This course is usually taught in the spring. This course covers properties of affine and projective varieties defined over algebraically closed fields rational mappings, birational geometry and divisors, especially on curves and surfaces introduction to the language of schemes and Riemann-Roch theorem for curves. This course is usually taught in the fall. It covers Riemann surfaces, projective algebraic curves, differential forms, integration, divisors of poles and zeroes, linear systems, the Riemann-Roch theorem, Serre duality, and applications. The course consists of an introduction to algebraic geometry in dimension 1 over the field of complex numbers. It will lead naturally into Math 511, Algebraic Geometry, its planned sequel Algebraic Geometry II, and various topics courses in algebraic geometry.
This course is designed to be an entry level course for algebraic geometry. Math 510, Riemann Surfaces and Algebraic Curves
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It thus enjoys exciting, vibrant interaction with those fields that is full of surprises. 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. 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. Algebraic Geometry investigates the dynamic interplay between algebraic equations and the intricate geometry of their solution sets, known as algebraic varieties.
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