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Contraction morphism

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536: 115: 302: 213: 371: 341: 253: 57: 432: 400: 577: 62: 570: 500: 451: 118: 601: 563: 164: 596: 266: 177: 492: 350: 551: 314: 222: 140:, any surjective projective morphism is a contraction morphism followed by a finite morphism. 30: 510: 405: 376: 8: 160: 137: 25: 543: 456: 130: 17: 516: 496: 148: 506: 547: 590: 255:= the real vector space of numerical equivalence classes of real 1-cycles on 144: 439: 126: 59:
between normal projective varieties (or projective schemes) such that
535: 117:
or, equivalently, the geometric fibers are all connected (
110:{\displaystyle f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{Y}} 521:
Positivity in Algebraic Geometry I: Classical Setting
408: 379: 353: 317: 269: 225: 180: 65: 33: 491:, Cambridge Tracts in Mathematics, vol. 134, 426: 394: 365: 335: 296: 247: 207: 109: 51: 215:the closure of the span of irreducible curves on 588: 438:gives rise to such a contraction morphism (cf. 571: 578: 564: 489:Birational geometry of algebraic varieties 486: 475: 311:, if it exists, is a contraction morphism 154: 159:The following perspective is crucial in 487:Kollár, János; Mori, Shigefumi (1998), 589: 347:such that for each irreducible curve 530: 434:. The basic question is which face 306:contraction morphism associated to 297:{\displaystyle {\overline {NS}}(X)} 208:{\displaystyle {\overline {NS}}(X)} 13: 96: 79: 14: 613: 452:Castelnuovo's contraction theorem 121:). It is also commonly called an 534: 119:Zariski's connectedness theorem 469: 415: 409: 389: 383: 327: 291: 285: 242: 236: 202: 196: 43: 1: 462: 550:. You can help Knowledge by 280: 191: 174:be a projective variety and 165:Mori's minimal model program 7: 445: 343:to some projective variety 10: 618: 529: 493:Cambridge University Press 402:is a point if and only if 366:{\displaystyle C\subset X} 125:, as it is an analog of a 602:Algebraic geometry stubs 336:{\displaystyle f:X\to Y} 248:{\displaystyle N_{1}(X)} 52:{\displaystyle f:X\to Y} 546:–related article is a 476:Kollár & Mori 1998 428: 396: 367: 337: 298: 249: 209: 155:Birational perspective 111: 53: 429: 427:{\displaystyle \in F} 397: 368: 338: 299: 250: 210: 123:algebraic fiber space 112: 54: 406: 395:{\displaystyle f(C)} 377: 351: 315: 267: 223: 178: 63: 31: 22:contraction morphism 161:birational geometry 138:Stein factorization 26:projective morphism 597:Algebraic geometry 544:algebraic geometry 478:, Definition 1.25. 457:Flip (mathematics) 424: 392: 363: 333: 294: 245: 205: 163:(in particular in 131:algebraic topology 107: 49: 18:algebraic geometry 559: 558: 517:Robert Lazarsfeld 502:978-0-521-63277-5 283: 194: 149:Mori fiber spaces 143:Examples include 609: 580: 573: 566: 538: 531: 513: 479: 473: 433: 431: 430: 425: 401: 399: 398: 393: 372: 370: 369: 364: 342: 340: 339: 334: 303: 301: 300: 295: 284: 279: 271: 254: 252: 251: 246: 235: 234: 214: 212: 211: 206: 195: 190: 182: 116: 114: 113: 108: 106: 105: 100: 99: 89: 88: 83: 82: 75: 74: 58: 56: 55: 50: 24:is a surjective 617: 616: 612: 611: 610: 608: 607: 606: 587: 586: 585: 584: 527: 503: 483: 482: 474: 470: 465: 448: 407: 404: 403: 378: 375: 374: 352: 349: 348: 316: 313: 312: 272: 270: 268: 265: 264: 259:. Given a face 230: 226: 224: 221: 220: 183: 181: 179: 176: 175: 157: 101: 95: 94: 93: 84: 78: 77: 76: 70: 66: 64: 61: 60: 32: 29: 28: 12: 11: 5: 615: 605: 604: 599: 583: 582: 575: 568: 560: 557: 556: 539: 525: 524: 514: 501: 481: 480: 467: 466: 464: 461: 460: 459: 454: 447: 444: 423: 420: 417: 414: 411: 391: 388: 385: 382: 362: 359: 356: 332: 329: 326: 323: 320: 293: 290: 287: 282: 278: 275: 244: 241: 238: 233: 229: 204: 201: 198: 193: 189: 186: 156: 153: 145:ruled surfaces 104: 98: 92: 87: 81: 73: 69: 48: 45: 42: 39: 36: 9: 6: 4: 3: 2: 614: 603: 600: 598: 595: 594: 592: 581: 576: 574: 569: 567: 562: 561: 555: 553: 549: 545: 540: 537: 533: 532: 528: 522: 518: 515: 512: 508: 504: 498: 494: 490: 485: 484: 477: 472: 468: 458: 455: 453: 450: 449: 443: 441: 437: 421: 418: 412: 386: 380: 360: 357: 354: 346: 330: 324: 321: 318: 310: 309: 288: 276: 273: 262: 258: 239: 231: 227: 218: 199: 187: 184: 173: 168: 166: 162: 152: 150: 146: 141: 139: 134: 132: 128: 124: 120: 102: 90: 85: 71: 67: 46: 40: 37: 34: 27: 23: 19: 552:expanding it 541: 526: 520: 488: 471: 440:cone theorem 435: 344: 307: 305: 260: 256: 216: 171: 169: 158: 142: 135: 122: 21: 15: 127:fiber space 591:Categories 463:References 419:∈ 358:⊂ 328:→ 281:¯ 192:¯ 72:∗ 44:→ 446:See also 511:1658959 136:By the 523:(2004) 509:  499:  304:, the 542:This 548:stub 497:ISBN 170:Let 147:and 20:, a 442:). 263:of 219:in 167:). 129:in 16:In 593:: 519:, 507:MR 505:, 495:, 373:, 151:. 133:. 579:e 572:t 565:v 554:. 436:F 422:F 416:] 413:C 410:[ 390:) 387:C 384:( 381:f 361:X 355:C 345:Y 331:Y 325:X 322:: 319:f 308:F 292:) 289:X 286:( 277:S 274:N 261:F 257:X 243:) 240:X 237:( 232:1 228:N 217:X 203:) 200:X 197:( 188:S 185:N 172:X 103:Y 97:O 91:= 86:X 80:O 68:f 47:Y 41:X 38:: 35:f

Index

algebraic geometry
projective morphism
Zariski's connectedness theorem
fiber space
algebraic topology
Stein factorization
ruled surfaces
Mori fiber spaces
birational geometry
Mori's minimal model program
cone theorem
Castelnuovo's contraction theorem
Flip (mathematics)
Kollár & Mori 1998
Cambridge University Press
ISBN
978-0-521-63277-5
MR
1658959
Robert Lazarsfeld
Stub icon
algebraic geometry
stub
expanding it
v
t
e
Categories
Algebraic geometry
Algebraic geometry stubs

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