On some ideals defined by an arithmetic sequence

Abstract

This paper investigates properties of ideals in the affine and homogeneous projective coordinate rings of the plane, defined using arithmetic sequence
$$\{a_{\ell }=a+\ell d|\ell \ge 0\}$$
for some positive integers $a$ and $d$. Specifically, we study two types of ideals:
$I(a,d)$ is generated by $D(a,d)$ in $K[x,y]$ and $J(a,d)$ is generated by $E(a,d)$ in $K[x,y,z]$ where
$$D(a,d)=\{f_{\ell }=x^{a_{\ell }}-y^{a_{\ell +1}}|\ell \ge 0\}$$
and
$$E(a,d)=\{F_{\ell }=x^{a_{\ell }}{z}^d-y^{a_{\ell +1}}|\ell \ge 0\}.$$
This paper provides detailed answers to several problems, including finding finite generating sets, describing the zero locus of these ideals, and determining their Hilbert functions. Finally, the Castelnuovo-Mumford regularity and the minimal free resolution of the homogeneous coordinate ring and multi secant line are discussed.