Algebraic curves (代数曲线)

Teaching information

Instructor: 杜佳宾（Jiabin DU）

Email: jiabin.du@simis.cn

Office: SIMIS 1619

Course information:

Course time: every Wednesday 10:00-11:45 a.m., at SIMIS 1210

Pre-requisite Courses

Modern Algebra, Commutative Algebra and Topology

Syllabus

Week 1-8

Goal: Basics on algebraic varieties (介绍代数簇的基本概念):



1. Affine and projective varieties (仿射和射影代数簇)
2. Projective plane curves (射影平面曲线)
3. Varieties, morphisms and rational maps (代数簇，态射和有理映射)
4. Resolution of singularities (奇异曲线的奇点消解)

Week 9-12

Goal: Riemann-Roch Theorem (介绍复曲线的 Riemann-Roch 定理):



1. Divisors and vector bundles on curves (除子和向量丛)
2. Cohomology (上同调)
3. Serre duality, Riemann-Roch Theorem and Riemann-Hurwitz formula

Week 13-15

Goal: Geometry of projective curves (介绍复射影曲线的几何)：



1. Embeddings into projective space and canonical maps (到射影空间的嵌入、典范映射)
2. Embeddings into abelian variety and Albanese morphisms (到阿贝尔簇的嵌入、Albanese 态射)
3. Clifford inequality and Castelnuovo inequality
4. *Brill-Noether theory (Brill-Noether 理论)

References:

1. Algebraic Curves , W. Fulton; 2008
2. Algebraic Geometry , Robin Hartshorne; Springer; 1977
3. Geometry of algebraic curves I, Arbarello-Cornalba-Griffiths-Harris;1985
4. Algebraic Geometry I: algebraic curves, algebraic manifolds and schemes ,I.R. Shafarevich (Ed.); 1994

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