Reduced Gr\"obner Bases of Certain Toric Varieties; A New Short Proof

Abstract

Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of positive integers. Let C be a toric variety in the affine (n+1)-space, defined parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a minimal Gr\"obner basis for the toric ideal which is the defining ideal of C and give sufficient and necessary conditions for this basis to be the reduced Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a much simpler proof than that of \cite{Ayy}.