Algebraic Curves

Bezout’s theorem: projective geometry, resultant, intersection multiplicities. Singular points: Jacobian criterion, branches of curves, Weierstrass’ preparation theorem, Hensel’s lemma, the Newton-Puiseux series.
Plucker formulae: Poncelet-Gergonne duality, polar curves, inflection points, the hessian. Max Noether’s fundamental theorem: Divisors, adjoint curves.
Cubic curves: modular invariant, group structure.
Resolution of Singularities: rational functions, blowing-up, quadratic transformations. Riemann-Roch theorem: Differentials, the Riemann-Hurwitz formula, Weierstrass points, hyperelliptical curves, curves of small genus or equal to 3. 

References:
ARBARELLO, E. – Geometry of Algebraic Curves, Vol. I, New York, Springer Verlag, 1985.
COOLIDGE, J.L. – A Treatise on Algebraic Plane Curves, Dover, 1959.
FULTON, W. – Algebraic Curves. New York, Benjamin, 1969.
WALKER, R. J. – Algebraic Curves. New York, Dover, 1950.

* Standard program. The teacher has the autonomy to make any changes