Georgia Institute of Technology College of Sciences

We will give an overview of divisor theory on curves and give definitions of the Picard group and the Jacobian of a compact Riemann surface. We will use these notions to prove Plucker’s formula for the genus of a smooth projective curve. In addition, we will discuss the various ways of defining the Jacobian of a curve and why these definitions are equivalent. We will also give an extension of these notions to schemes, in which we define the Picard group of a scheme in terms of the group of invertible sheaves and in terms of sheaf cohomology.