# 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.

## Key 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 ${\mathfrak p}$.

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Source: https://tomesphere.com/paper/1904.05797