# Algebra and Algebraic Geometry Seminar Spring 2020

## Spring 2020 Schedule

date | speaker | title | host(s) |
---|---|---|---|

January 24 | Xi Chen (Alberta) | Rational Curves on K3 Surfaces | Michael K |

January 31 | Janina Letz (Utah) | TBD | Daniel and Michael B |

February 7 | Jonathan Montaño (New Mexico State) | TBD | Daniel |

February 14 | |||

February 21 | Erika Ordog (Duke) | TBD | Daniel |

February 28 | |||

March 6 | |||

March 13 | |||

March 20 | |||

March 27 | Patrick McFaddin (Fordham) | TBD | Michael B |

April 3 | |||

April 10 | Ruijie Yang (Stony Brook) | TBD | Michael K |

April 17 | |||

April 24 | |||

May 1 | Lazarsfeld Distinguished Lectures | ||

May 8 |

## Abstracts

### Xi Chen

**Rational Curves on K3 Surfaces**

It is conjectured that there are infinitely many rational curves on every projective K3 surface. A large part of this conjecture was proved by Jun Li and Christian Liedtke, based on the characteristic p reduction method proposed by Bogomolov-Hassett-Tschinkel. They proved that there are infinitely many rational curves on every projective K3 surface of odd Picard rank. Over complex numbers, there are a few remaining cases: K3 surfaces of Picard rank two excluding elliptic K3's and K3's with infinite automorphism groups and K3 surfaces with two particular Picard lattices of rank four. We have settled these leftover cases and also generalized the conjecture to the existence of curves of high genus. This is a joint work with Frank Gounelas and Christian Liedtke.