Divisors and Differential Forms

Abstract

In algebraic geometry, divisors are codimension 1 subvarieties, whose significance is to serve as the locus of zeros and poles of rational functions. The chapter discusses Weil divisors and locally principal divisors (or Cartier divisors), and linear equivalence between them. A divisor gives rise to a linear system, that one visualises as a family of codimension 1 subvarieties parametrised by a vector space. Algebraic groups are varieties with a group law given by regular maps. This includes linear algebraic groups (or matrix groups), but also elliptic curves and their higher dimensional generalisations, the Abelian varieties. Differential forms are dual to tangent fields, and have many applications. A principal aim is to discuss the canonical class of a variety. The chapter includes a complete discussion and proof of the Riemann–Roch theorem for curves.