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Explore advanced topics in algebraic curves through this doctoral-level lecture focusing on Bézout's theorem, projective geometry, resultants, and intersection multiplicities. Delve into the mathematical foundations of singular points using Jacobi's criterion, curve branches, Weierstrass preparation theorem, Hensel's lemma, and Newton-Puiseux series. Master Plücker formulas through the study of Poncelet-Gergonne duality, polar curves, inflection points, and Hessian concepts. Examine Max Noether's fundamental theorem covering divisors and adjoint curves, then investigate cubic curves including 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 theoretical foundations essential for advanced research in algebraic geometry.
