SYLLABUS MATH 531: Algebraic Geometry

  • Algebraic Geometry is the study of solution sets of systems of polynomial equations in several variables with complex coefficients (or more generally with coefficients in an algebraically closed field). The solution sets are viewed as geometric/topological objects and geometric/topological ideas play a central role in the subject. Today Algebraic Geometry is one of the central subjects of pure mathematics, at the confluence of number theory, differential geometry, and theoretical physics. The prerequisites are minimal: only a course in graduate (or even undergraduate) abstract algebra is needed. I will follow Shafarevich's book "Basic Algebraic Geometry, Vol 1" and I will complement it with some material from Serre's "Algebraic groups and class fields". None of these books is required; they are merely recommended. I will cover: affine and projective varieties, normalization, projections, blow up, divisors, differentials, Weil's proof of Riemann-Roch for curves.
  • Grading will be based upon attendance plus a set of homework problems to be assigned in class.
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