Contents:

• Topological Preliminaries — Singular Homology and Relative Homology
• Cellular Homology
• De Rham Cohomology
• Commutative Algebra — An Introduction — Closed Ideals and Varieties
• Coordinate Rings
• Dimension Theory
• Intersection Numbers
• Singular Plane Curves — The Classical Plücker Formulae
• Divisors on a Compact Complex Manifold — Divisors and Holomorphic Line Bundles
• Linear Systems on a Compact Riemann Surface and Holomorphic Maps
• Compact Riemann Surfaces — The Jacobian Variety and Abel's Theorem
• The Riemann-Roch Theorem and the Canonical Embedding
• Hyperelliptic Riemann Surfaces and the Weierstrass Points
• Geometry of Projective Curves — The Complex Flag Manifold
• Metric Geometry of Projective Curves
• Plücker Formulae for Projective Algebraic Curves
• Harmonic Maps from a Compact Riemann Surface
• A Brief Look at Algebraic Surfaces — The Intersection Form
• Blow-Ups and Rational Maps
• The Kodaira Dimension of an Algebraic Surface