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Explore advanced topics in algebraic curves through this doctoral-level lecture that delves into Bézout's theorem, covering projective geometry, resultants, and intersection multiplicities. Master the analysis of singular points using Jacobi's criterion, curve branches, Weierstrass preparation theorem, Hensel's lemma, and Newton-Puiseux series. Study Plücker formulas including Poncelet-Gergonne duality, polar curves, inflection points, and Hessian concepts. Examine Max Noether's fundamental theorem through divisors and adjoint curves, then investigate cubic curves focusing on modular invariants and group structures. Learn resolution of singularities techniques involving rational functions, blowing-up procedures, and quadratic transformations. Conclude with the Riemann-Roch theorem, exploring differentials, the Riemann-Hurwitz formula, Weierstrass points, hyperelliptic curves, and curves of genus less than or equal to 3. This comprehensive mathematical treatment draws from classical references including works by Arbarello, Coolidge, Fulton, and Walker, providing rigorous foundations in algebraic geometry essential for doctoral-level understanding of curve theory.
