Algebraic Plane Curves

Code	14804
Year	3
Semester	S2
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	NA
Learning outcomes	1. recognize and classify algebraic curves defined by polynomials in two real variables, namely as to the number of connected components, self-intersections and singularities. 2. generalize algebraic curves to projective spaces to include points at infinity and deal with asymptotic behavior. 3. understand the advantages of working with complex curves. 4. understand the concept of intersection multiplicity of plane projective curves and use the resultant to prove Bézout’s Theorem. 5. recognize a non-singular projective algebraic curve as a compact Riemann surface and demonstrate the degree-genus formula for surfaces from the Riemann-Hurwitz theorem. 6. describe the zeros and poles of meromorphic functions and differentials on a surface in terms of divisors and apply the Theorem of Riemann-Roch to calculate the dimension of the corresponding spaces. 7. understand in detail the particular case of elliptic curves.
Syllabus	1. Introduction – preliminary definitions and examples 2. Affine and Projective Spaces – the projective line and the projective plane 3. Complex Projective Algebraic Curves: affine and projective curves; tangents and singularities. 4. Plane Curves Intersection and Multiplicities - Bézout’s Theorem 5. Riemann Surfaces: Riemann-Hurwitz theorem; the degree-genus formula. 6. The Riemann-Roch Theorem 7. Elliptic curves
Main Bibliography	1. E. Kunz, Introduction to Plane Algebraic Curves, Birkhauser (2005) 2. F. Kirwan, Complex Algebraic Curves, Cambridge University Press (1995) 3. G. Fischer, Plane Algebraic Curves, American Mathematical Association (2000) 4. I. Vainsencher, Introdução às Curvas Algébricas Planas, IMPA (2009) 5. K. Kendig, Plane Algebraic Curves, Mathematical American Association (2011)
Language	Portuguese. Tutorial support is available in English.