Algebraic Curves and Riemann Surfaces

• Grant number: 08/04470-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: August 01, 2008
• End date: December 31, 2009
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Claudio Gorodski
• Grantee: Lucas Kaufmann Sacchetto
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 07/03192-7 - Submanifold geometry and Morse theory in finite and infinite dimensions, AP.TEM

Abstract

1. Riemann Surfaces: basic definitions, examples, projective curves.2. Functions and maps: holomorphic e meromorphic functions, Laurent series, holomorphic and meromorphic maps between Riemann surfaces and global properties.3. More Example of Riemann surfaces: conics, hyperellitic surfaces, genus, group actions, monodromy, basic projective geometry.4. Integration on Riemann surfaces: differential forms, Poincaré and Dolbeault lemmas, Stoke's and residue theorems5. Divisors and meromorphic functions: basic definitions, linear equivalence of divisors, Bézout's theorem, Divisors and Maps to Projective Space.6. Riemann-Roch Theorem: algebraic curves, Mittag-Leffler problems, Serre duality.7. Apllications of Riemann-Roch: Clifford's Theorem, the canonical map, the geometric form of Riemann-Roch, degree of projective curves, inflection points and Weierstrass points.8. Abel's Theorem: homology, periods an Jacobian, traces, group law on cubics.