*STAG: Students Teaching Algebraic Geometry* is a student-driven algebraic geometry discussion and learning seminar. We meet **Wednesdays** and **Fridays** at **10:00-10:30** in **SCEN 349**. The seminar is open to all mathematics students interested in algebraic geometry or related fields. Please come check us out!

Our format consists of short expository talks with questions encouraged, in addition to collaborative problem-solving sessions and presentations to prepare attendees to do mathematics research and give cogent talks. In the fall semester we learn the foundations of algebraic geometry and in the spring semester we transition into exploring interesting topics and open problems. The topics are attendee-guided, so participants are encouraged to suggest interesting directions and contribute.

On days where we present and discuss problems there will be pizza or other foods! Come with full minds and empty bellies.

Here is our calendar of topics and links to posted notes. NB: any links to posted notes may be works in progress and may be updated silently at any time without warning.

Date | Speaker(s) | Topic | Notes |
---|---|---|---|

Wed 15 Sept | Duncan | Motivation for Algebraic Geometry | Introduction to Algebraic Geometry §1 (pp. 1-2) |

Fri 17 Sept | Eric | The Nullstellensatz | Ibid. §2 (pp. 2-3) |

Wed 22 Sept | Eric | Krull Dimension, Topologically | Ibid. §2 (pp. 4-5) |

Fri 24 Sept | Duncan | Motivation and Construction of Projective Space | Ibid. §2 (pp. 5-6) |

Wed 29 Sept | Eric | Affine Charts of Projective Space, Introduction to Presheaves | Ibid. §2 (pp. 6-7), §3 (pp. 7-8) |

Fri 1 Oct | Duncan | Sheaves, Localization of Rings | Ibid. §3 (pp. 8-10) |

Wed 6 Oct | Eric | Regular Functions and the Structure Sheaf on an Affine/Projective Variety | Ibid. §3 (pp. 10-11) |

Fri 8 Oct | Duncan | Sections and Stalks of the Structure Sheaf | Ibid. §3 (pp. 11-12) |

Wed 13 Oct | Eric | Objects and Arrows in the Category of Locally Ringed Spaces | Ibid. §3 (pp. 12-14) |

Fri 15 Oct | Duncan | §3 Cleanup, Equivalence between Varieties and Integral Domains over k | Ibid. §3 (pp. 14-15) |

Wed 20 Oct | Eric | Rational Maps | Ibid. §4 (pp. 15-17) |

Thurs 21 Oct | Everyone | Problem-Solving Session | Pulled pork / jackfruit tacos |

Fri 22 Oct | Duncan | Dominant Rational Maps, Birationality, and an Equivalence of Varieties | Ibid. §5 (pp. 17-19) |

Wed 27 Oct | Eric | Proving the Equivalence of Categories | Ibid. §5 (pp. 19-20) |

Fri 29 Oct | Duncan | The Case of Curves | Silverman, The Arithmetic of Elliptic Curves §II.2 (pp. 19-21) |

Wed 3 Nov | Eric | Motivation and Definition of Blowing Up | ItAG §6 (pp. 21-23) |

Fri 5 Nov | Duncan | Blowing Up, Continued, and Hironaka's Theorem | Ibid. §6 (pp. 23-25) |

Wed 10 Nov | Eric | Motivating Affine Schemes, Definition Part 1 | Ibid. §7 (pp. 25-27), §8 (pp. 27-28) |

Fri 12 Nov | Duncan | Examples and Morphisms of Affine Schemes, Definition Part 2 | Ibid. §8 (pp. 30-33) |

Wed 17 Nov | Eric | Affine Schemes of Any Ring, Equivalence of Categories | Ibid. §8 (pp. 30-31, 33) |

Thurs 18 Nov | Everyone | Problem-Solving Session | Pineapple Thai curry |

Fri 19 Nov | Duncan | Noetherian Schemes, Spectra of Finite Type k-Algebras | Ibid. §8 (pp. 31-32) |

Wed 1 Dec | Eric | Schemes | Ibid. §9 (pp. 34-35) |

Fri 3 Dec | Eric | Simplicial Objects and Homotopy | Weibel, An introduction to homological algebra §8.1, 8.3 (pp. 254-257, 263-264) |

Wed 8 Dec | Everyone | Plans for next semester | None |

For additional information or questions, contact Duncan at asduncan@uark.edu.

*Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.* — David Mumford