A new addition to the zoo of isolated symplectic singularities

TL;DR

This paper introduces a novel 4-dimensional isolated symplectic singularity with unique non-reduced tangent cone properties, and classifies all its $Q$-factorial terminalizations, expanding understanding of symplectic singularities.

Contribution

It presents a new isolated symplectic singularity with distinctive non-reduced tangent cone and provides a complete classification of its $Q$-factorial terminalizations.

Findings

Discovery of a new isolated symplectic singularity in $C^4/G_5$

Identification of the non-reduced projective tangent cone

Classification of all 12 $Q$-factorial terminalizations

Abstract

We give details of a new isolated symplectic singularity found in an affine chart in a crepant partial resolution of $\mathbb{C}^4/G_5$, which is 4-dimensional, isolated, and locally simply-connected. We distinguish the new singularity among all known such by the fact that the projective tangent cone at the singularity is non-reduced. We also find all 12 of its $\mathbb{Q}$-factorial terminalisations, in the process finding 24 for the quotient singularity $\mathbb{C}^4/G_5$.