Algebraic Geometry is the study of geometric figures cut out by polynomial
equations. Such figures can be studied using a mixture of algebraic and
geometric methods. A central theme of the subject is that problems can be
translated back and forth between algebraic and geometric viewpoints. The main
goal of the course will be to develop the beautiful theory of algebraic
varieties over an algebraically closed field. Topics will include affine,
projective, and complete varieties, products of varieties, dimension, rational
maps, non-singular varieties, divisors and line bundles, coherent sheaves,
algebraic curves, and the Riemann-Roch theorem. At some point we will introduce
schemes, which makes it possible to assign geometry to any commutative ring.
Hartshorne: Algebraic Geometry / Vakil: The Rising Sea (the books will not be followed closely)
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