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Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [ABSTRACT FROM AUTHOR]

Let A be a noetherian local k-algebra with residue field k and associated graded ring G(A). Let  be the conductor of A in its normalization [image omitted], which we assume to be regular and finite over A. Assume also that Proj(G(A)) is reduced and [image omitted]. Let r be the embedding dimension of A and [image omitted] be the Hilbert function of [image omitted]. We show that if the Hilbert function of Proj(G(A)) is equal to [image omitted], then G(A) is reduced. This allows us to compute, in the general case, the conductor of the local ring of singular isolated points on surfaces whose tangent cone is multiplanar; that is, it consists—as a set—of planes. We explicitly construct a class of such surfaces in the rational case. [ABSTRACT FROM AUTHOR]