The main textbook for this course isAlgebraic Curves and Riemann Surfacesby Rick Miranda, Graduate Studies in Mathematics, Volume 5, AMS (1995). The lectures will cover the following topics:

**Riemann Surfaces: basic definitions****First few examples**: Riemann Sphere, complex tori, graphs of holomorphic functions, and smooth affine plane curves**Projective curves**: smooth projective plane curves and (local) complete intersections**Functions on Riemann surfaces**: holomorphic and meromorphic functions**Examples of meromorphic functions**: Riemann Sphere, complex tori, smooth plane, and smooth projective curves.**Holomorphic maps between Riemann surfaces**: isomorphism and automorphisms, Euler number, Hurwitz's formula**Further examples examples**: lines and conics, hyperelliptic Riemann surfaces, resolving nodes of a plane curve, resolving monomial singularities, cyclic coverings of the line.**Group actions on Riemann surfaces**: quotient map and quotient Riemann surface, ramification of the quotient map, Hurwitz's theorem on automorphisms**Monodromy**: holomorphic maps and coverings via monodromy representations, examples**Differential forms on Riemann surfaces**: holomorphic and meromorphic 1- and 2-forms**Differential calculus on forms**: differentials and wedge products**Integration on a Riemann surface**:residue of a meromorphic 1-form, Stoke's theorem.

**Divisors**:principal and canonical divisors, degree of a divisor, divisors of holomorphic maps, intersection divisors, and hyperplane divisors**Linear equivalence of divisors**: degree of a smooth projective curve, Bezout's theorem, Pascal's mystic hexagon, Plücker's formula**The space L(D)**: complete linear systems and linear systems, pencils and webs**Divisors and holomorphic maps to projective spaces**: defining a holomorphic map via a linear system, criteria for the holomorphic map associated to the complete linear system |D| to be an embedding**Algebraic curves**: function field on an algebraic curve, Laurent tail divisors and Mittag-Leffler problem**Riemann-Roch theorem and Serre duality**: the equality of the three genera**Applications of Riemann-Roch theorem**: genus zero, one, and two curves, Clifford theorem, canonical map and the geometric form of Riemann-Roch, dimension of the moduli space of algebraic curves of genus g, degree of projective curves, general position lemma, Castelnuovo bound, inflection points and Weierstrass points**Abel's theorem**: homology, periods and the Jacobian, the Able-Jacobi map, residue theorem, proof of Abel's theorem, Abel's theorem for genus one curves

**Complex Algebraic Curves**, by Frances Kirwan, Cambridge University Press (1992).**Basic Algebraic Geometry, Volumes 1&2**, by Igor R. Shafarevich, Springer-Verlag (1994).**Conics and Cubics, A Concrete Introduction to Algebraic Curves**, by Robert Bix, Springer-Verlag (1998).**Algebraic Geometry, A First Course**, by Joe Harris, Springer-Verlag (1992).**Algebraic Geometry**, by Robin Hartshorne, Springer-Verlag (1987).**Principles of Algebraic Geometry**, by Philip Griffiths and Joseph Harris, John Wiley (1978).**Geometry of Algebraic Curves, Volume I**, by E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Springer-Verlag 1985.**Rational Curves on Algebraic Varieties**, by János Kollár, Springer-Verlag (1996).**Algebraic Geometry. V**, A. N. Parishin, and I. R. Shafarevich, Eds.,*Encyclopedia of Mathematical Sciences, 47*, Springer-Verlag (1999).