by Daniel Bump (Stanford University)

Daniel Bump received his PhD in 1982 under Professor Walter Baily at the University of Chicago. His research is in automorphic forms and representation theory. After teaching for two years at the University of Texas at Austin, and spending a year at the Institute for Advanced Study, he came to Stanford University, where he is now a Professor of Mathematics.
 

This is a graduate-level text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebra which is used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth. The book culminates with a selection of topics from the theory of algebraic curves, including the Riemann–Roch theorem, elliptic curves, the zeta function of a curve over a finite field, and the Riemann hypothesis for elliptic curves.

• Affine Algebraic Sets and Varieties
• The Extension Theorem
• Maps of Affine Varieties
• Dimensions and Products
• Local Algebra
• Properties of Affine Varieties
• Varieties
• Complete Nonsingular Curves
• Ramification
• Completions
• Differentials and Residues
• The Riemann–Roch Theorem
• Elliptic Curves and Abelian Varieties
• The Zeta Function of a Curve