Algebraic Curves by William Fulton

An Introduction to Algebraic Geometry

A concise, accessible introduction to the geometry and function-field theory of algebraic curves that develops plane projective curves, singularities and normalization, divisors and linear systems, genus and the Riemann–Roch theorem, differentials and ramification, and maps between curves. Emphasizing concrete examples (plane curves, rational and elliptic cases), computations and exercises, the text connects the algebraic, geometric and complex-analytic viewpoints and prepares readers for further study of birational classification and the correspondence between curves and their function fields.