The ring of differential operators on a monomial curve is a Hopf algebroid

TL;DR

This paper investigates the structure of differential operators on certain algebraic curves, showing they form cocommutative Hopf algebroids, and characterizes when they admit an antipode based on the symmetry of the underlying semigroup.

Contribution

It demonstrates that the rings of differential operators on cuspidal monomial curves are cocommutative Hopf algebroids and identifies conditions for the existence of an antipode.

Findings

Rings of differential operators are cocommutative and conilpotent Hopf algebroids.

Symmetric semigroups lead to Gorenstein curves with full Hopf algebroid structures.

Provides a general descent result for Hopf algebroid structures.

Abstract

This article considers cuspidal curves whose coordinate rings are numerical semigroup algebras. Using a general result about descent of Hopf algebroid structures, their rings of differential operators are shown to be cocommutative and conilpotent left Hopf algebroids. If the semigroups are symmetric so that the curves are Gorenstein, they are full Hopf algebroids (admit an antipode).