**Time and Place :** MW 10-11:20; Math 4-130

**Instructor :** Radu Laza

Office : Math Tower 4-121

Email : radu.laza@stonybrook.edu

Office Hours: Wed 1-3pm

**References**:

- [B] A. Beauville, Complex Algebraic Surfaces -- this is the main reference that we will use during the semester.
- [F] R. Friedman, Algebraic Surfaces and Holomorphic Vector Bundles -- a complementary reference (for some topics we will use this).
- [B et al.] W. Barth, K. Hulek, C. Peters, A.van de Ven - Compact Complex Surfaces - a more analytic point of view.
- [KM] J. Kollár, S. Mori - Birational Geometry of Algebraic Varieties - the standard reference for birational geometry/MMP (this will come handy occasionally, and it is a natural follow-up text)
- [H] D. Huybrechts - Lectures on K3 Surfaces - more details on the very important topic of K3 surfaces.

**Topics Discussed**: The purpose of the course is to discuss the Castelnuovo-Enriques classification of surfaces. We will follow mostly Beauville's book, but with a greater emphasis on the MMP point of view (i.e. more focus on the behavior of the canonical class). We will spend some extra time on some special classes of surfaces, such as del Pezzo and K3 surfaces. With an eye on the higher dimensional case, I will introduce various classes of singularities and I will explain how they naturally occur.

Dates | Topic | References |

(1) Jan 22, 24 |
Invariants of surfaces Divisors and Intersection pairing |
[F] [B] -- Ch. 1 |

(2) Jan 29, 31 | Riemann-Roch et al. Ample, Big, Nef divisors |
[B] -- Ch. 1 [F] / [KM] |

(3) Feb 5, 7 | Birational Maps |
[B] -- Ch. 2 |

(4) Feb 12, 14 | Minimal models and Kodaira dimension Ruled surfaces |
[B] -- Ch. 3 |

(5) Feb 19, 21 | Ruled surfaces | [B] -- Ch. 3 |

(6) Feb 26, 28 | Rational Surfaces Hirzebruch and del Pezzo surfaces |
[B] -- Ch. 4 |

(7) Mar 5, 7 | Castelnuovo's Theorem | [B] -- Ch. 5 |

(8) Mar 19, 21 | ||

(9) Mar 26, 28 | ||

(10) Apr 2, 4 | ||

(11) Apr 9, 11 | ||

(12) Apr 16, 18 | ||

(13) Apr 23, 25 | ||

(14) Apr 30, May 2 |