THE GEOMETRY OF ALGEBRAIC FERMI CURVES
CONTENTS INTRODUCTION WHY YOU WANT TO READ THIS BOOK WHY WE WROTE THIS BOOK WHAT’S WITH THE TITLE? WHAT’S IN THIS BOOK ?EXERCISES AND HINTS ?RELATION OF THIS BOOK TO OTHER TEXTS PREREQUISITES, NOTATION AND CONVENTIONS ?COMMUTATIVE ALGEBRA ?PROJECTIVE GEOMETRY ?SHEAVES AND COHOMOLOGY CHAPTER 1 LINEAR SERIES AND MORPHISMS TO PROJECTIVE SPACE DIVISORS DIVISORS AND RATIONAL FUNCTIONS ?GENERALIZATIONS ?DIVISORS OF FUNCTIONS ?INVERTIBLE SHEAVES ?INVERTIBLE SHEAVES AND LINE BUNDLES LINEAR SERIES AND MAPS TO PROJECTIVE SPACE THE GEOMETRY OF LINEAR SERIES ?AN UPPER BOUND ON ?0(?) ?INCOMPLETE LINEAR SERIES ?SUMS OF LINEAR SERIES ?WHICH LINEAR SERIES DEFINE EMBEDDINGS? EXERCISES CHAPTER 2 THE RIEMANN–ROCH THEOREM HOW MANY SECTIONS? ?RIEMANN–ROCH WITHOUT DUALITY THE MOST INTERESTING LINEAR SERIES ?THE ADJUNCTION FORMULA ?HURWITZ’S THEOREM RIEMANN–ROCH WITH DUALITY ?RESIDUES ?ARITHMETIC GENUS AND GEOMETRIC GENUS THE CANONICAL MORPHISM ?GEOMETRIC RIEMANN–ROCH ?LINEAR SERIES ON A HYPERELLIPTIC CURVE CLIFFORD’S THEOREM CURVES ON SURFACES ?THE INTERSECTION PAIRING ?THE RIEMANN–ROCH THEOREM FOR SMOOTH SURFACES ?BLOWUPS OF SMOOTH SURFACES QUADRICS IN ?3 AND THE CURVES THEY CONTAIN ?THE CLASSIFICATION OF QUADRICS ?SOME CLASSES OF CURVES ON QUADRICS CHAPTER 3 CURVES OF GENUS RATIONAL NORMAL CURVES OTHER RATIONAL CURVES ?SMOOTH RATIONAL QUARTICS ?SOME OPEN PROBLEMS ABOUT RATIONAL CURVES THE COHEN–MACAULAY PROPERTY CHAPTER 4 SMOOTH PLANE CURVES AND CURVES OF GENUS RIEMANN, CLEBSCH, BRILL AND NOETHER SMOOTH PLANE CURVES DIFFERENTIALS ON A SMOOTH PLANE CURVE LINEAR SERIES ON A SMOOTH PLANE CURVE THE CAYLEY–BACHARACH–MACAULAY THEOREM CURVES OF GENUS 1 AND THE GROUP LAW OF AN ELLIPTIC CURVE LOW DEGREE DIVISORS ON CURVES OF GENUS ?THE DIMENSION OF FAMILIES ?PLANE CUBICS CHAPTER 5 JACOBIANS SYMMETRIC PRODUCTS AND THE UNIVERSAL DIVISOR ?FINITE GROUP QUOTIENTS THE PICARD VARIETIES JACOBIANS ABEL’S THEOREM THE ??+ 3 THEOREM THE SCHEMES ???? ??(??) EXAMPLES IN LOW GENUS ?GENUS MARTENS’ THEOREM CHAPTER 6 HYPERELLIPTIC CURVES AND CURVES OF GENUS 2 AND HYPERELLIPTIC CURVES ?THE EQUATION OF A HYPERELLIPTIC CURVE ?DIFFERENTIALS ON A HYPERELLIPTIC CURVE BRANCHED COVERS WITH SPECIFIED BRANCHING CURVES OF GENUS ?THE DIMENSION OF THE FAMILY OF GENUS 2 CURVES ?OTHER REPRESENTATIONS OF A CURVE OF GENUS THETA CHARACTERISTICS ?COUNTING THETA CHARACTERISTICS (PROOF OF THEOREM 6 8) CHAPTER 7 FINE MODULI SPACES WHAT IS A MODULI PROBLEM? WHAT IS A SOLUTION TO A MODULI PROBLEM? HILBERT SCHEMES ?7 3 1 THE TANGENT SPACE TO THE HILBERT SCHEME PARAMETRIZING TWISTED CUBICS CONSTRUCTION OF THE HILBERT SCHEME IN GENERAL GRASSMANNIANS EQUATIONS DEFINING THE HILBERT SCHEME BOUNDING THE NUMBER OF MAPS BETWEEN CURVES CHAPTER 8 MODULI OF CURVES ???1 IS A COARSE MODULI SPACE ?THE GOOD NEWS HIGHER GENUS ?STABLE, SEMISTABLE, UNSTABLE STABLE CURVES ?HOW WE DEAL WITH THE FACT THAT ????IS NOT FINE CAN ONE WRITE DOWN A GENERAL CURVE OF GENUS ??? HURWITZ SPACES ?THE DIMENSION OF ???? ?IRREDUCIBILITY OF ???? THE SEVERI VARIETY ?LOCAL GEOMETRY OF THE SEVERI VARIETY CHAPTER 9 CURVES OF GENUS 4 AND ?THE CANONICAL MODEL ?MAPS TO PROJECTIVE SPACE CANONICAL CURVES OF GENUS ?FIRST CASE: THE INTERSECTION OF THE QUADRICS HAS DIMENSION ?SECOND CASE: THE INTERSECTION OF THE QUADRICS IS A SURFACE CHAPTER 10 HYPERPLANE SECTIONS OF A CURVE LINEARLY GENERAL POSITION CASTELNUOVO’S THEOREM ?PROOF OF CASTELNUOVO’S BOUND ?CONSEQUENCES AND SPECIAL CASES OTHER APPLICATIONS OF LINEARLY GENERAL POSITION ?EXISTENCE OF GOOD PROJECTIONS ?THE CASE OF EQUALITY IN MARTENS’ THEOREM ?THE ??+ 2 THEOREM CHAPTER 11 MONODROMY OF HYPERPLANE SECTIONS UNIFORM POSITION AND MONODROMY ?THE MONODROMY GROUP OF A GENERICALLY FINITE MORPHISM ?UNIFORM POSITION FLEXES AND BITANGENTS ARE ISOLATED ?NOT EVERY TANGENT LINE IS TANGENT AT A FLEX ?NOT EVERY TANGENT IS BITANGENT PROOF OF THE UNIFORM POSITION LEMMA ?UNIFORM POSITION FOR HIGHER-DIMENSIONAL VARIETIES APPLICATIONS OF UNIFORM POSITION ?IRREDUCIBILITY OF FIBER POWERS ?NUMERICAL UNIFORM POSITION ?NODES OF PLANE CURVES CHAPTER 12 BRILL–NOETHER THEORY AND APPLICATIONS TO GENUS WHAT LINEAR SERIES EXIST? BRILL–NOETHER THEORY A BRILL–NOETHER INEQUALITY REFINEMENTS OF THEBRILL–NOETHER THEOREM LINEAR SERIES ON CURVES OF GENUS GENERAL CURVES OF GENUS DEL PEZZO SURFACES THE CANONICAL IMAGE OF A GENERAL CURVE OF GENUS CLASSIFICATION OF CURVES OF GENUS ?|??| HAS A BASEPOINT ???IS NOT TRIGONAL AND THE IMAGE OF ????IS TWO-TO-ONE ONTO A PLANE CURVE OF DEGREE 3 CHAPTER 13 INFLECTION POINTS INFLECTION POINTS, PLÜCKER FORMULAS AND WEIERSTRASS POINTS ?DEFINITIONS ?THE PLÜCKER FORMULA ?FLEXES OF PLANE CURVES ?WEIERSTRASS POINTS ?ANOTHER CHARACTERIZATION OF WEIERSTRASS POINTS FINITENESS OF THE AUTOMORPHISM GROUP CURVES WITH AUTOMORPHISMS ARE SPECIAL ?SCHUBERT CYCLES ?SPECIAL SCHUBERT CYCLES AND PIERI’S FORMULA ?CONCLUSION CHAPTER 14 PROOF OF THE BRILL–NOETHER THEOREM CASTELNUOVO’S APPROACH ?UPPER BOUND ON THE CODIMENSION OF ???? ??(??) SPECIALIZING TO A ??-CUSPIDAL CURVE ?CONSTRUCTING CURVES WITH CUSPS ?SMOOTHING A CUSPIDAL CURVE THE FAMILY OF PICARD VARIETIES ?THE PICARD VARIETY OF A CUSPIDAL CURVE ?THE RELATIVE PICARD VARIETY ?LIMITS OF INVERTIBLE SHEAVES PUTTING IT ALL TOGETHER ?NONEXISTENCE ?EXISTENCE BRILL–NOETHER WITH INFLECTION CHAPTER 15 USING A SINGULAR PLANE MODEL NODAL PLANE CURVES DIFFERENTIALS ON A NODAL PLANE CURVE LINEAR SERIES ON A NODAL PLANE CURVE ARBITRARY PLANE CURVES ?THE CONDUCTOR IDEAL AND LINEAR SERIES ON THE NORMALIZATION ?DIFFERENTIALS CHAPTER 16 LINKAGE AND THE CANONICAL SHEAVES OF SINGULAR CURVES INTRODUCTION LINKAGE OF TWISTED CUBICS DEGREE AND GENUS OF LINKED CURVES ?DUALIZING SHEAVES FOR SINGULAR CURVES THE CONSTRUCTION OF DUALIZING SHEAVES THE LINKAGE EQUIVALENCE RELATION COMPARING THE CANONICAL SHEAF WITH THAT OF THE NORMALIZATION A GENERAL RIEMANN–ROCH THEOREM 16 10 EXERCISES ?ROPES AND RIBBONS ?GENERAL ADJUNCTION CHAPTER 17 SCROLLS AND THE CURVES THEY CONTAIN SOME CLASSICAL GEOMETRY 1-GENERIC MATRICES AND THE EQUATIONS OF SCROLLS SCROLLS AS IMAGES OF PROJECTIVE BUNDLES CURVES ON A 2-DIMENSIONAL SCROLL ?FINDING A SCROLL CONTAINING A GIVEN CURVE ?FINDING CURVES ON A GIVEN SCROLL CHAPTER 18 FREE RESOLUTIONS AND CANONICAL CURVES FREE RESOLUTIONS ?THE CLASSIFICATION OF 1-GENERIC 2 × ??MATRICES ?HOW TO LOOK AT A RESOLUTION ?WHEN IS A FINITE FREE COMPLEX A RESOLUTION? DEPTH AND THE COHEN–MACAULAY PROPERTY ?THE GORENSTEIN PROPERTY THE EAGON–NORTHCOTT COMPLEX ?THE CASE RANK G = 1: THE KOSZUL COMPLEX ?THE CASE RANK F = RANK G + 1: THE HILBERT–BURCH COMPLEX ?THE HILBERT–BURCH THEOREM ?THE GENERAL CASE OF THE EAGON–NORTHCOTT COMPLEX GREEN’S CONJECTURE CHAPTER 19 HILBERT SCHEMES DEGREE EXTRANEOUS COMPONENTS THE EXPECTED DIMENSION OF ?? ??,??,?? SOME OPEN PROBLEMS ?BRILL–NOETHER IN LOW CODIMENSION ?MAXIMALLY SPECIAL CURVES ?RIGID CURVES? DEGREE 8, GENUS 19 10 DEGREE 9, GENUS 19 11 ESTIMATING THE DIMENSION OF THE RESTRICTED HILBERT SCHEMES USING THE BRILL–NOETHER THEOREM 19 12 EXERCISES