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MTH 405/505: Algebraic Curves


Algebraic Curves is a two-semester 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. 

Homework:








Lectures: 


Part I   (10 lectures; Summer course)

  1. 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
  2. Foundations
    • Complex algebraic curves in C^2
    • Complex projective spaces
    • Complex projective curves in P_2
    • Affine and projective curves
    • Exercises
  3. Bezut's theorem
    • Bezut's theorem
    • Exercises
  4. Points of inflection and cubic curves
    • Points of inflections on curves
    • Cubic curves
    • Exercises 
  5. The degree-genus formula
    • The degree-genus formula
      • First method of proof
      • Second method of proof
    • Exercises
  6. Branched covers of the projective line
    • Coverings and their properties
    • Branch points and branch locus
    • Proof of the degree-genus formula
    • Exercises
  7. Riemann surfaces
    • The Weierstrass rho-function
    • Riemann Surfaces
    • Exercises
  8. Differentials on Riemann surfaces, Abel's theorem
    • Holomorphic differentials on Riemann surfaces
    • Abel's theorem
    • Exercises
  9. Riemann-Roch theorem 
    • The Riemann-Roch theorem
    • Exercises
  10. Singular curves
    • Resolution of singularities
    • Newton polygons and Puiseux expansions
    • Singular curves
    • Exercises. 
  11. 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

Ċ
Tony Shaska,
Jul 6, 2012, 5:47 AM
Ċ
hw_2.pdf
(113k)
Tony Shaska,
Jul 5, 2012, 2:13 AM