# 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.

## Key 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 $n \geq 1$.

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Source: https://tomesphere.com/paper/1904.00556