课程大纲

COURSE SYLLABUS

课程代码/名称

Course Code/Title

MAT7076 代数曲线 Algebraic Curves

课程性质

Compulsory/Elective

选修 Elective

课程学分/学时

Course Credit/Hours

3 学分/48 学时

授课语言

Teaching Language

英文 English

授课教师

Instructor(s)

胡勇 Hu Yong

是否面向本科生开放

Open to undergraduates

or not

是 Yes

先修要求

Pre-requisites

（如面向本科生开放，请注明区分内容。 If the course is open to

undergraduates, please indicate the difference.）

MA323 拓扑学 Topology, MAT8020 抽象代数 II Abstract Algebra II

教学目标

Course Objectives

（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the

difference.）

介绍代数几何的基本概念和基本研究方法和工具。引入层的上同调，详细讲述代数曲线，特别是平面代数

曲线的基本知识。如有时间，还计划讲述与黎曼曲面之间的联系。

The course will introduce basic concepts and fundamental methods and tools in algebraic geometry. Main

topics covered include sheaves and their cohomology and basic theory of algebraic curves, especially

plane algebraic curves. If time permits, we will also talk about relationship with the theory of

Riemann surfaces.

教学方法

Teaching Methods

（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the

difference.）

理论课程，课堂讲授为主。

This is a theoretical course, taught by classroom lectures.

10.

教学内容

Course Contents

（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the

difference.）

Section 1

仿射代数集 Affine algebraic sets

Section 2

射影代数集 Projective algebraic sets

Section 3

层论 Sheaf theory

Section 4

代数簇 Algebraic varieties

Section 5

维数理论 Dimension theory

Section 6

切空间和奇点 Tangent spaces and singular points

Section 7

层的上同调 Cohomology of sheaves

Section 8

亏格与 Riemann-Roch 定理 Genus and Riemann-Roch theorem

Section 9

有理映射 Rational maps

Section 10

几何亏格 Geometric genus

Section 11

有理曲线 Rational curves

Section 12

其他专题 Other topics

…………

11.

课程考核

Course Assessment

（

○

考核形式 Form of examination；

○

.分数构成 grading policy；

○

如面向本科生开放，请注明区分内容。

If the course is open to undergraduates, please indicate the difference.）

(1) 考试，十三级等级记分制；Written exam, letter grading

(2) 分数构成：平时作业 20%；课程报告 30%；期中考试：20%，期末考试：30%

Grading policy: homework 20%; project/presentation 30%; midterm 20%, final exam: 30%

12.

教材及其它参考资料

Textbook and Supplementary Readings

教材 Textbook:

教师自编讲义 Lecture notes by the instructor

其他参考资料 Supplementary readings:

Daniel Perrin, Algebraic Geometry: an introduction, Springer, 2008, ISBN: 978-1-84800-055-1 e-ISBN: 978-

1-84800-056-8

W. Fulton, Algebraic curves: an introduction to algebraic geometry, ISBN: 201510103, 9780201510102,

online version available at http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf