Tight closure and plus closure for cones over elliptic curves

TL;DR

This paper characterizes the tight closure of graded primary ideals in coordinate rings over elliptic curves using numerical conditions, and shows equivalence with plus closure in positive characteristic.

Contribution

It provides a numerical characterization of tight closure for ideals over elliptic curves and establishes their equivalence with plus closure in positive characteristic.

Findings

Tight closure characterized by numerical conditions.

Equivalence of tight closure and plus closure in positive characteristic.

Applicable to coordinate rings over elliptic curves.

Abstract

We characterize the tight closure of graded primary ideals in a homogeneous coordinate ring over an elliptic curve by numerical conditions and we show that it is in positive characteristic the same as the plus closure.