Global Matric Massey Products and the Compactified Jacobian of the E6-Singularity

Abstract

In this paper we compute the compactified Jacobian of the singularity E6. By G. M. Greuel and H. Knörrer (1985, Math. Ann.270, 417–425) this singularity has only a finite number of isomorphism classes of rank 1 torsionfree modules. Using the theory of Matric Massey products, in an earlier work we computed the local formal moduli with its local versal family for each local module, and we studied the degeneracy of each local module. Here we prove a result showing how the local theory connects to the global theory; i.e., we prove that the morphism from the local formal moduli of a local module to the local ring at the point corresponding to the module on the compactified Jacobian is a smooth morphism. In the case where M=E6, i.e., the normalization, this morphism is an isomorphism. Thus the degeneracy (stratification) diagram for the compactified Jacobian can be found from the degeneracy of the normalization in the local case.