Symbolic Blowup algebras of monomial curves in ${\mathbb A}^3$ defined by arithmetic sequence

TL;DR

This paper proves that the symbolic blowup algebras of certain monomial curves in three-dimensional affine space, defined by an arithmetic sequence, are Gorenstein, providing a simplified proof of this property.

Contribution

It offers a straightforward proof that the symbolic blowup algebras of specific monomial curves are Gorenstein, expanding understanding of their algebraic structure.

Findings

Symbolic blowup algebras are Gorenstein for the given monomial curves.

Provides a simplified proof of the Gorenstein property.

Focuses on curves parameterized by an arithmetic sequence with gcd condition.

Abstract

In this paper, we consider monomial curves in ${\mathbb A}_k^3$ parameterized by $t \rightarrow (t^{2q +1}, t^{2q +1 + m}, t^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. The symbolic blowup algebras of these monomial curves is Gorenstein (\cite{goto-nis-shim}, \cite{goto-nis-shim-2}). We give a simple proof for the the Gorenstein property for the symbolic blowup algebras of these curves.