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

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.