Basic Algebraic Geometry II 2nd ed.

Author: I.R. Shafarevich
Publisher:
Publication Date: 1998-03-01
Features: Books 2 and 3 correspond to Chapters V-IX of the first edition. They study schemes and complex manifolds, two notions that generalize in different directions the varieties in projective space studied in Book 1. Introducing them leads as well to new results in the theory of projective varieties. For example, it is within the framework of the theory of schemes and abstract varieties that we find the natural proof of the adjunction formula for the genus of a curve, which we have already stated and applied in Chapter IV, 2.3. The theory of complex analytic manifolds leads to the study of the topology of projective varieties over the field of complex numbers. For some questions, it is only here that the natural and historical logic of the subject can be asserted; for example, differential forms were constructed in order to be integrated, a process which only makes sense for varieties over the (maior) complex fields. Changes from the First Edition