INTRODUCTION TO ALGEBRAIC GEOMETRY

By J. HIUNY

CONTENTS GUIDING PROBLEMS IMPLICITIZATION IDEAL MEMBERSHIP INTERPOLATION EXERCISES DIVISION ALGORITHM AND GR¨OBNER BASES MONOMIAL ORDERS GR¨OBNER BASES AND THE DIVISION ALGORITHM NORMAL FORMS EXISTENCE AND CHAIN CONDITIONS BUCHBERGER’S CRITERION SYZYGIES AFFINE VARIETIES IDEALS AND VARIETIES CLOSED SETS AND THE ZARISKI TOPOLOGY COORDINATE RINGS AND MORPHISMS RATIONAL MAPS RESOLVING RATIONAL MAPS RATIONAL AND UNIRATIONAL VARIETIES ELIMINATION PROJECTIONS AND GRAPHS IMAGES OF RATIONAL MAPS SECANT VARIETIES, JOINS, AND SCROLLS RESULTANTS COMMON ROOTS OF UNIVARIATE POLYNOMIALS THE RESULTANT AS A FUNCTION OF THE ROOTS RESULTANTS AND ELIMINATION THEORY REMARKS ON HIGHER-DIMENSIONAL RESULTANTS IRREDUCIBLE VARIETIES EXISTENCE OF THE DECOMPOSITION IRREDUCIBILITY AND DOMAINS DOMINANT MORPHISMS ALGORITHMS FOR INTERSECTIONS OF IDEALS DOMAINS AND ?ELD EXTENSIONS NULLSTELLENSATZ STATEMENT OF THE NULLSTELLENSATZ CLASSI?CATION OF MAXIMAL IDEALS TRANSCENDENCE BASES INTEGRAL ELEMENTS PROOF OF NULLSTELLENSATZ I APPLICATIONS DIMENSION PRIMARY DECOMPOSITION IRREDUCIBLE IDEALS QUOTIENT IDEALS PRIMARY IDEALS UNIQUENESS OF PRIMARY DECOMPOSITION AN APPLICATION TO RATIONAL MAPS PROJECTIVE GEOMETRY INTRODUCTION TO PROJECTIVE SPACE HOMOGENIZATION AND DEHOMOGENIZATION PROJECTIVE VARIETIES EQUATIONS FOR PROJECTIVE VARIETIES PROJECTIVE NULLSTELLENSATZ MORPHISMS OF PROJECTIVE VARIETIES PRODUCTS ABSTRACT VARIETIES PROJECTIVE ELIMINATION THEORY 101 HOMOGENEOUS EQUATIONS REVISITED 102 PROJECTIVE ELIMINATION IDEALS 103 COMPUTING THE PROJECTIVE ELIMINATION IDEAL 104 IMAGES OF PROJECTIVE VARIETIES ARE CLOSED 105 FURTHER ELIMINATION RESULTS 106 EXERCISES PARAMETRIZING LINEAR SUBSPACES 111 DUAL PROJECTIVE SPACES 112 TANGENT SPACES AND DUAL VARIETIES 113 GRASSMANNIANS: ABSTRACT APPROACH 114 EXTERIOR ALGEBRA 115 GRASSMANNIANS AS PROJECTIVE VARIETIES 116 EQUATIONS FOR THE GRASSMANNIAN 117 EXERCISES HILBERT POLYNOMIALS AND THE BEZOUT THEOREM 121 HILBERT FUNCTIONS DE?NED 122 HILBERT POLYNOMIALS AND ALGORITHMS 123 INTERSECTION MULTIPLICITIES 124 BEZOUT THEOREM 125 INTERPOLATION PROBLEMS REVISITED 126 CLASSI?CATION OF PROJECTIVE VARIETIES 127 EXERCISES