Syllabus - Mathematics 137: Algebraic Geometry; Spring 2008

Mathematics 137: Algebraic Geometry Harvard University Spring 2008
Course homepage
Meets: MWF at 2pm in SC 216
Instructed by: John F. Duncan Harvard University Department of Mathematics One Oxford Street Cambridge, MA 02138 U.S.A.
Office: Phone: Web: Email:	SC 320 (617) 495 5377 math.harvard.edu/~jfd/ duncan (at) math
Course Assistant: Silas Richelson Email: sirichel (at) fas
Page last revised: January 30th, 2008

Course Outline
Fundamentals:	Algebraic curves Affine and projective algebraic varieties Complex manifolds and Riemann surfaces Stokes' Theorem on Riemann surfaces The Residue Theorem on Riemann surfaces The Poincaré-Hopf formula Mapping compact Riemann surfaces into complex projective spaces
Normalization Theorem:	Riemann surface structure on the smooth set of an irreducible algebraic curve The Weierstrass Preparation Theorem Local structure of an algebraic curve Existence and uniqueness of normalizations of an irreducible algebraic curve Bezout's Theorem The Riemann-Hurwitz Formula The genus formula
Riemann-Roch Theorem:	Brill-Noether reciprocity Principal parts and the Mittag-Leffler problem Dimension of the space of holomorphic forms on a compact Riemann surface Singular and de Rham cohomology groups of a compact Riemann surface The Hodge and de Rham Theorems The Riemann-Roch Theorem Applications of The Riemann-Roch Theorem to compact Riemann surfaces of low genus
Abel-Jacobi Theorem:	Jacobian variety of a compact Riemann surface Abel-Jacobi map Picard variety of a compact Riemann surface The Torelli Theorem The Abel-Jacobi Theorem Classification of compact Riemann surfaces of genus 1 Elliptic curves and elliptic integrals
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Mathematics 137: Algebraic Geometry