Fiber cones of rational normal scrolls are Cohen-Macaulay

Abstract

In this short paper, we show that the fiber cones of rational normal scrolls are Cohen-Macaulay. As an application, we compute their Castelnuovo-Mumford regularities and $\mathbf{a}$-invariants, as well as the reduction number of the defining ideals of the rational normal scrolls. We also characterize the Gorensteinness of the fiber cone.

arXiv commentFollowing the suggestion of experts, we split the original paper in arXiv:2009.03484v1 into two parts. And this is the first part, showing up as arXiv:2009.03484v2, with title changed. The second part regarding the regularity of the secant variety appears as arXiv:2107.04168. A result regarding Gorensteinness is new. Comments are welcome!