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Tropical compactification

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268: 460: 194: 39:, introduced by Jenia Tevelev. Given an algebraic torus and a connected closed subvariety of that torus, a compactification of the subvariety is defined as a closure of it in a 418: 83: 230: 112: 275: 61: 43:
of the original torus. The concept of a tropical compactification arises when trying to make compactifications as "nice" as possible. For a torus
501: 120: 343: 296: 520: 494: 525: 487: 28: 394: 439: 374: 66: 429: 370: 352: 305: 256: 20: 206: 88: 391:; Ranganathan, Dhruv (2017). "Tropical compactification and the Gromov–Witten theory of 323: 241: 32: 362: 315: 267: 36: 366: 443: 471: 388: 46: 514: 251: 200: 40: 246: 459: 291: 327: 319: 467: 189:{\displaystyle \Phi :T\times {\bar {X}}\to \mathbb {P} ,\ (t,x)\to tx} 341:
Brugallé, Erwan; Shaw, Kristin (2014). "A Bit of Tropical Geometry".
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Hannah Markwig, Aaron Bertram, and Renzo Cavalieri, 2012 at the
475: 397: 209: 123: 91: 69: 49: 412: 224: 188: 106: 77: 55: 495: 8: 292:"Compactifications of subvarieties of tori" 502: 488: 433: 404: 400: 399: 396: 356: 309: 211: 210: 208: 152: 151: 137: 136: 122: 93: 92: 90: 71: 70: 68: 48: 282: 7: 456: 454: 375:10.4169/amer.math.monthly.121.07.563 367:10.4169/amer.math.monthly.121.07.563 124: 14: 344:The American Mathematical Monthly 458: 413:{\displaystyle \mathbb {P} ^{1}} 297:American Journal of Mathematics 216: 177: 174: 162: 148: 142: 98: 1: 290:Tevelev, Jenia (2007-08-07). 474:. You can help Knowledge by 78:{\displaystyle \mathbb {P} } 542: 453: 225:{\displaystyle {\bar {X}}} 107:{\displaystyle {\bar {X}}} 114:is tropical when the map 25:tropical compactification 85:, the compactification 27:is a compactification ( 470:-related article is a 414: 278: 226: 190: 108: 79: 57: 415: 320:10.1353/ajm.2007.0029 270: 227: 191: 109: 80: 58: 29:projective completion 395: 207: 121: 89: 67: 63:and a toric variety 47: 16:Mathematical concept 444:2014arXiv1410.2837C 422:Selecta Mathematica 410: 387:Cavalieri, Renzo; 279: 257:Toroidal embedding 222: 186: 104: 75: 53: 21:algebraic geometry 521:Tropical geometry 483: 482: 242:Tropical geometry 219: 161: 145: 101: 56:{\displaystyle T} 533: 504: 497: 490: 462: 455: 447: 437: 419: 417: 416: 411: 409: 408: 403: 379: 378: 360: 338: 332: 331: 313: 304:(4): 1087–1104. 287: 231: 229: 228: 223: 221: 220: 212: 195: 193: 192: 187: 159: 155: 147: 146: 138: 113: 111: 110: 105: 103: 102: 94: 84: 82: 81: 76: 74: 62: 60: 59: 54: 541: 540: 536: 535: 534: 532: 531: 530: 511: 510: 509: 508: 451: 398: 393: 392: 389:Markwig, Hannah 386: 383: 382: 340: 339: 335: 289: 288: 284: 265: 238: 205: 204: 201:faithfully flat 119: 118: 87: 86: 65: 64: 45: 44: 37:algebraic torus 17: 12: 11: 5: 539: 537: 529: 528: 523: 513: 512: 507: 506: 499: 492: 484: 481: 480: 463: 449: 448: 407: 402: 381: 380: 351:(7): 563–589. 333: 281: 280: 264: 261: 260: 259: 254: 249: 244: 237: 234: 218: 215: 197: 196: 185: 182: 179: 176: 173: 170: 167: 164: 158: 154: 150: 144: 141: 135: 132: 129: 126: 100: 97: 73: 52: 15: 13: 10: 9: 6: 4: 3: 2: 538: 527: 526:Algebra stubs 524: 522: 519: 518: 516: 505: 500: 498: 493: 491: 486: 485: 479: 477: 473: 469: 464: 461: 457: 452: 445: 441: 436: 431: 428:: 1027–1060. 427: 423: 405: 390: 385: 384: 376: 372: 368: 364: 359: 354: 350: 346: 345: 337: 334: 329: 325: 321: 317: 312: 307: 303: 299: 298: 293: 286: 283: 277: 273: 269: 262: 258: 255: 253: 252:Chow quotient 250: 248: 245: 243: 240: 239: 235: 233: 213: 202: 183: 180: 171: 168: 165: 156: 139: 133: 130: 127: 117: 116: 115: 95: 50: 42: 41:toric variety 38: 34: 30: 26: 22: 476:expanding it 465: 450: 425: 421: 348: 342: 336: 311:math/0412329 301: 295: 285: 271: 247:GIT quotient 198: 24: 18: 232:is proper. 515:Categories 272:From left: 263:References 33:subvariety 435:1410.2837 358:1311.2360 328:1080-6377 217:¯ 178:→ 149:→ 143:¯ 134:× 125:Φ 99:¯ 236:See also 468:algebra 440:Bibcode 31:) of a 373:  326:  160:  35:of an 466:This 430:arXiv 371:JSTOR 353:arXiv 306:arXiv 472:stub 324:ISSN 203:and 23:, a 420:". 363:doi 349:121 316:doi 302:129 276:MFO 199:is 19:In 517:: 438:. 426:23 424:. 369:. 361:. 347:. 322:. 314:. 300:. 294:. 503:e 496:t 489:v 478:. 446:. 442:: 432:: 406:1 401:P 377:. 365:: 355:: 330:. 318:: 308:: 214:X 184:x 181:t 175:) 172:x 169:, 166:t 163:( 157:, 153:P 140:X 131:T 128:: 96:X 72:P 51:T

Index

algebraic geometry
projective completion
subvariety
algebraic torus
toric variety
faithfully flat
Tropical geometry
GIT quotient
Chow quotient
Toroidal embedding

MFO
"Compactifications of subvarieties of tori"
American Journal of Mathematics
arXiv
math/0412329
doi
10.1353/ajm.2007.0029
ISSN
1080-6377
The American Mathematical Monthly
arXiv
1311.2360
doi
10.4169/amer.math.monthly.121.07.563
JSTOR
10.4169/amer.math.monthly.121.07.563
Markwig, Hannah
arXiv
1410.2837

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