## Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 248A: Algebraic Geometry

**Approved:**2009-05-01, Eric Babson, Motohico Mulase, Brian Osserman

**Units/Lecture:**

Fall, alternate years; 4 units; lecture/extensive problem solving

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Algebraic Geometry, Robin Hartshorne, ISBN 978-0-387-90244-9 ($69); Alternate: Basic Algebraic Geometry 1, Shafarevich, ISBN-13: 978-3540548126 ($69)

**Prerequisites:**

MAT 250ABC

**Course Description:**

Affine varieties and radical ideals. Projective varieties. Abstract varieties. Morphisms and rational maps. Smoothness. Algebraic curves and the Riemann-Roch theorem. Special topics.

**Suggested Schedule:**

Lectures | Sections | Topics/Comments |
---|---|---|

1 - 4 | I.1 - I.2 | Classical affine and projective varieties, including affine varities over arbitrary fields. |

5 -12 | I.3 - I.4 | The Zariski topology, gluing varieties, abstract varieties. Morphisms and rational maps. Smoothness. |

13 - 20 | IV.1 - IV.6 | Divisors and differentials on smooth projective curves, the Riemann-Roch theorem, and applications. |

21 - 28 | N/A | Special topics |

**Additional Notes:**

Possibilities for special topics include: group varieties, elliptic curves, toric varieties, Grassmannians, and real algebraic geometry. Most of these topics, as well as several subtopics above, require supplemental material beyond that in Hartshorne.