Projective Geometry

I. Algebraic Varieties in a Projective Space.- I. Fundamental Concepts.- § 1. Plane Algebraic Curves.- 1. Rational Curves.- 2. Connections with the Theory of Fields.- 3. Birational Isomorphism of Curves.- Exercises.- §2. Closed Subsets of Affine Spaces.- 1. Definition of Closed Subset.- 2. Regular Functions on a Closed Set.- 3. Regular Mappings.- Exercises.- § 3. Rational Functions.- 1. Irreducible Sets.- 2. Rational Functions.- 3. Rational Mappings.- Exercises.- § 4. Quasiprojective Varieties.- 1. Closed Subsets of a Projective Space.- 2. Regular Functions.- 3. Rational Functions.

Authors: Diverio, Simone, Rousseau, Erwan
Offers an updated, fresh view of hyperbolicity-type results about projective hypersurfaces
Presents new and classical concepts, like the basics of Kobayashi hyperbolicity and algebraic hyperbolicity
Is as self-contained as possible, and uses straightforward language

The main topics discussed at the D. M. V. Seminar were the connectedness theorems of Fulton and Hansen, linear normality and subvarieties of small codimension in projective spaces. They are closely related; thus the connectedness theorem can be used to prove the inequality-part of Hartshorne's conjecture on linear normality, whereas Deligne's generalisation of the connectedness theorem leads to a refinement of Barth's results on the topology of varieties with small codimension in a projective space. The material concerning the connectedness theorem itself (including the highly surprising application to tamely ramified coverings of the projective plane) can be found in the paper by Fulton and the first author: W. Fulton, R. Lazarsfeld, Connectivity and its applications in algebraic geometry, Lecture Notes in Math. 862, p. 26-92 (Springer 1981).