MODULI OF CURVES

By KENNETH MILLER

CONTENTS PARAMETER SPACES: CONSTRUCTIONS AND EXAMPLES A PARAMETERS AND MODULI B CONSTRUCTION OF THE HILBERT SCHEME C TANGENT SPACE TO THE HILBERT SCHEME D EXTRINSIC PATHOLOGIES MUMFORD’S EXAMPLE OTHER EXAMPLES E DIMENSION OF THE HILBERT SCHEME F SEVERI VARIETIES G HURWITZ SCHEMES BASIC FACTS ABOUT MODULI SPACES OF CURVES A WHY DO ?NE MODULI SPACES OF CURVES NOT EXIST? B MODULI SPACES WE’LL BE CONCERNED WITH C CONSTRUCTIONS OF MG THE TEICHM¨ULLER APPROACH THE HODGE THEORY APPROACH THE GEOMETRIC INVARIANT THEORY (G I T ) APPROACH D GEOMETRIC AND TOPOLOGICAL PROPERTIES BASIC PROPERTIES LOCAL PROPERTIES COMPLETE SUBVARIETIES OF MG COHOMOLOGY OF MG: HARER’S THEOREMS COHOMOLOGY OF THE UNIVERSAL CURVE COHOMOLOGY OF HILBERT SCHEMES STRUCTURE OF THE TAUTOLOGICAL RING WITTEN’S CONJECTURES AND KONTSEVICH’S THEOREM E MODULI SPACES OF STABLE MAPS TECHNIQUES A BASIC FACTS ABOUT NODAL AND STABLE CURVES DUALIZING SHEAVES AUTOMORPHISMS B DEFORMATION THEORY OVERVIEW DEFORMATIONS OF SMOOTH CURVES VARIATIONS ON THE BASIC DEFORMATION THEORY PLAN UNIVERSAL DEFORMATIONS OF STABLE CURVES DEFORMATIONS OF MAPS C STABLE REDUCTION RESULTS EXAMPLES D INTERLUDE: CALCULATIONS ON THE MODULI STACK DIVISOR CLASSES ON THE MODULI STACK EXISTENCE OF TAUTOLOGICAL FAMILIES E GROTHENDIECK-RIEMANN-ROCH AND PORTEOUS GROTHENDIECK-RIEMANN-ROCH CHERN CLASSES OF THE HODGE BUNDLE CHERN CLASS OF THE TANGENT BUNDLE PORTEOUS’ FORMULA THE HYPERELLIPTIC LOCUS IN M3 RELATIONS AMONGST STANDARD COHOMOLOGY CLASSES DIVISOR CLASSES ON HILBERT SCHEMES F TEST CURVES: THE HYPERELLIPTIC LOCUS IN M3 BEGUN G ADMISSIBLE COVERS H THE HYPERELLIPTIC LOCUS IN M3 COMPLETED CONSTRUCTION OF MG A BACKGROUND ON GEOMETRIC INVARIANT THEORY THE G I T STRATEGY FINITE GENERATION OF AND SEPARATION BY INVARIANTS THE NUMERICAL CRITERION STABILITY OF PLANE CURVES B STABILITY OF HILBERT POINTS OF SMOOTH CURVES THE NUMERICAL CRITERION FOR HILBERT POINTS GIESEKER’S CRITERION STABILITY OF SMOOTH CURVES C CONSTRUCTION OF MG VIA THE POTENTIAL STABILITY THEOREM THE PLAN OF THE CONSTRUCTION AND A FEW COROLLARIES THE POTENTIAL STABILITY THEOREM LIMIT LINEAR SERIES AND BRILL-NOETHER THEORY A INTRODUCTORY REMARKS ON DEGENERATIONS B LIMITS OF LINE BUNDLES C LIMITS OF LINEAR SERIES: MOTIVATION AND EXAMPLES D LIMIT LINEAR SERIES: DE?NITIONS AND APPLICATIONS LIMIT LINEAR SERIES SMOOTHING LIMIT LINEAR SERIES LIMITS OF CANONICAL SERIES AND WEIERSTRASS POINTS E LIMIT LINEAR SERIES ON ?AG CURVES INEQUALITIES ON VANISHING SEQUENCES THE CASE ? = PROOF OF THE GIESEKER-PETRI THEOREM GEOMETRY OF MODULI SPACES: SELECTED RESULTS A IRREDUCIBILITY OF THE MODULI SPACE OF CURVES B DIAZ’ THEOREM THE IDEA: STRATIFYING THE MODULI SPACE THE PROOF C MODULI OF HYPERELLIPTIC CURVES FIDDLING AROUND THE CALCULATION FOR AN (ALMOST) ARBITRARY FAMILY THE PICARD GROUP OF THE HYPERELLIPTIC LOCUS D AMPLE DIVISORS ON MG AN INEQUALITY FOR GENERICALLY HILBERT STABLE FAMILIES PROOF OF THE THEOREM AN INEQUALITY FOR FAMILIES OF POINTED CURVES AMPLE DIVISORS ON MG E IRREDUCIBILITY OF THE SEVERI VARIETIES INITIAL REDUCTIONS ANALYZING A DEGENERATION AN EXAMPLE COMPLETING THE ARGUMENT F KODAIRA DIMENSION OF MG WRITING DOWN GENERAL CURVES BASIC IDEAS PULLING BACK THE DIVISORS DRS DIVISORS ON MG THAT MISS J(M2,1 \ W) DIVISORS ON MG THAT MISS I(M0,G) FURTHER DIVISOR CLASS CALCULATIONS CURVES DE?NED OVER Q