Students often find, in setting out to study algebraic geometry, that most of the serious textbooks on the subject require knowledge of ring theory, field theory, local rings, and transcendental field extensions, and even sheaf theory. Often the expected background goes well beyond college mathematics. This book, aimed at senior undergraduates and graduate students, grew out of Miyanishi's attempt to lead students to an understanding of algebraic surfaces while presenting the necessary background along the way. Originally published in Japanese in 1990, it presents a self-contained introduction to the fundamentals of algebraic geometry. This book begins with background on commutative algebras, sheaf theory, and related cohomology theory. The next part introduces schemes and algebraic varieties, the basic language of algebraic geometry. The last section brings readers to a point at which they can start to learn about the classification of algebraic surfaces.
Part I. Preliminaries: Theorem of Luroth Theory of sheaves and cohomologies Part II. Schemes and algebraic varieties: Affine schemes and algebraic varieties Schemes and algebraic varieties Projective schemes and projective algebraic varieties Nonsingular algebraic varieties Part III. Algebraic surfaces: Algebraic curves Intersection theory on algebraic surfaces Pencils of curves The Riemann-Roch Theorem for algebraic surfaces Minimal algebraic surfaces Ruled surfaces and rational surfaces Solutions to problems List of notation Bibliography Index.