Symbolic blowup algebras and invariants associated to certain monomial curves in ${\mathbb P}^3$

TL;DR

This paper explicitly describes the symbolic powers of certain monomial curves in projective 3-space, proves their symbolic blowup algebra is Noetherian and Gorenstein, and computes key invariants like resurgence, Waldschmidt constant, and Castelnuovo-Mumford regularity.

Contribution

It provides explicit formulas for symbolic powers, invariants, and properties of a class of monomial curves in ${f P}^3$, including their symbolic blowup algebra being Noetherian and Gorenstein.

Findings

Symbolic powers are explicitly described for the curves.

Symbolic blowup algebra is shown to be Noetherian and Gorenstein.

Formulas for resurgence, Waldschmidt constant, and Castelnuovo-Mumford regularity are derived.

Abstract

In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$ in ${\mathbb P}^3$ parametrized by $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd(d,m)=1$. The defining ideal of these curves is a set-theoretic complete intersection. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal ${\mathfrak p}:={\mathfrak p}_{ { \mathcal C}(q,m) }$ defining the curve ${\mathcal C}(q,m)$ is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers ${\mathfrak p}^{(n)}$ for all $n \geq 1$.