Introduces basic notions in algebraic geometry, as applied to the study of algebraic curves. Topics include affine varieties and coordinate rings, projective space, projective varieties and homogeneous coordinates, regular and rational maps, dimension, singularities, and smoothness. Illustrates these ideas with examples of algebraic varieties including conics and plane cubics, quadric surfaces and their lines, and the Segre and Veronese embeddings. Concludes with a detailed discussion of algebraic curves including differentials, genus, linear systems and divisors, maps to projective space, and the Riemann-Roch theorem.