Castelnuovo-Mumford regularity and reduction number of smooth monomial curves

TL;DR

This paper investigates the relationship between Castelnuovo-Mumford regularity and reduction number in smooth monomial projective curves, showing they coincide in some cases and establishing a stronger inequality than the Eisenbud-Goto conjecture.

Contribution

It provides a comparison of regularity and reduction number for smooth monomial curves, including an example of their difference and a proof of a stronger inequality.

Findings

Regularity and reduction number coincide for certain monomial curves

An example where these two invariants differ is presented

A stronger inequality than the Eisenbud-Goto conjecture is proved for smooth monomial curves

Abstract

We compare, for smooth monomial projective curves, the Castel- nuovo-Mumford regularity and the reduction number; we present an example where these two numbers differ. However, we show they coin- cide for a certain class of monomial curves. Furthermore, for smooth monomial curves we prove an inequality which is stronger than the one from the Eisenbud-Goto conjecture.