Introduction to Special Divisors

Abstract

In this chapter we shall begin our study of linear series on smooth curves. Our central question is: What are the limitations on the dimension r(D) of a complete linear series | D | ? The first result in this direction is the classical Clifford theorem. After discussing this, in a somewhat different vein, we shall prove Castelnuovo’s bound on the genus of a curve in projective r-space. This will lead to Max Noether’s theorem on the projective normality of canonical curves, to a detailed study of extremal curves in r-space and to a brief presentation of Petri’s theory.