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Explore advanced topics in algebraic curves through this doctoral-level lecture from the 2026 Summer School at Instituto de Matemática Pura e Aplicada. Delve into Bézout's theorem covering projective geometry, resultants, and intersection multiplicities, then examine 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 Hessians, followed by Max Noether's fundamental theorem involving divisors and adjoint curves. Investigate cubic curves with their modular invariants and group structures, then progress to singularity resolution through rational functions, blowing-up techniques, and quadratic transformations. Conclude with the Riemann-Roch theorem encompassing differentials, the Riemann-Hurwitz formula, Weierstrass points, hyperelliptic curves, and curves of genus less than or equal to 3. Master these fundamental concepts in algebraic geometry essential for doctoral-level understanding of curve theory and its applications in pure mathematics.
