On coherent sheaves of small length on the affine plane

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.