Ulrich ideals on rational triple points of dimension two

TL;DR

This paper classifies Ulrich ideals on two-dimensional rational triple point singularities and quotient singularities, showing the canonical trace ideal is Ulrich and identifying unique Ulrich ideals in certain cases.

Contribution

It proves the canonical trace ideal is Ulrich for rational triple points and classifies all Ulrich ideals in these and quotient singularities.

Findings

Canonical trace ideal trace(omega_A) is Ulrich for rational triple points

All Ulrich ideals are classified for rational triple points and quotient singularities

Unique Ulrich ideal m when multiplicity e ≥ 4 in quotient singularities

Abstract

In this paper, we prove the canonical trace ideal trace(omega_A) is an Ulrich ideal for any two-dimensional rational triple point A. Using this, we classify all Ulrich ideals on rational triple points. Moreover, we show that if (A, m) is a two-dimensional quotient singularity with the multiplicity e \ge 4 then m is the unique Ulrich ideal of A. As a result, we can classify all Ulrich ideals of A if A is either a rational triple point or a quotient singularity.