**Algebraic geometry II · Math 632**

This is the second course in the algebraic geometry sequence. We will continue to study varieties, schemes, and coherent sheaves. A large portion of this course will focus on the development of sheaf cohomology -- especially the properties and computation of cohomology of coherent sheaves.

** · course information ·**

**instructor ·** David Stapleton
**email ·** dajost@umich.edu
**office ·** East Hall 4839
**office hours · **Mon: 11:30-12:30pm. Tues, Thurs: 10-11am.

**· schedule ·**

Tuesday | Thursday |
---|---|

·1/4· |
·1/6· First day of class. Overview of class. Review of Proj. |

·1/11· Linear systems and maps to projective space. |
·1/13· Relative proj, projective bundles and blowing up. |

·1/18· Projective bundles and blowing up. |
·1/20· Blow-ups,
and intro to differentials. HW1 due |

·1/25· The module of Kahler differentials |
·1/27· The cotangent sheaf and the Euler sequence |

·2/1· The Euler sequence and smoothness criteria |
·2/3· HW2 due Bertini's theorem |

·2/8· Line bundles on curves and Riemann-Hurwitz |
·2/10· Hilbert polynomials |

·2/15· Riemann's Theorem and Riemann Roch |
·2/17· Introduction to Čech cohomology |

·2/22·Even more Čech cohomology HW3 due |
·2/24· |

·3/1· Spring Break! No class. |
·3/3· Spring Break! No class. |

·3/8· Cohomology of projective space. |
·3/11· Intro to derived functors. |

·3/15· Intro to spectral sequences. |
·3/17· Ext groups and sheaves |

·3/22·HW4 due Properties of Ext sheaves |
·3/24· Aside on ampleness |

·3/29· Serre duality |
·3/31·HW5 due |

·4/12· |
·4/14· |

·4/19· Last day of class. |
·4/21· |

**· index of topics ·**

· Proj, linear series, ample line bundles, and
blowing up · (1/6-1/20) Notes |

· Differentials and Bertini's theorem · (1/20-2/8) Notes |

· Line bundles on curves and Riemann-Roch · (2/8-2/15) Notes |

· Intro to Čech cohomology · (2/15-3/15) Notes |

· Derived functors, spectral sequences · (2/17-...) |

**· course information ·**

**prerequisites**

Math 631 or instructors approval.

**recommended textbooks**

*Algebraic geometry* by Robin Hartshorne,

*Foundations of Algebraic Geometry* by Ravi Vakil.

*Algebraic geometry II* by David Mumford and Tadao Oda

We will use all the above texts in tandem. It is not too difficult to find pdfs online of these texts. For Hartshorne, it is possible to download the text through Springer using umich credentials. For Mumford and Oda, there is a *penultimate draft* available online.

**grades**

Your grade will be determined as follows. Homework will count towards 50% of your grade, the midterm will count towards 20% of your grade, and the final exam will count towards 30% of your grade.

**homework**

We will have homework due approximately every one to two weeks, handed in at the beginning of class. Homework will be posted on the course website with a pdf link by the date in the calendar that it is due.