Vector bundles on curves (曲线上的向量丛)

Teaching information

Instructor: 杜佳宾（Jiabin DU）

Email: jiabin.du@simis.cn

Office: SIMIS 1619

Course information:

Course time: every Monday 9:55-12:30 a.m., at SIMIS 1210

Pre-requisite Courses

Algebraic Geometry

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Syllabus (planned topics)





Part 1: Vector bundles

1.1 Sheaf theory and vector bundles
1.2 Riemann-Roch formula
1.3 Positivity of vector bundles
1.4 Ample vector bundles on curves
1.5 Geometric applications



Part 2: Moduli spaces of vector bundles on curves

2.1 Hilbert schemes
2.2 Semi-stability
2.3 Jordan-Hölder filtrations and Harder-Narasimhan filtrations
2.4 Invariant geometry
2.5 Construction of M(r,d)
2.6 Geometry of M(r,d)



Part 3: Koszul cohomology

3.1 Koszul complex and minimal resolutions
3.2 Kernel bundles and sheaf regularity
3.3 Syzygy schemes
3.4 Brill-Noether theory, gonality of curves and Conjectures of Green and Green-Lazarsfeld
3.5 Koszul cohomology and Hilbert schemes
3.6 Syzygy conjectures
3.7 Slopes of fibred surfaces



References:

Koszul cohomology and Algebraic Geometry, Marian Aprodu and Jan Nagel
Algebraic Geometry, Robin Hartshorne
The Geometry of Moduli Spaces of Sheaves, Daniel Huybrechts and Manfred Lehn
Positivity in Algebraic Geometry, Robert Lazarsfeld
Lectures on Vector Bundles, J. Le Potier


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