**Contemporary Mathematics**

Volume: 564;
2012;
249 pp;
Softcover

MSC: Primary 14;

Print ISBN: 978-0-8218-6899-7

Product Code: CONM/564

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-0-8218-8537-6
Product Code: CONM/564.E**

List Price: $89.00

Individual Member Price: $71.20

# Compact Moduli Spaces and Vector Bundles

Share this page *Edited by *
*Valery Alexeev; Angela Gibney; Elham Izadi; János Kollár; Eduard Looijenga*

This book contains the proceedings of the conference on Compact Moduli and Vector Bundles, held from October 21–24, 2010, at the University of Georgia.

This book is a mix of survey papers and original research articles on two related subjects: Compact Moduli spaces of algebraic varieties, including of higher-dimensional stable varieties and pairs, and Vector Bundles on such compact moduli spaces, including the conformal block bundles. These bundles originated in the 1970s in physics; the celebrated Verlinde formula computes their ranks.

Among the surveys are those that examine compact moduli spaces of surfaces of general type and others that concern the GIT constructions of log canonical models of moduli of stable curves.

The original research articles include, among others, papers on a formula for the Chern classes of conformal classes of conformal block bundles on the moduli spaces of stable curves, on Looijenga's conjectures, on algebraic and tropical Brill–Noether theory, on Green's conjecture, on rigid curves on moduli of curves, and on Steiner surfaces.

#### Table of Contents

# Table of Contents

## Compact Moduli Spaces and Vector Bundles

- Preface vii8 free
- Talks at the UGA conference ix10 free
- Compact moduli spaces of surfaces of general type 112 free
- 1. Introduction 112
- 2. Moduli of stable surfaces 213
- 3. Infinitesimal study of the moduli stack 415
- 4. Wahl singularities 516
- 5. Orbifold double normal crossing singularities 617
- 6. Surfaces with boundary 718
- 7. Plane curves 819
- 8. Exceptional vector bundles associated to degenerations of surfaces 1122
- 9. Boundary divisors of the moduli space of stable surfaces 1324
- 10. Relation with Donaldson theory 1425
- 11. Other examples of boundary divisors 1627
- References 1627

- Rigid curves on \𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒{𝑀}_{0,𝑛} and arithmetic breaks 1930
- 1. Introduction 1930
- 2. The Keel-M\textsuperscript{c}Kernan theorem 2233
- 3. Surfaces in \𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒{𝑀}_{0,𝑛} from configurations of points in ℙ² 2435
- 4. The hypergraph construction 2738
- 5. The dual Hesse configuration and a rigid curve on \𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒{𝑀}_{0,12} 3041
- 6. The “Two Conics” construction 3445
- 7. Arithmetic break of a hypergraph curve 3647
- 8. Rigid matroids 4960
- 9. Arithmetic break of a “Two Conics” curve - part I 5162
- 10. Arithmetic break of a “Two Conics” curve - part II 6071
- References 6677

- Algebraic and combinatorial Brill-Noether theory 6980
- GIT constructions of log canonical models of \𝑜𝑣𝑒𝑟𝑙𝑖𝑛𝑒{𝑀}_{𝑔} 8798
- 1. Introduction 8798
- 2. Parameter spaces 89100
- 3. Finite Hilbert Stability 91102
- 4. Unstable curves: Degenerate and non-reduced curves 94105
- 5. Unstable curves: Badly singular curves and special subcurves 95106
- 6. Local study of the moduli spaces of c-semistable and of h-semistable curves 100111
- References 105116

- The geometry of the ball quotient model of the moduli space of genus four curves 107118
- Introduction 107118
- 1. Preliminaries on canonical genus 4 curves and cubic 3-folds 110121
- 2. GIT for 𝑔=4 curves via cubic 3-folds 115126
- 3. Stability for canonical genus 4 curves 121132
- 4. Hassett–Keel Program 124135
- 5. Ball quotient model for the moduli of genus 4 curves 126137
- 6. The comparison of the GIT and ball quotient models 131142
- References 134145

- Two remarks on the Weierstrass flag 137148
- Chern classes of conformal blocks 145156
- Restrictions of stable bundles 177188
- Orthogonal bundles, theta characteristics and symplectic strange duality 185196
- The splitting principle and singularities 195206
- Igusa quartic and Steiner surfaces 205216
- Green’s conjecture for general covers 211222
- 1. Introduction 211222
- 2. Koszul cohomology 213224
- 3. Syzygy conjectures for general étale double covers 215226
- 4. Syzygies of sections of Nikulin surfaces 218229
- 5. Green’s conjecture for general covers of plane curves 221232
- 6. Green’s conjecture for general triple covers of elliptic curves 223234
- 7. Syzygies of double covers of curves of Clifford dimension 3 224235
- References 225236

- Spaces of sections of quadric surface fibrations over curves 227238
- 1. Introduction 227238
- Acknowledgments 228239
- 2. Quadratic forms, discriminants, and heights 228239
- 3. Reduction to the discriminant for quadric surface fibrations 230241
- 4. Census of quadric fibrations over ℙ¹ 234245
- 5. Hecke correspondences and elementary transformations 237248
- 6. Stable bundles and cohomology 238249
- 7. Projective bundles over limits of hyperelliptic curves 239250
- 8. Limits of sections and Néron models of intermediate Jacobians 242253
- 9. Stability and discriminant curves 244255
- 10. Arithmetic applications 246257
- References 247258

#### Readership

Graduate students and research mathematicians interested in algebraic geometry and moduli theory.