Algebraic curves
Bezout’s Theorem: Projective geometry, resultants, intersection multiplicities. Singular points: Jacobi’s criterion, branches of curves, Weierstrass preparation theorem, Hensel’s lemma, Newton-Puiseux series. Plücker formulas: Poncelet-Gergonne duality, polar curve, inflection points, hessian. Max Noether’s Fundamental Theorem: Divisors, adjoint curves. Cubic curves: Modular invariant, group structure. Solving Singularities: Rational functions, blowing-up, quadratic transformations. Riemann-Roch Theorem: Differentials, Riemann-Hurwitz formula, Weierstrass points, hyperelliptic curves, curves of genus less than 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.
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.
* Basic syllabus. The teacher has the autonomy to make any changes.