Resurgence and Castelnuovo-Mumford regularity of certain monomial curves in ${\mathbb A}^3$

TL;DR

This paper investigates the algebraic properties of specific monomial curves in three-dimensional affine space, focusing on their resurgence, Waldschmidt constant, and Castelnuovo-Mumford regularity of symbolic powers, providing new insights into their algebraic structure.

Contribution

It computes the resurgence, Waldschmidt constant, and Castelnuovo-Mumford regularity for the defining ideals of certain monomial curves, advancing understanding of their algebraic invariants.

Findings

Resurgence of the ideal is explicitly calculated.

Waldschmidt constant of the ideal is determined.

Regularity of symbolic powers is established.

Abstract

Let ${\mathfrak p}$ be the defining ideal of the monomial curve ${\mathcal C}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space ${\mathbb A}_k^3$ parameterized by $(x^{2q +1}, x^{2q +1 + m}, x^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. In this paper we compute the resurgence of ${\mathfrak p}$, the Waldschmidt constant of ${\mathfrak p}$ and the Castelnuovo-Mumford regularity of the symbolic powers of ${\mathfrak p}$.