Regularity and symbolic defect of points on rational normal curves

TL;DR

This paper investigates the algebraic properties of point ideals on rational normal curves, providing explicit formulas for regularity and analyzing the differences between symbolic and ordinary powers.

Contribution

It offers a new explicit formula for Castelnuovo-Mumford regularity of powers of ideals of points on rational normal curves and compares symbolic and ordinary powers.

Findings

Explicit formula for regularity of powers

Conditions for non-zero symbolic defect

Comparison of symbolic and ordinary powers

Abstract

In this paper we study ideals of points lying on rational normal curves defined in projective plane and projective $ 3 $-space. We give an explicit formula for the value of Castelnuovo-Mumford regularity for their ordinary powers. Moreover, we compare the $ m $-th symbolic and ordinary powers for such ideals in order to show whenever the $ m $-th symbolic defect is non-zero.