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Zariski geometry

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543: 51:, but richer in closed sets defined by equations that mix two sets of variables. The result described gives that a very definite meaning, applying to 496:
Geometrically this says there are enough curves to separate points (I), and to connect points (K); and that such curves can be taken from a single
106: 535: 504: 580: 585: 56: 265: 213: 575: 76: 226:(D) There is a uniform bound on the number of elements of a fiber in a projection of any closed set in 44: 145:
defined by projecting out certain coordinates and setting others as constants are all continuous.
497: 552: 521: 52: 48: 36: 514: 511: 167: 40: 28: 569: 32: 557: 20: 524:
is isomorphic to the given one. In short, the geometry can be algebraized.
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Then Hrushovski and Zilber prove that under these conditions there is an
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Some standard terminology for Noetherian spaces will now be assumed.
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coordinates are set equal, has all components of dimension at least
128: 439:, selected from outside a proper closed subset, there is some 43:, and all its powers. The Zariski topology on a product of 268:). It is assumed there is an irreducible closed subset 520:, such that its Zariski geometry of powers and their 245:, when intersected with a diagonal subset in which 27:consists of an abstract structure introduced by 410:is an irreducible closed set of dimension 1. 35:, in order to give a characterisation of the 8: 544:Journal of the American Mathematical Society 230:, other than the cases where the fiber is 556: 260:The further condition required is called 534:Hrushovski, Ehud; Zilber, Boris (1996). 123:, the subsets defined by equality in an 7: 276:, and an irreducible closed subset 237:(E) A closed irreducible subset of 378:outside a proper closed subset of 14: 288:, with the following properties: 1: 558:10.1090/S0894-0347-96-00180-4 208:is a proper closed subset of 107:Noetherian topological space 186:) lies between its closure 98:satisfying certain axioms. 602: 505:algebraically closed field 216:, at an abstract level.) 131:are closed. The mappings 57:compact Riemann surfaces 109:, of dimension at most 447:such that the set of ( 266:very ample line bundle 214:quantifier elimination 148:(B) For a projection 16:Concept in mathematics 77:topological structure 536:"Zariski Geometries" 413:(K) For all pairs ( 79:on each of the sets 47:is very rarely the 45:algebraic varieties 71:consists of a set 498:parametric family 291:(I) Given pairs ( 53:projective curves 593: 581:Algebraic curves 562: 560: 540: 522:Zariski topology 491: 484: 433: 426: 369: 362: 311: 304: 223:is irreducible. 206: 199: 101:(N) Each of the 69:Zariski geometry 49:product topology 37:Zariski topology 25:Zariski geometry 601: 600: 596: 595: 594: 592: 591: 590: 566: 565: 538: 533: 530: 515:algebraic curve 489: 482: 431: 424: 367: 360: 309: 302: 241:, of dimension 204: 197: 65: 59:in particular. 41:algebraic curve 29:Ehud Hrushovski 17: 12: 11: 5: 599: 597: 589: 588: 586:Vector bundles 583: 578: 568: 567: 564: 563: 529: 526: 382:, the set of ( 325:, the set of ( 170:closed subset 164: 163: 143: 142: 96: 95: 64: 61: 15: 13: 10: 9: 6: 4: 3: 2: 598: 587: 584: 582: 579: 577: 574: 573: 571: 559: 554: 550: 546: 545: 537: 532: 531: 527: 525: 523: 519: 516: 513: 509: 506: 501: 499: 494: 492: 485: 478: 474: 470: 466: 462: 458: 454: 450: 446: 442: 438: 434: 427: 420: 416: 411: 409: 405: 401: 397: 393: 389: 385: 381: 377: 372: 370: 363: 356: 352: 348: 344: 340: 336: 332: 328: 324: 320: 316: 312: 305: 298: 294: 289: 287: 283: 279: 275: 271: 267: 263: 258: 256: 252: 248: 244: 240: 235: 233: 229: 224: 222: 217: 215: 211: 207: 200: 193: 189: 185: 181: 177: 173: 169: 162: 158: 154: 151: 150: 149: 146: 141: 137: 134: 133: 132: 130: 126: 122: 117: 114: 112: 108: 104: 99: 93: 89: 85: 82: 81: 80: 78: 74: 70: 62: 60: 58: 54: 50: 46: 42: 38: 34: 30: 26: 22: 576:Model theory 548: 542: 517: 512:non-singular 507: 502: 495: 487: 480: 476: 472: 468: 464: 460: 456: 452: 448: 444: 440: 436: 429: 422: 418: 414: 412: 407: 403: 399: 395: 391: 387: 383: 379: 375: 373: 365: 358: 354: 350: 346: 342: 338: 334: 330: 326: 322: 318: 314: 307: 300: 296: 292: 290: 285: 281: 277: 273: 269: 261: 259: 254: 250: 246: 242: 238: 236: 231: 227: 225: 220: 218: 209: 202: 195: 191: 187: 183: 179: 175: 171: 165: 160: 156: 152: 147: 144: 139: 135: 124: 120: 119:(A) In each 118: 115: 110: 102: 100: 97: 91: 87: 83: 72: 68: 66: 33:Boris Zilber 24: 18: 551:(1): 1–56. 353:) but not ( 317:, for some 212:. (This is 168:irreducible 21:mathematics 570:Categories 528:References 463:includes ( 341:includes ( 262:very ample 63:Definition 510:, and a 374:(J) For 272:of some 253:− 475:) and ( 166:and an 435:) in 313:) in 201:where 75:and a 39:on an 539:(PDF) 459:) in 406:) in 390:) in 337:) in 264:(cf. 257:+ 1. 129:tuple 105:is a 94:, ... 421:), ( 394:, ( 299:), ( 219:(C) 190:and 55:and 31:and 23:, a 553:doi 493:). 443:in 321:in 280:of 174:of 19:In 572:: 547:. 541:. 500:. 486:, 479:, 471:, 467:, 455:, 451:, 428:, 417:, 402:, 398:, 386:, 371:) 364:, 357:, 349:, 345:, 333:, 329:, 306:, 295:, 284:× 234:. 194:\ 178:, 159:→ 155:: 138:→ 113:. 90:, 86:, 67:A 561:. 555:: 549:9 518:C 508:K 490:′ 488:y 483:′ 481:x 477:t 473:y 469:x 465:t 461:Q 457:v 453:u 449:t 445:P 441:t 437:X 432:′ 430:y 425:′ 423:x 419:y 415:x 408:Q 404:y 400:x 396:t 392:X 388:y 384:x 380:P 376:t 368:′ 366:y 361:′ 359:x 355:t 351:y 347:x 343:t 339:Q 335:v 331:u 327:t 323:P 319:t 315:X 310:′ 308:y 303:′ 301:x 297:y 293:x 286:X 282:P 278:Q 274:X 270:P 255:s 251:r 247:s 243:r 239:X 232:X 228:X 221:X 210:Z 205:′ 203:Z 198:′ 196:Z 192:Z 188:Z 184:Y 182:( 180:p 176:X 172:Y 161:X 157:X 153:p 140:X 136:X 127:- 125:n 121:X 111:n 103:X 92:X 88:X 84:X 73:X

Index

mathematics
Ehud Hrushovski
Boris Zilber
Zariski topology
algebraic curve
algebraic varieties
product topology
projective curves
compact Riemann surfaces
topological structure
Noetherian topological space
tuple
irreducible
quantifier elimination
very ample line bundle
parametric family
algebraically closed field
non-singular
algebraic curve
Zariski topology
"Zariski Geometries"
Journal of the American Mathematical Society
doi
10.1090/S0894-0347-96-00180-4
Categories
Model theory
Algebraic curves
Vector bundles

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