a field and \(A=\{0, a_1, \dots , a_d, \alpha \}\) a subset of \(\mathbb N\), where \(d\in \mathbb N^+\) and \(0 < a_1 <\dots < a_d < \alpha\) is a sequence of relatively prime integers. The ring \(K[A]=K[t^{\alpha}_0, t^{\alpha-a_1}_0 t_1^{a_1},\dots, t^{\alpha-a_d}_0 t_1^{a_d}, t_1^{\alpha}] \subseteq K[t_0, t_1]\) is uniquely determined by \(A\) and isomorphic to the coordinate ring of a projective monomial curve of degree \(\alpha\) in \(\mathbb P^{d+1}\). In view of [\textit{L. T. Hoa} and \textit{J. Stuckrad}, J. Algebra 259, No. 1, 127--146 (2003; Zbl 1080.13504)], the property \(\mathrm{reg} K[A] = r(K[A])\) is very interesting, since the Eisenbud-Goto conjecture [\textit{D. Eisenbud} and \textit{S. Goto}, J. Algebra 88, No. 1, 89--133 (1984; Zbl 0531.13015)] holds in this situation. But even for smooth projective monomial curves, this equality does not hold in general. In the paper under review, the author computes explicitly the Castelnuovo-Mumford regularity \(\mathrm{reg} K[A]\) and the reduction number \(r(K[A])\) of coordinate rings of smooth projective monomial curves in \(\mathbb P^{5}\), and shows that these numbers are equal.NEWLINENEWLINELet \(A=\{0, 1, b, c, \alpha-1, \alpha \}\) with \(A\subset \mathbb N\) and \(1<b<c<\alpha-1\). The author proves that \(\mathrm{reg} K[A] = r(K[A])=\max\{\lfloor{\frac{c}{b}}\rfloor+b-2, \lfloor{\frac{\alpha-b}{\alpha-c}}\rfloor+\alpha-c-2\}\).