فضاهای مدولی فشرده و بسته‌های برداری: کنفرانس در مورد فضاهای مدولی فشرده و بسته‌های برداری

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. Preface 8 Talks at the UGA conference 10 Compact moduli spaces of surfaces of general type 12 1. Introduction 12 2. Moduli of stable surfaces 13 3. Infinitesimal study of the moduli stack 15 4. Wahl singularities 16 5. Orbifold double normal crossing singularities 17 6. Surfaces with boundary 18 7. Plane curves 19 8. Exceptional vector bundles associated to degenerations of surfaces 22 9. Boundary divisors of the moduli space of stable surfaces 24 10. Relation with Donaldson theory 25 11. Other examples of boundary divisors 27 References 27 Rigid curves on \overline{M}_{0,n} and arithmetic breaks 30 1. Introduction 30 2. The Keel-M\textsuperscript{c}Kernan theorem 33 3. Surfaces in \overline{M}_{0,n} from configurations of points in P2 35 4. The hypergraph construction 38 5. The dual Hesse configuration and a rigid curve on \overline{M}_{0,12} 41 6. The “Two Conics” construction 45 7. Arithmetic break of a hypergraph curve 47 8. Rigid matroids 60 ऀ⸀ 䄀爀椀琀栀洀攀琀椀挀 戀爀攀愀欀 漀昀 愀†ᰀ吀眀漀 䌀漀渀椀挀猠ᴀ 挀甀爀瘀攀 ⴀ 瀀愀爀琀 62 10. Arithmetic break of a “Two Conics” curve - part II 71 References 77 Algebraic and combinatorial Brill-Noether theory 80 1. Introduction 80 2. Algebro-geometric preliminaries 82 3. Divisor theory on graphs 84 4. Baker specialization lemma refined 86 5. Specialization for graphs with loops 89 6. On the emptyness of Brill-Noether loci 92 References 95 GIT constructions of log canonical models of \overline{M}_{g} 98 1. Introduction 98 2. Parameter spaces 100 3. Finite Hilbert Stability 102 4. Unstable curves: Degenerate and non-reduced curves 105 5. Unstable curves: Badly singular curves and special subcurves 106 6. Local study of the moduli spaces of c-semistable and of h-semistable curves 111 References 116 The geometry of the ball quotient model of the moduli space of genus four curves 118 Introduction 118 1. Preliminaries on canonical genus 4 curves and cubic 3-folds 121 2. GIT for g=4 curves via cubic 3-folds 126 3. Stability for canonical genus 4 curves 132 4. Hassett–Keel Program 135 5. Ball quotient model for the moduli of genus 4 curves 137 6. The comparison of the GIT and ball quotient models 142 References 145 Two remarks on the Weierstrass flag 148 1. Introduction 148 2. Linear series of Weierstrass type 149 3. On the Weierstrass flags 152 References 155 Chern classes of conformal blocks 156 1. Introduction 156 2. Conformal blocks 158 3. The case g=0 165 4. g=0, g=sl2 171 5. g=0, arbitrary g and l=1 176 6. The case g>0 181 7. Questions 185 References 186 Restrictions of stable bundles 188 References 195 Orthogonal bundles, theta characteristics and symplectic strange duality 196 1. Introduction 196 2. Moduli spaces 199 3. Hitchin’s connection and the geometric Segal-Sugawara tensor 201 4. The proof of Proposition 2.2 202 References 204 The splitting principle and singularities 206 1. Introduction 206 2. The Abstract Splitting Principle 209 3. Applications 213 References 214 Igusa quartic and Steiner surfaces 216 1. Self-morphism of degree 8 217 2. Proof of Proposition 1 218 3. Proof of Theorem 2 219 References 221 Green’s conjecture for general covers 222 1. Introduction 222 2. Koszul cohomology 224 3. Syzygy conjectures for general étale double covers 226 4. Syzygies of sections of Nikulin surfaces 229 5. Green’s conjecture for general covers of plane curves 232 6. Green’s conjecture for general triple covers of elliptic curves 234 7. Syzygies of double covers of curves of Clifford dimension 3 235 References 236 Spaces of sections of quadric surface fibrations over curves 238 1. Introduction 238 Acknowledgments 239 2. Quadratic forms, discriminants, and heights 239 3. Reduction to the discriminant for quadric surface fibrations 241 4. Census of quadric fibrations over P1 245 5. Hecke correspondences and elementary transformations 248 6. Stable bundles and cohomology 249 7. Projective bundles over limits of hyperelliptic curves 250 8. Limits of sections and Néron models of intermediate Jacobians 253 9. Stability and discriminant curves 255 10. Arithmetic applications 257 References 258