"Algebraic Geometry in simplest terms is the study of polynomial equations and the geometry of their solutions. It is an old subject with a rich classical history, while the modern theory is is built on a more technical but rich and beautiful foundation. The future looks very bright indeed with promising new directions for research being undertaken, many of which connect algebraic geometry to other areas of mathematics as well as to physics."  From  Algebraic Geometry, Department of Mathematics, University of Illinois at Urbana-Champaign

"Algebraic Geometry investigates the dynamic interplay between algebraic equations and the intricate geometry of their solution sets, known as algebraic varieties.  The field has seen tremendous advances in subtle internal questions concerning the classification of algebraic varieties, their topology, and the structure of their singularities, but deep and fundamental questions remain.  At the same time, algebraic geometry provides basic examples, tools, and insights for commutative algebra, differential geometry, complex analysis, representation theory, number theory, and mathematical physics.  It thus enjoys exciting, vibrant interaction with those fields that is full of surprises." From Graduate Study in Algebraic Geometry, Department of Mathematics, University of Illinois at Urbana-Champaign

Books on algebraic geometry can be found in the Mathemathics Library Stacks shelved under call number range 512.1 to 512.9.