Erratum to "The Homogeneous Coordinate Ring of a Toric Variety", along with the original paper

TL;DR

This paper provides an erratum correcting a proof error in a foundational work on the Cox ring of toric varieties, reaffirming the original results and clarifying the structure of automorphisms.

Contribution

The paper corrects a key proof in the original 1995 paper on the Cox ring of toric varieties, ensuring the validity of the original results and clarifying the automorphism group structure.

Findings

Corrected proof of Proposition 4.3

Clarification of the automorphism group of a toric variety

Validation of the original paper's main results

Abstract

This submission consists of two papers: 1) an erratum that corrects an error in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of a Toric Variety", and 2) the original (unchanged) version of the paper, published in 1995. The original paper introduced the homogeneous coordinate ring of a toric variety (now called the total coordinate ring or Cox ring) and gave a quotient construction. The paper also studied sheaves on a toric variety, and in Section 4 described its automorphism group. The error in the proof of Proposition 4.3 resulted from the faulty assumption that a certain set of graded endomorphisms forms a ring; rather, it is a monoid under composition. The erratum notes this error and gives a correct proof of the proposition.