MWF 9:05-9:55am, LGRT 1234.

Office: LGRT 1235E.

Office hours MWF 1:30-3:30

E-mail: tevelev(at)math.umass.edu

Class webpage: http://people.math.umass.edu/~tevelev/797_2017/

All lecture notes in one file

Class Meeting | Topic | Assignments |

Jan 23 | Grassmannian as a complex manifold. | |

Jan 25 | Grassmannian as a quotient. Stiefel and Plucker coordinates. | |

Jan 27 | Grassmannian as a projective variety. | |

Jan 30 | Homogeneous ideal of the Grassmannian. | |

Feb 1 | Hilbert function and degree of the Grassmannian. Topics of presentations | |

Feb 3 | Application to a Schubert calculus problem. | |

Feb 6 | Representable functors | |

Feb 8 | Grassmannian as a fine moduli space | |

Feb 10 | Moduli of algebraic curves. | |

Feb 13 | Snow day | |

Feb 15 | Riemann surfaces. Meromorphic forms. Genus. | |

Feb 17 | Divisors. Canonical divisor. Riemann-Hurwitz. | Homework 1 due. Notes on the Grassmannian and Homework 1 |

Feb 22 | Linear systems and Riemann-Roch | |

Feb 24 | Curves of genus 1 | |

Feb 27 | Complex tori | |

Mar 1 | J-invariant | |

Mar 3 | Elliptic fibrations | Homework 2 due. Notes on algebraic curves and Homework 2 |

Mar 6 | Cartier divisors | |

Mar 8 | Morphisms with reduced fibers | |

Mar 10 | Pushforwards and derived pushforwards | |

SPRING BREAK | ||

Mar 20 | Cohomology and base change | |

Mar 22 | Riemann-Roch in families. | |

Mar 24 | Invariants of finite groups | |

Mar 27 | Properties of quotients of finite groups | |

Mar 29 | Quotient singularities | |

Mar 31 | Icosahedral singularity | Homework 3 due. Notes on moduli of elliptic curves and Homework 3 |

Apr 3 | Linear algebraic groups | |

Apr 5 | Reductive groups | |

Apr 7 | Geometric and categorical quotient | |

Apr 10 | Weighted projective space | |

Apr 12 | Projective spectrum | |

Apr 14 | GIT quotients and stability | Homework 4 due. Notes on families of algebraic varieties, invariants of finite group and Homework 4 |

Apr 18 | Toric Varieties as Quotients of Affine Space (by Johnson, Li) | |

Apr 19 | Stable Curves and Stable Reduction (by Stern, Torres) | |

Apr 21 | Examples of GIT stability | |

Apr 24 | Moduli of stable vector bundles (by T. Nakamura, Simonetti). | |

Apr 26 | Mnev's universality and applications (by Cabrera, Fu, K. Nakamura). | |

Apr 28 | Nagata's Counterexample (by Day, Hart). | |

May 1 | Stability of smooth hypersurfaces | Homework 5 due on May 3. Notes on quotients by reductive groups and Homework 5 |

To get an A+ you have to accumulate 100 points by the end of the semester.

Homework problems can be presented in two ways. An ideal method is to come to my office and explain your solution to me. You can do it during office hours, by scheduling an appointment, or by just showing up in my office assuming I am not too busy. If you can show me a correct solution at the blackboard, you wonâ€™t have to turn in the problem in the written form. If your solution does not work, there will be no penalty and I will probably give you a hint.

Each homework will have a two-week deadline. Problems not discussed in my office to my satisfaction will have to be written down and turned in by the end of the two-week period.