3 Picard Groups of Non-Singular Curves 7

We will develop a purely algebraic definition for the Picard group of an affine variety. We will then develop computational techniques for Picard groups of affine curves. If the curve is non-singular, the technique will require some geometry. However, in the singular case, we will find a purely algebraic tool. Along the way we will obtain an explicit description of the Picard group for every non-singular affine curve over C as well as affine curves with a single cusp or node singularity