  Deformation quantization on varieties with singularities offers perspectives
that are not found on manifolds. Essential deformations are classified by the
Harrison component of Hochschild cohomology, that vanishes on smooth manifolds
and reflects information about singularities. The Harrison 2-cochains are
symmetric and are interpreted in terms of abelian $*$-products. This paper
begins a study of abelian quantization on plane curves over $\Crm$, being
algebraic varieties of the form R2/I where I is a polynomial in two variables;
that is, abelian deformations of the coordinate algebra C[x,y]/(I).
  To understand the connection between the singularities of a variety and
cohomology we determine the algebraic Hochschild (co-)homology and its
Barr-Gerstenhaber-Schack decomposition. Homology is the same for all plane
curves C[x,y]/(I), but the cohomology depends on the local algebra of the
singularity of I at the origin.
