On coherent sheaves of small length on the affine plane

TL;DR

This paper classifies small-length coherent modules supported at the origin on the affine plane and relates this classification to the motivic class of the moduli stack, revealing a connection with skew Ferrers diagrams.

Contribution

It provides a complete classification of coherent modules of length at most 4 supported at a point on the affine plane and links this to motivic invariants via torus fixed points.

Findings

Classification of modules of length ≤4 supported at the origin

Identification of fixed points with skew Ferrers diagrams

Connection between module classification and motivic classes

Abstract

We classify coherent modules on $k[x,y]$ of length at most $4$ and supported at the origin. We compare our calculation with the motivic class of the moduli stack parametrizing such modules, extracted from the Feit-Fine formula. We observe that the natural torus action on this stack has finitely many fixed points, corresponding to connected skew Ferrers diagrams.