## Algebraic Geometry: A First CourseThis book is based on one-semester courses given at Harvard in 1984, at Brown in 1985, and at Harvard in 1988. It is intended to be, as the title suggests, a first introduction to the subject. Even so, a few words are in order about the purposes of the book. Algebraic geometry has developed tremendously over the last century. During the 19th century, the subject was practiced on a relatively concrete, down-to-earth level; the main objects of study were projective varieties, and the techniques for the most part were grounded in geometric constructions. This approach flourished during the middle of the century and reached its culmination in the work of the Italian school around the end of the 19th and the beginning of the 20th centuries. Ultimately, the subject was pushed beyond the limits of its foundations: by the end of its period the Italian school had progressed to the point where the language and techniques of the subject could no longer serve to express or carry out the ideas of its best practitioners. |

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### Contents

V | 3 |

VI | 5 |

VII | 6 |

VIII | 8 |

XI | 9 |

XII | 10 |

XIII | 11 |

XV | 12 |

CXIII | 136 |

CXV | 138 |

CXVI | 142 |

CXVIII | 143 |

CXIX | 146 |

CXX | 148 |

CXXII | 149 |

CXXIII | 151 |

XVI | 14 |

XVII | 16 |

XIX | 17 |

XX | 18 |

XXI | 20 |

XXII | 21 |

XXIII | 24 |

XXV | 25 |

XXVI | 27 |

XXVII | 28 |

XXVIII | 29 |

XXIX | 30 |

XXXI | 32 |

XXXII | 33 |

XXXIII | 34 |

XXXIV | 37 |

XXXV | 38 |

XXXVI | 39 |

XXXVII | 41 |

XXXVIII | 42 |

XXXIX | 43 |

XL | 44 |

XLI | 45 |

XLII | 47 |

XLIII | 48 |

XLIV | 50 |

XLV | 51 |

XLVI | 53 |

XLVII | 54 |

XLVIII | 55 |

XLIX | 56 |

L | 57 |

LI | 60 |

LII | 63 |

LIII | 66 |

LIV | 67 |

LV | 68 |

LVII | 69 |

LVIII | 70 |

LX | 72 |

LXI | 73 |

LXII | 75 |

LXIII | 77 |

LXIV | 78 |

LXV | 79 |

LXVII | 80 |

LXVIII | 81 |

LXIX | 82 |

LXX | 84 |

LXXI | 85 |

LXXII | 87 |

LXXIII | 88 |

LXXIV | 89 |

LXXV | 90 |

LXXVII | 91 |

LXXVIII | 92 |

LXXIX | 93 |

LXXX | 94 |

LXXXI | 95 |

LXXXIII | 96 |

LXXXIV | 97 |

LXXXV | 98 |

LXXXVI | 99 |

LXXXVIII | 100 |

LXXXIX | 103 |

XC | 105 |

XCI | 109 |

XCII | 111 |

XCIII | 112 |

XCV | 114 |

XCVI | 115 |

XCVII | 116 |

C | 117 |

CI | 119 |

CII | 120 |

CIII | 121 |

CIV | 122 |

CVI | 123 |

CVII | 124 |

CVIII | 125 |

CIX | 126 |

CXI | 127 |

CXII | 133 |

CXXIV | 152 |

CXXV | 155 |

CXXVII | 156 |

CXXVIII | 157 |

CXXIX | 158 |

CXXX | 159 |

CXXXI | 161 |

CXXXIV | 163 |

CXXXV | 166 |

CXXXVII | 168 |

CXXXVIII | 170 |

CXXXIX | 171 |

CXL | 172 |

CXLI | 174 |

CXLII | 177 |

CXLIII | 181 |

CXLIV | 184 |

CXLV | 186 |

CXLVI | 188 |

CXLVII | 189 |

CXLVIII | 190 |

CXLIX | 193 |

CL | 195 |

CLI | 196 |

CLII | 200 |

CLIII | 202 |

CLIV | 203 |

CLV | 204 |

CLVII | 206 |

CLVIII | 208 |

CLIX | 209 |

CLX | 211 |

CLXI | 213 |

CLXII | 214 |

CLXIII | 215 |

CLXIV | 216 |

CLXV | 219 |

CLXVI | 222 |

CLXVII | 224 |

CLXVIII | 227 |

CLXIX | 229 |

CLXX | 231 |

CLXXI | 233 |

CLXXII | 234 |

CLXXIII | 235 |

CLXXIV | 237 |

CLXXV | 239 |

CLXXVI | 240 |

CLXXVII | 241 |

CLXXVIII | 242 |

CLXXIX | 243 |

CLXXX | 244 |

CLXXXI | 245 |

CLXXXII | 247 |

CLXXXIII | 251 |

CLXXXIV | 256 |

CLXXXV | 258 |

CLXXXVI | 260 |

CLXXXVII | 264 |

CLXXXVIII | 266 |

CLXXXIX | 268 |

CXC | 273 |

CXCI | 275 |

CXCII | 278 |

CXCIII | 279 |

CXCIV | 282 |

CXCV | 283 |

CXCVI | 284 |

CXCVII | 285 |

CXCVIII | 287 |

CXCIX | 289 |

CC | 290 |

CCI | 291 |

CCII | 293 |

CCIII | 295 |

CCV | 296 |

CCVI | 297 |

CCVII | 299 |

CCVIII | 301 |

CCIX | 308 |

314 | |

317 | |

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### Common terms and phrases

affine variety automorphism birational blow-up closure codimension common zero locus complete intersection cone conic curve construction containing coordinate ring deduce defined definition describe determinantal variety disjoint dual exactly Example Exercise fact Fano variety fc-planes fiber finite Gauss map given graph Grassmannian Hilbert polynomial homogeneous polynomials hyperplane section ideal incidence correspondence inverse image isomorphic l)-plane Lecture linear forms linear space linear subspace matrix Note open subset pair parameter space parametrized plane conic point p e polynomials of degree projection map projective space projective tangent space projective variety projectively equivalent proof Proposition quadric hypersurface quadric Q quadric surface quotient rank rational map rational normal curve rational normal scroll regular functions regular map secant variety Segre variety Show smooth point smooth quadric span subvariety tangent line Theorem topology TP(X transversely twisted cubic twisted cubic curve union vanishing vector space Veronese map Veronese surface Zariski tangent space