PROJECTIVE GEOMETRY AND ALGEBRAIC STRUCTURES

By EMMA WILSON

CONTENTS 1 BASICS ON COMMUTATIVE ALGEBRA IDEALS AND OPERATIONS ON IDEALS UFDS AND PIDS POLYNOMIAL RINGS POLYNOMIALS IN D[X], WHERE D A UFD THE CASE D = K A ?ELD RESULTANT OF TWO POLYNOMIALS IN D[X] RESULTANT IN D[X1, , XN] AND ELIMINATION NOETHERIAN RINGS AND THE HILBERT BASIS THEOREM R-MODULES, R-ALGEBRAS AND FINITENESS CONDITIONS INTEGRALITY ZARISKI’S LEMMA TRANSCENDENCE DEGREE TENSOR PRODUCTS OF R-MODULES AND OF R-ALGEBRAS RESTRICTION AND EXTENSION OF SCALARS TENSOR PRODUCT OF ALGEBRAS GRADED RINGS AND MODULES, HOMOGENEOUS IDEALS HOMOGENEOUS POLYNOMIALS GRADED MODULES AND GRADED MORPHISMS LOCALIZATION LOCAL RINGS AND LOCALIZATION KRULL-DIMENSION OF A RING EXERCISES 2 ALGEBRAIC A?NE SETS ALGEBRAIC A?NE SETS AND IDEALS HILBERT “NULLSTELLENSATZ” SOME CONSEQUENCES OF HILBERT “NULLSTELLENSATZ” AND OF ELIMINATION THEORY STUDY’S PRINCIPLE INTERSECTIONS OF A?NE PLANE CURVES 3 ALGEBRAIC PROJECTIVE SETS ALGEBRAIC PROJECTIVE SETS HOMOGENEOUS “HILBERT NULLSTELLENSATZ” FUNDAMENTAL EXAMPLES AND REMARKS POINTS COORDINATE LINEAR SUBSPACES HYPERPLANES AND THE DUAL PROJECTIVE SPACE FUNDAMENTAL A?NE OPEN SETS (OR A?NE CHARTS) OF PN PROJECTIVE CLOSURE OF A?NE SETS PROJECTIVE SUBSPACES AND THEIR IDEALS PROJECTIVE AND A?NE SUBSPACES HOMOGRAPHIES, PROJECTIVITIES AND A?NITIES PROJECTIVE CONES PROJECTIVE HYPERSURFACES AND PROJECTIVE CLOSURE OF A?NE HYPERSURFACES PROPER CLOSED SUBSETS OF P2 A?NE AND PROJECTIVE TWISTED CUBICS 4 TOPOLOGICAL PROPERTIES AND ALGEBRAIC VARIETIES IRREDUCIBLE TOPOLOGICAL SPACES COORDINATE RINGS, IDEALS AND IRREDUCIBILITY ALGEBRAIC VARIETIES NOETHERIAN SPACES: IRREDUCIBLE COMPONENTS COMBINATORIAL DIMENSION 5 REGULAR AND RATIONAL FUNCTIONS ON ALGEBRAIC VARIETIES BASICS ON SHEAVES REGULAR FUNCTIONS RATIONAL FUNCTIONS CONSEQUENCES OF THE FUNDAMENTAL THEOREM ON REGULAR AND RATIONAL FUNCTIONS EXAMPLES 6 MORPHISMS OF ALGEBRAIC VARIETIES MORPHISMS MORPHISMS WITH (QUASI) A?NE TARGET MORPHISMS WITH (QUASI) PROJECTIVE TARGET LOCAL PROPERTIES OF MORPHISMS: A?NE OPEN COVERINGS OF AN ALGEBRAIC VARIETY VERONESE MORPHISM: DIVISORS AND LINEAR SYSTEMS VERONESE MORPHISM AND CONSEQUENCES DIVISORS AND LINEAR SYSTEMS 7 PRODUCTS OF ALGEBRAIC VARIETIES PRODUCTS OF A?NE VARIETIES PRODUCTS OF PROJECTIVE VARIETIES SEGRE MORPHISM AND THE PRODUCT OF PROJECTIVE SPACES PRODUCTS OF ALGEBRAIC VARIETIES PRODUCTS OF MORPHISMS DIAGONALS, GRAPH OF A MORPHISM AND FIBER-PRODUCTS 8 RATIONAL MAPS OF ALGEBRAIC VARIETIES RATIONAL AND BIRATIONAL MAPS SOME PROPERTIES AND SOME EXAMPLES OF (BI)RATIONAL MAPS UNIRATIONAL AND RATIONAL VARIETIES STEREOGRAPHIC PROJECTION OF A RANK-FOUR QUADRIC SURFACE MONOIDS BLOW-UP OF PN AT A POINT BLOW-UPS AND RESOLUTION OF SINGULARITIES 9 COMPLETENESS OF PROJECTIVE VARIETIES COMPLETE ALGEBRAIC VARIETIES THE MAIN THEOREM OF ELIMINATION THEORY CONSEQUENCES OF THE MAIN THEOREM OF ELIMINATION THEORY 10 DIMENSION OF ALGEBRAIC VARIETIES DIMENSION OF AN ALGEBRAIC VARIETY COMPARISON ON VARIOUS DE?NITIONS OF “DIMENSION” DIMENSION AND INTERSECTIONS COMPLETE INTERSECTIONS 11 FIBER-DIMENSION: SEMICONTINUITY FIBERS OF A DOMINANT MORPHISM SEMICONTINUITY 12 TANGENT SPACES: SMOOTHNESS OF ALGEBRAIC VARIETIES TANGENT SPACE AT A POINT OF AN A?NE VARIETY: SMOOTHNESS TANGENT SPACE AT A POINT OF A PROJECTIVE VARIETY: SMOOTHNESS ZARISKI TANGENT SPACE OF AN ALGEBRAIC VARIETY: INTRINSIC DE?NITION OF SMOOTHNESS SOLUTIONS TO EXERCISES