MATH3321/MATH4011

**Lecturer :** Norbert Peyerimhoff

**Term :** Epiphany 2014

- Wednesdays, 11:00 in CM107
- Thursday, 9:00 in CM107

**Literature**

Relevant Books:

- L. Flatto,
*Poncelet's Theorem, American Mathematical Society, Providence 2009.* - C.G. Gibson,
*Elementary Geometry of Algebraic Curves, An Undergraduate Introduction, Cambridge 1998.* - G.A. Jones and D. Singerman,
*Complex Functions, An algebraic and geometric viewpoint, Cambridge 1987.* - F. Kirwan,
*Complex Algebraic Curves, LMS Student Texts 23, Cambridge 1992.* - M. Reid,
*Undergraduate Algebraic Geometry, LMS Student Texts 12, Cambridge 1988.*

**Assignments**

Exercise | Date | Hand in | Solutions | |

Exercise 1 pdf | 22.1.2014 | --- | Solution 1 pdf | |

Exercise 2 pdf | 29.1.2014 | 5.2.2014 | Solution 2 pdf | |

Exercise 3 pdf | 5.2.2014 | --- | Solution 3 pdf | |

Exercise 4 pdf | 12.2.2014 | 19.2.2014 | Solution 4 pdf | |

Exercise 5 pdf | 19.2.2014 | --- | Solution 5 pdf | |

Exercise 6 pdf | 26.2.2014 | 5.3.2014 | Solution 6 pdf | |

Exercise 7 pdf | 5.3.2014 | --- | Solution 7 pdf | |

Exercise 8 pdf | 12.3.2014 | 20.3.2014 | Solution 8 pdf | |

Exercise 9 pdf | 19.3.2014 | --- | Solution 9 pdf |

**Content of Lectures**

Date | Content |

Wednesday, 22 January 2014 (Week 11) | Recap of group structure on a non-singular projective cubic. Roadway towards the law of associativity. |

Thursday, 23 October 2014 (Week 11) | Completion of the proof that the addition law on non-singular projective cubics is associative. |

Wednesday, 29 January 2014 (Week 12) | Roadway towards complex manifolds: topological spaces, Hausdorff property, homeomorphisms, definition of a complex manifold. |

Thursday, 30 January 2014 (Week 12) | Examples of complex manifolds, Riemann surfaces, holomorphic maps between Riemann surfaces, square root function example. |

Wednesday, 5 February 2014 (Week 13) | Continuation of square root function example, topological classification of compact Riemann surfaces, statement of the degree-genus formula, recap of implicit function theorem. |

Thursday, 6 February 2014 (Week 13) | Proof that non-singular algebraic curves are Riemann surfaces, example of a singular algebraic curve which is not a Riemann surface, Euler number and genus of triangulations. |

Wednesday, 12 February 2014 (Week 14) | Key example of a branched covering (algebraic curve defined by Y^2=XZ over complex projective line), visualization of branched coverings, definition of ramification index and ramification point, geometric interpretation of ramification indices. |

Thursday, 13 February 2014 (Week 14) | Problem Class |

Wednesday, 19 February 2014 (Week 15) | Relations between ramification indices, ramification points and d-fold branched coverings. |

Thursday, 20 February 2014 (Week 15) | Start of proof of the degree genus formula, together with various topological concepts and arguments. |

Wednesday, 26 February 2014 (Week 16) | Continuation of proof of the degree genus formula, Riemann Hurwitz formula, and examples. |

Thursday, 27 February 2014 (Week 16) | More examples to the degree genus formula, non-singular model of an irreducible projective algebraic curve, local concept of a blowup, exceptional divisor. |

Wednesday, 5 March 2014 (Week 17) | Strict transform of a function, connection between preimages of blow-up above (0,0) and tangent lines of algebraic curve at (0,0), examples. |

Thursday, 6 March 2014 (Week 17) | Discussion of the local nature of blow-ups, repeated blow-ups in singularities to obtain non-singular models, examples: desingularisations of irreducible affine and projective curves. |

Wednesday, 12 March 2014 (Week 18) | Continuation of desingularisation of an example of a projective curve, method to calculate the genus of a non-singular model of a singular irreducible projective algebraic curve, start with Problem 1 for Problem Class. |

Thursday, 13 March 2014 (Week 18) | Problem Class: Continuation of Problem 1, Problem 2. |

Wednesday, 19 March 2014 (Week 19) | Complex projective version of Pascal's Mystic Hexagon with proof, Poncelet's Theorem a la Griffith/Harris. |

Thursday, 20 March 2014 (Week 19) | Continuation of the proof of Poncelet's Theorem a la Griffith/Harris. |