Fiber cones of rational normal scrolls are Cohen-Macaulay

TL;DR

This paper proves that fiber cones of rational normal scrolls are Cohen-Macaulay and computes their algebraic invariants, providing insights into their structure and Gorenstein properties.

Contribution

It establishes the Cohen-Macaulayness of fiber cones of rational normal scrolls and characterizes their Gorensteinness, including calculations of regularity, a-invariants, and reduction numbers.

Findings

Fiber cones of rational normal scrolls are Cohen-Macaulay.

Computed Castelnuovo-Mumford regularities and a-invariants.

Characterized Gorensteinness of the fiber cone.

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.