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 .......................................... 24
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 Кр,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 ................... 60
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 а КЗ surface ................... 75
6.1. The Serre construction, and vector bundles on КЗ
surfaces .................................................. 75
6.2. Brill-Noether theory of curves on КЗ surfaces ............. 76
6.3. Voisin's proof of Green's generic conjecture: even
genus ..................................................... 79
6.4. The odd-genus case (outline) .............................. 88
6.5. Notes and comments ........................................ 90
Chapter 7. Specific versions of the syzygy conjectures ......... 91
7.1. Nodal curves on КЗ surfaces ............................... 91
7.2. The specific Green conjecture ............................. 92
7.3. Stable curves with extra-syzygies ......................... 95
7.4. Curves with small Brill-Noether loci ...................... 97
7.5. Further evidence for the Green-Lazarsfeld conjecture ..... 100
7.6. Exceptional curves ....................................... 103
7.7. Notes and comments ....................................... 103
Chapter 8. Applications ....................................... 105
8.1. Koszul cohomology and Hodge theory ....................... 105
8.2. Koszul divisors of small slope on the moduli space ....... 1ll
8.3. Slopes of fibered surfaces ............................... 115
8.4. Notes and comments ....................................... 116
Bibliography .................................................. 119
Index ......................................................... 125
|
