Contents
Preface vii
Chapter 1. Basic definitions 1
1.1. The Koszul complex 1
1.2. Definitions in the algebraic context 2
1.3. Minimal resolutions 4
1.4. Definitions in the geometric context 5
1.5. Functorial properties 7
1.6. Notes and comments 11
Chapter 2. Basic results 13
2.1. Kernel bundles 13
2.2. Projections and linear sections 15
2.3. Duality 20
2.4. Koszul cohomology versus usual cohomology 22
2.5. Sheaf regularity
2.6. Vanishing theorems 25
2.7. Notes and comments 27
Chapter 3. Syzygy schemes 29
3.1. Basic definitions 29
3.2. Koszul classes of low rank 37
3.3. The Kp,1 theorem 39
3.4. Rank-2 bundles and Koszul classes 43
3.5. The curve case 46
3.6. Notes and comments 50
Chapter 4. The conjectures of Green and Green–Lazarsfeld 53
4.1. Brill-Noether theory 53
4.2. Numerical invariants of curves 55
4.3. Statement of the conjectures 57
4.4. Generalizations of the Green conjecture
4.5. Notes and comments 63
Chapter 5. Koszul cohomology and the Hilbert scheme 65
5.1. Voisin’s description 65
5.2. Examples 68
5.3. Vanishing via base change 72
Chapter 6. Koszul cohomology of a K3 surface 75
6.1. The Serre construction, and vector bundles on K3 surfaces 75
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