Igor R. Shafarevich 
Basic Algebraic Geometry 1 [PDF ebook] 
Varieties in Projective Space

Shafarevich’s Basic Algebraic Geometry has been a classic and universally used introduction to the subject since its first appearance over 40 years ago. As the translator writes in a prefatory note, «For all [advanced undergraduate and beginning graduate] students, and for the many specialists in other branches of math who need a liberal education in algebraic geometry, Shafarevich’s book is a must.» The third edition, in addition to some minor corrections, now offers a new treatment of the Riemann—Roch theorem for curves, including a proof from first principles.Shafarevich’s book is an attractive and accessible introduction to algebraic geometry, suitable for beginning students and nonspecialists, and the new edition is set to remain a popular introduction to the field.