Algebraic Curves is a twosemester course which is offered to students who have had at least one year of abstract algebra (including some commutative algebra)
Course objective:
The objective of this course is to introduce students to the area of algebraic curves and algebraic geometry. Algebraic curves are some of the most classical objects studied in mathematics. Algebraic curves provide the most intuitive approach to introducing students to algebraic geometry. Furthermore, they have many applications in areas of coding theory, cryptography, etc. This course will be a valuable tool of introducing students to these areas.
Textbooks
Detailed lecture notes will be provided during class.
Lectures:
Part I (10 lectures; Summer course)  Introduction and background on algebraic curves
 Conics
 An exercise in trigonometry
 Hyperbola shadows
 Linear families of conics
 The cross ratio
 Rational points on conics
 Abelian integrals
 The method of partial fractions
 Elliptic integrals
 Abelian integrals
 Jacobi's theta functions
 Exercises
 Foundations
 Complex algebraic curves in C^2
 Complex projective spaces
 Complex projective curves in P_2
 Affine and projective curves
 Exercises
 Bezut's theorem
 Points of inflection and cubic curves
 Points of inflections on curves
 Cubic curves
 Exercises
 The degreegenus formula
 The degreegenus formula
 First method of proof
 Second method of proof
 Exercises
 Branched covers of the projective line
 Coverings and their properties
 Branch points and branch locus
 Proof of the degreegenus formula
 Exercises
 Riemann surfaces
 The Weierstrass rhofunction
 Riemann Surfaces
 Exercises
 Differentials on Riemann surfaces, Abel's theorem
 Holomorphic differentials on Riemann surfaces
 Abel's theorem
 Exercises
 RiemannRoch theorem
 The RiemannRoch theorem
 Exercises
 Singular curves
 Resolution of singularities
 Newton polygons and Puiseux expansions
 Singular curves
 Exercises.
 Projects
Part II
Part III Other topics (will not be covered during lecture) Automorphisms of curves The automorphism group of a curve
 The Hurwitz bound
 Determining the automorphism group of an algebraic curve
 Finding the equation of the curve from the automorphism group
Jacobians of curves Divisors
 Linear equivalence and the Picard group
 Analytic description of the Jacobian
 Abel's theorem
 Schotky's problem
Moduli space of curves Construction of the moduli space
 Fine space versus coarse space
 Describing the moduli points
