by Kichoon Yang (Arkansas State Univ.)

This volume is an introduction to the theory of Compact Riemann Surfaces and algebraic curves. It gives a concise account of the elementary aspects of different viewpoints in curve theory. Foundational results on divisors and compact Riemann surfaces are also stated and proved.

• 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