Changeset 11d225 in git for Singular/LIB/schubert.lib
 Timestamp:
 Nov 6, 2013, 4:34:15 PM (9 years ago)
 Branches:
 (u'jengelhdatetime', 'ceac47cbc86fe4a15902392bdbb9bd2ae0ea02c6')(u'spielwiese', 'a800fe4b3e9d37a38c5a10cc0ae9dfa0c15a4ee6')
 Children:
 2ea781ea236dc6d65399e46b262e9954538749a1
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 da7eaf561933d677968dbf5e5b17b8a9e33117d5
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Singular/LIB/schubert.lib
rda7eaf r11d225 1 1 //////////////////////////////////////////////////////////////////////////////// 2 version="version schubert.lib 4.0.0.0 Jun_2013 "; // $Id$2 version="version schubert.lib 4.0.0.0 Nov_2013 "; // $Id$ 3 3 category="Algebraic Geometry"; 4 4 info=" 5 LIBRARY: Schubert.lib Proceduces for Intersection Theory6 7 AUTHOR: Hiep Dang, email: hiep@mathematik.unikl.de5 LIBRARY: schubert.lib Proceduces for Intersection Theory 6 7 AUTHOR: Hiep Dang, email: hiep@mathematik.unikl.de 8 8 9 9 OVERVIEW: 10 10 11 11 We implement new classes (variety, sheaf, stack, graph) and methods for 12 computing with them. Here a variety is represented by a nonnegative integer13 which is its dimension and a graded ring which is its Chow ring. A sheaf is14 represented by a variety and a polynomial which is its Chern character.15 In particular, we implement the concrete varieties such as projective spaces16 , Grassmannians, and projective bundles.12 computing with them. An abstract variety is represented by a nonnegative 13 integer which is its dimension and a graded ring which is its Chow ring. 14 An abstract sheaf is represented by a variety and a polynomial which is its 15 Chern character. In particular, we implement the concrete varieties such as 16 projective spaces, Grassmannians, and projective bundles. 17 17 18 18 An important task of this library is related to the computation of … … 24 24 Kontsevich, and the graphs corresponding to the fixed point components of a 25 25 torus action on the moduli spaces of stable maps. 26 27 As an insightful example, the numbers of rational curves on general complete 28 intersection CalabiYau threefolds in projective spaces are computed up to 29 degree 6. The results are all in agreement with predictions made from mirror 30 symmetry computations. 26 31 27 32 REFERENCES: … … 95 100 dualPartition(list) compute the dual of a partition 96 101 97 KEYWORDS: Intersection Theory; Enumerative Geometry; Schubert Calculus;98 Bott's formula .102 KEYWORDS: Intersection theory; Enumerative geometry; Schubert calculus; 103 Bott's formula; GromovWitten invariants. 99 104 100 105 "; … … 217 222 " 218 223 { 219 int i,j,k,h,m,n,p ;224 int i,j,k,h,m,n,p,q; 220 225 list l; 221 226 int d = M.degreeCurve; … … 228 233 if (i <> j) 229 234 { 230 l = insert(l,list(graph1(d,i,j),2*d),size(l));235 l[size(l)+1] = list(graph1(d,i,j),2*d); 231 236 } 232 237 } … … 242 247 if (i <> j and j <> k) 243 248 { 244 l = insert(l,list(graph2(list(1,1),i,j,k),2),size(l));249 l[size(l)+1] = list(graph2(list(1,1),i,j,k),2); 245 250 } 246 251 } … … 258 263 if (i <> j and j <> k) 259 264 { 260 l = insert(l,list(graph2(list(2,1),i,j,k),2),size(l));265 l[size(l)+1] = list(graph2(list(2,1),i,j,k),2); 261 266 for (h=0;h<=r;h++) 262 267 { 263 268 if (h <> k) 264 269 { 265 l = insert(l,list(graph31(list(1,1,1),i,j,k,h),2),size(l));270 l[size(l)+1] = list(graph31(list(1,1,1),i,j,k,h),2); 266 271 } 267 272 if (h <> j) 268 273 { 269 l = insert(l,list(graph32(list(1,1,1),i,j,k,h),6),size(l));274 l[size(l)+1] = list(graph32(list(1,1,1),i,j,k,h),6); 270 275 } 271 276 } … … 285 290 if (i <> j and j <> k) 286 291 { 287 l = insert(l,list(graph2(list(3,1),i,j,k),3),size(l));288 l = insert(l,list(graph2(list(2,2),i,j,k),8),size(l));292 l[size(l)+1] = list(graph2(list(3,1),i,j,k),3); 293 l[size(l)+1] = list(graph2(list(2,2),i,j,k),8); 289 294 for (h=0;h<=r;h++) 290 295 { 291 296 if (h <> k) 292 297 { 293 l = insert(l,list(graph31(list(2,1,1),i,j,k,h),2),size(l));294 l = insert(l,list(graph31(list(1,2,1),i,j,k,h),4),size(l));298 l[size(l)+1] = list(graph31(list(2,1,1),i,j,k,h),2); 299 l[size(l)+1] = list(graph31(list(1,2,1),i,j,k,h),4); 295 300 } 296 301 if (h <> j) 297 302 { 298 l = insert(l,list(graph32(list(2,1,1),i,j,k,h),4),size(l));303 l[size(l)+1] = list(graph32(list(2,1,1),i,j,k,h),4); 299 304 } 300 305 for (m=0;m<=r;m++) … … 302 307 if (k <> h and m <> h) 303 308 { 304 l = insert(l,list(graph41(list(1,1,1,1),i,j,k,h,m),2),size(l));309 l[size(l)+1] = list(graph41(list(1,1,1,1),i,j,k,h,m),2); 305 310 } 306 311 if (k <> h and m <> k) 307 312 { 308 l = insert(l,list(graph42(list(1,1,1,1),i,j,k,h,m),2),size(l));313 l[size(l)+1] = list(graph42(list(1,1,1,1),i,j,k,h,m),2); 309 314 } 310 315 if (h <> j and m <> j) 311 316 { 312 l = insert(l,list(graph43(list(1,1,1,1),i,j,k,h,m),24),size(l));317 l[size(l)+1] = list(graph43(list(1,1,1,1),i,j,k,h,m),24); 313 318 } 314 319 } … … 329 334 if (i <> j and j <> k) 330 335 { 331 l = insert(l,list(graph2(list(4,1),i,j,k),4),size(l));332 l = insert(l,list(graph2(list(3,2),i,j,k),6),size(l));336 l[size(l)+1] = list(graph2(list(4,1),i,j,k),4); 337 l[size(l)+1] = list(graph2(list(3,2),i,j,k),6); 333 338 for (h=0;h<=r;h++) 334 339 { 335 340 if (k <> h) 336 341 { 337 l = insert(l,list(graph31(list(3,1,1),i,j,k,h),3),size(l));338 l = insert(l,list(graph31(list(1,3,1),i,j,k,h),6),size(l));339 l = insert(l,list(graph31(list(2,2,1),i,j,k,h),4),size(l));340 l = insert(l,list(graph31(list(2,1,2),i,j,k,h),8),size(l));342 l[size(l)+1] = list(graph31(list(3,1,1),i,j,k,h),3); 343 l[size(l)+1] = list(graph31(list(1,3,1),i,j,k,h),6); 344 l[size(l)+1] = list(graph31(list(2,2,1),i,j,k,h),4); 345 l[size(l)+1] = list(graph31(list(2,1,2),i,j,k,h),8); 341 346 } 342 347 if (j <> h) 343 348 { 344 l = insert(l,list(graph32(list(3,1,1),i,j,k,h),6),size(l));345 l = insert(l,list(graph32(list(2,2,1),i,j,k,h),8),size(l));349 l[size(l)+1] = list(graph32(list(3,1,1),i,j,k,h),6); 350 l[size(l)+1] = list(graph32(list(2,2,1),i,j,k,h),8); 346 351 } 347 352 for (m=0;m<=r;m++) … … 349 354 if (k <> h and h <> m) 350 355 { 351 l = insert(l,list(graph41(list(2,1,1,1),i,j,k,h,m),2),size(l));352 l = insert(l,list(graph41(list(1,2,1,1),i,j,k,h,m),2),size(l));356 l[size(l)+1] = list(graph41(list(2,1,1,1),i,j,k,h,m),2); 357 l[size(l)+1] = list(graph41(list(1,2,1,1),i,j,k,h,m),2); 353 358 } 354 359 if (k <> h and k <> m) 355 360 { 356 l = insert(l,list(graph42(list(2,1,1,1),i,j,k,h,m),4),size(l));357 l = insert(l,list(graph42(list(1,2,1,1),i,j,k,h,m),4),size(l));358 l = insert(l,list(graph42(list(1,1,2,1),i,j,k,h,m),2),size(l));361 l[size(l)+1] = list(graph42(list(2,1,1,1),i,j,k,h,m),4); 362 l[size(l)+1] = list(graph42(list(1,2,1,1),i,j,k,h,m),4); 363 l[size(l)+1] = list(graph42(list(1,1,2,1),i,j,k,h,m),2); 359 364 } 360 365 if (j <> h and j <> m) 361 366 { 362 l = insert(l,list(graph43(list(2,1,1,1),i,j,k,h,m),12),size(l));367 l[size(l)+1] = list(graph43(list(2,1,1,1),i,j,k,h,m),12); 363 368 } 364 369 for (n=0;n<=r;n++) … … 366 371 if (k <> h and h <> m and m <> n) 367 372 { 368 l = insert(l,list(graph51(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l));373 l[size(l)+1] = list(graph51(list(1,1,1,1,1),i,j,k,h,m,n),2); 369 374 } 370 375 if (k <> h and h <> m and h <> n) 371 376 { 372 l = insert(l,list(graph52(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l));377 l[size(l)+1] = list(graph52(list(1,1,1,1,1),i,j,k,h,m,n),2); 373 378 } 374 379 if (k <> h and k <> m and k <> n) 375 380 { 376 l = insert(l,list(graph53(list(1,1,1,1,1),i,j,k,h,m,n),6),size(l));381 l[size(l)+1] = list(graph53(list(1,1,1,1,1),i,j,k,h,m,n),6); 377 382 } 378 383 if (j <> h and h <> m and h <> n) 379 384 { 380 l = insert(l,list(graph54(list(1,1,1,1,1),i,j,k,h,m,n),8),size(l));385 l[size(l)+1] = list(graph54(list(1,1,1,1,1),i,j,k,h,m,n),8); 381 386 } 382 387 if (k <> h and k <> m and h <> n) 383 388 { 384 l = insert(l,list(graph55(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l));389 l[size(l)+1] = list(graph55(list(1,1,1,1,1),i,j,k,h,m,n),2); 385 390 } 386 391 if (j <> h and j <> m and j <> n) 387 392 { 388 l = insert(l,list(graph56(list(1,1,1,1,1),i,j,k,h,m,n),120),size(l));393 l[size(l)+1] = list(graph56(list(1,1,1,1,1),i,j,k,h,m,n),120); 389 394 } 390 395 } … … 406 411 if (i <> j and j <> k) 407 412 { 408 l = insert(l,list(graph2(list(5,1),i,j,k),5),size(l));409 l = insert(l,list(graph2(list(4,2),i,j,k),8),size(l));410 l = insert(l,list(graph2(list(3,3),i,j,k),18),size(l));413 l[size(l)+1] = list(graph2(list(5,1),i,j,k),5); 414 l[size(l)+1] = list(graph2(list(4,2),i,j,k),8); 415 l[size(l)+1] = list(graph2(list(3,3),i,j,k),18); 411 416 for (h=0;h<=r;h++) 412 417 { 413 418 if (k <> h) 414 419 { 415 l = insert(l,list(graph31(list(4,1,1),i,j,k,h),4),size(l));416 l = insert(l,list(graph31(list(1,4,1),i,j,k,h),8),size(l));417 l = insert(l,list(graph31(list(3,2,1),i,j,k,h),6),size(l));418 l = insert(l,list(graph31(list(3,1,2),i,j,k,h),6),size(l));419 l = insert(l,list(graph31(list(1,3,2),i,j,k,h),6),size(l));420 l = insert(l,list(graph31(list(2,2,2),i,j,k,h),16),size(l));420 l[size(l)+1] = list(graph31(list(4,1,1),i,j,k,h),4); 421 l[size(l)+1] = list(graph31(list(1,4,1),i,j,k,h),8); 422 l[size(l)+1] = list(graph31(list(3,2,1),i,j,k,h),6); 423 l[size(l)+1] = list(graph31(list(3,1,2),i,j,k,h),6); 424 l[size(l)+1] = list(graph31(list(1,3,2),i,j,k,h),6); 425 l[size(l)+1] = list(graph31(list(2,2,2),i,j,k,h),16); 421 426 } 422 427 if (j <> h) 423 428 { 424 l = insert(l,list(graph32(list(4,1,1),i,j,k,h),8),size(l));425 l = insert(l,list(graph32(list(3,2,1),i,j,k,h),6),size(l));426 l = insert(l,list(graph32(list(2,2,2),i,j,k,h),48),size(l));429 l[size(l)+1] = list(graph32(list(4,1,1),i,j,k,h),8); 430 l[size(l)+1] = list(graph32(list(3,2,1),i,j,k,h),6); 431 l[size(l)+1] = list(graph32(list(2,2,2),i,j,k,h),48); 427 432 } 428 433 for (m=0;m<=r;m++) … … 430 435 if (k <> h and h <> m) 431 436 { 432 l = insert(l,list(graph41(list(3,1,1,1),i,j,k,h,m),3),size(l));433 l = insert(l,list(graph41(list(1,3,1,1),i,j,k,h,m),3),size(l));434 l = insert(l,list(graph41(list(2,2,1,1),i,j,k,h,m),4),size(l));435 l = insert(l,list(graph41(list(2,1,2,1),i,j,k,h,m),4),size(l));436 l = insert(l,list(graph41(list(2,1,1,2),i,j,k,h,m),8),size(l));437 l = insert(l,list(graph41(list(1,2,2,1),i,j,k,h,m),8),size(l));437 l[size(l)+1] = list(graph41(list(3,1,1,1),i,j,k,h,m),3); 438 l[size(l)+1] = list(graph41(list(1,3,1,1),i,j,k,h,m),3); 439 l[size(l)+1] = list(graph41(list(2,2,1,1),i,j,k,h,m),4); 440 l[size(l)+1] = list(graph41(list(2,1,2,1),i,j,k,h,m),4); 441 l[size(l)+1] = list(graph41(list(2,1,1,2),i,j,k,h,m),8); 442 l[size(l)+1] = list(graph41(list(1,2,2,1),i,j,k,h,m),8); 438 443 } 439 444 if (k <> h and k <> m) 440 445 { 441 l = insert(l,list(graph42(list(3,1,1,1),i,j,k,h,m),6),size(l));442 l = insert(l,list(graph42(list(1,3,1,1),i,j,k,h,m),6),size(l));443 l = insert(l,list(graph42(list(1,1,3,1),i,j,k,h,m),3),size(l));444 l = insert(l,list(graph42(list(2,2,1,1),i,j,k,h,m),8),size(l));445 l = insert(l,list(graph42(list(1,1,2,2),i,j,k,h,m),8),size(l));446 l = insert(l,list(graph42(list(2,1,2,1),i,j,k,h,m),4),size(l));447 l = insert(l,list(graph42(list(1,2,2,1),i,j,k,h,m),4),size(l));446 l[size(l)+1] = list(graph42(list(3,1,1,1),i,j,k,h,m),6); 447 l[size(l)+1] = list(graph42(list(1,3,1,1),i,j,k,h,m),6); 448 l[size(l)+1] = list(graph42(list(1,1,3,1),i,j,k,h,m),3); 449 l[size(l)+1] = list(graph42(list(2,2,1,1),i,j,k,h,m),8); 450 l[size(l)+1] = list(graph42(list(1,1,2,2),i,j,k,h,m),8); 451 l[size(l)+1] = list(graph42(list(2,1,2,1),i,j,k,h,m),4); 452 l[size(l)+1] = list(graph42(list(1,2,2,1),i,j,k,h,m),4); 448 453 } 449 454 if (j <> h and j <> m) 450 455 { 451 l = insert(l,list(graph43(list(3,1,1,1),i,j,k,h,m),18),size(l));452 l = insert(l,list(graph43(list(2,2,1,1),i,j,k,h,m),16),size(l));456 l[size(l)+1] = list(graph43(list(3,1,1,1),i,j,k,h,m),18); 457 l[size(l)+1] = list(graph43(list(2,2,1,1),i,j,k,h,m),16); 453 458 } 454 459 for (n=0;n<=r;n++) … … 456 461 if (k <> h and h <> m and m <> n) 457 462 { 458 l = insert(l,list(graph51(list(2,1,1,1,1),i,j,k,h,m,n),2),size(l));459 l = insert(l,list(graph51(list(1,2,1,1,1),i,j,k,h,m,n),2),size(l));460 l = insert(l,list(graph51(list(1,1,2,1,1),i,j,k,h,m,n),4),size(l));463 l[size(l)+1] = list(graph51(list(2,1,1,1,1),i,j,k,h,m,n),2); 464 l[size(l)+1] = list(graph51(list(1,2,1,1,1),i,j,k,h,m,n),2); 465 l[size(l)+1] = list(graph51(list(1,1,2,1,1),i,j,k,h,m,n),4); 461 466 } 462 467 if (k <> h and h <> m and h <> n) 463 468 { 464 l = insert(l,list(graph52(list(2,1,1,1,1),i,j,k,h,m,n),4),size(l));465 l = insert(l,list(graph52(list(1,2,1,1,1),i,j,k,h,m,n),4),size(l));466 l = insert(l,list(graph52(list(1,1,2,1,1),i,j,k,h,m,n),4),size(l));467 l = insert(l,list(graph52(list(1,1,1,2,1),i,j,k,h,m,n),2),size(l));469 l[size(l)+1] = list(graph52(list(2,1,1,1,1),i,j,k,h,m,n),4); 470 l[size(l)+1] = list(graph52(list(1,2,1,1,1),i,j,k,h,m,n),4); 471 l[size(l)+1] = list(graph52(list(1,1,2,1,1),i,j,k,h,m,n),4); 472 l[size(l)+1] = list(graph52(list(1,1,1,2,1),i,j,k,h,m,n),2); 468 473 } 469 474 if (k <> h and k <> m and k <> n) 470 475 { 471 l = insert(l,list(graph53(list(2,1,1,1,1),i,j,k,h,m,n),12),size(l));472 l = insert(l,list(graph53(list(1,2,1,1,1),i,j,k,h,m,n),12),size(l));473 l = insert(l,list(graph53(list(1,1,2,1,1),i,j,k,h,m,n),4),size(l));476 l[size(l)+1] = list(graph53(list(2,1,1,1,1),i,j,k,h,m,n),12); 477 l[size(l)+1] = list(graph53(list(1,2,1,1,1),i,j,k,h,m,n),12); 478 l[size(l)+1] = list(graph53(list(1,1,2,1,1),i,j,k,h,m,n),4); 474 479 } 475 480 if (j <> h and h <> m and h <> n) 476 481 { 477 l = insert(l,list(graph54(list(2,1,1,1,1),i,j,k,h,m,n),4),size(l));478 l = insert(l,list(graph54(list(1,1,2,1,1),i,j,k,h,m,n),16),size(l));482 l[size(l)+1] = list(graph54(list(2,1,1,1,1),i,j,k,h,m,n),4); 483 l[size(l)+1] = list(graph54(list(1,1,2,1,1),i,j,k,h,m,n),16); 479 484 } 480 485 if (k <> h and k <> m and h <> n) 481 486 { 482 l = insert(l,list(graph55(list(2,1,1,1,1),i,j,k,h,m,n),2),size(l));483 l = insert(l,list(graph55(list(1,2,1,1,1),i,j,k,h,m,n),2),size(l));484 l = insert(l,list(graph55(list(1,1,1,2,1),i,j,k,h,m,n),4),size(l));487 l[size(l)+1] = list(graph55(list(2,1,1,1,1),i,j,k,h,m,n),2); 488 l[size(l)+1] = list(graph55(list(1,2,1,1,1),i,j,k,h,m,n),2); 489 l[size(l)+1] = list(graph55(list(1,1,1,2,1),i,j,k,h,m,n),4); 485 490 } 486 491 if (j <> h and j <> m and j <> n) 487 492 { 488 l = insert(l,list(graph56(list(2,1,1,1,1),i,j,k,h,m,n),48),size(l));493 l[size(l)+1] = list(graph56(list(2,1,1,1,1),i,j,k,h,m,n),48); 489 494 } 490 495 for (p=0;p<=r;p++) … … 492 497 if (k <> h and h <> m and m <> n and n <> p) 493 498 { 494 l = insert(l,list(graph61(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l));499 l[size(l)+1] = list(graph61(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2); 495 500 } 496 501 if (k <> h and h <> m and m <> n and m <> p) 497 502 { 498 l = insert(l,list(graph62(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l));503 l[size(l)+1] = list(graph62(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2); 499 504 } 500 505 if (k <> h and h <> m and h <> n and n <> p) 501 506 { 502 l = insert(l,list(graph63(list(1,1,1,1,1,1),i,j,k,h,m,n,p),1),size(l));507 l[size(l)+1] = list(graph63(list(1,1,1,1,1,1),i,j,k,h,m,n,p),1); 503 508 } 504 509 if (k <> h and h <> m and h <> n and h <> p) 505 510 { 506 l = insert(l,list(graph64(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6),size(l));511 l[size(l)+1] = list(graph64(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6); 507 512 } 508 513 if (k <> h and k <> m and k <> n and n <> p) 509 514 { 510 l = insert(l,list(graph65(list(1,1,1,1,1,1),i,j,k,h,m,n,p),4),size(l));515 l[size(l)+1] = list(graph65(list(1,1,1,1,1,1),i,j,k,h,m,n,p),4); 511 516 } 512 517 if (k <> h and k <> m and m <> p and h <> n) 513 518 { 514 l = insert(l,list(graph66(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6),size(l));519 l[size(l)+1] = list(graph66(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6); 515 520 } 516 521 if (j <> h and h <> m and m <> n and m <> p) 517 522 { 518 l = insert(l,list(graph67(list(1,1,1,1,1,1),i,j,k,h,m,n,p),8),size(l));523 l[size(l)+1] = list(graph67(list(1,1,1,1,1,1),i,j,k,h,m,n,p),8); 519 524 } 520 525 if (j <> h and h <> m and h <> n and h <> p) 521 526 { 522 l = insert(l,list(graph68(list(1,1,1,1,1,1),i,j,k,h,m,n,p),12),size(l));527 l[size(l)+1] = list(graph68(list(1,1,1,1,1,1),i,j,k,h,m,n,p),12); 523 528 } 524 529 if (j <> h and h <> m and h <> n and n <> p) 525 530 { 526 l = insert(l,list(graph69(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l));531 l[size(l)+1] = list(graph69(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2); 527 532 } 528 533 if (k <> h and k <> m and k <> n and k <> p) 529 534 { 530 l = insert(l,list(graph610(list(1,1,1,1,1,1),i,j,k,h,m,n,p),24),size(l));535 l[size(l)+1] = list(graph610(list(1,1,1,1,1,1),i,j,k,h,m,n,p),24); 531 536 } 532 537 if (j <> h and j <> m and j <> n and j <> p) 533 538 { 534 l = insert(l,list(graph611(list(1,1,1,1,1,1),i,j,k,h,m,n,p),720),size(l));539 l[size(l)+1] = list(graph611(list(1,1,1,1,1,1),i,j,k,h,m,n,p),720); 535 540 } 536 541 } … … 548 553 { 549 554 "EXAMPLE:"; echo=2; 550 ring r = 0, (x),dp;555 ring r = 0,x,dp; 551 556 variety P = projectiveSpace(4); 552 557 stack M = moduliSpace(P,2); … … 573 578 for (i=0;i<=n;i++) 574 579 { 575 l = insert(l,number( 5^i),size(l));580 l = insert(l,number(10^i),size(l)); 576 581 } 577 582 return (l); … … 580 585 { 581 586 "EXAMPLE:"; echo=2; 582 ring r = 0, (x),dp;587 ring r = 0,x,dp; 583 588 variety P = projectiveSpace(4); 584 589 def L = torusList(P); … … 657 662 { 658 663 "EXAMPLE:"; echo=2; 659 ring r = 0, (x),dp;664 ring r = 0,x,dp; 660 665 variety P = projectiveSpace(4); 661 666 stack M = moduliSpace(P,2); … … 691 696 int d = e[j][3]; 692 697 number c = (1)^d*factorial(d)^2; 693 number y = c*(L[e[j][1]+1]L[e[j][2]+1])^(2*d)/( d^(2*d));698 number y = c*(L[e[j][1]+1]L[e[j][2]+1])^(2*d)/(number(d)^(2*d)); 694 699 for (k=0;k<=n;k++) 695 700 { … … 749 754 { 750 755 "EXAMPLE:"; echo=2; 751 ring r = 0, (x),dp;756 ring r = 0,x,dp; 752 757 variety P = projectiveSpace(4); 753 758 stack M = moduliSpace(P,2); … … 788 793 { 789 794 "EXAMPLE:"; echo=2; 790 ring r = 0, (x),dp;795 ring r = 0,x,dp; 791 796 multipleCover(1); 792 797 multipleCover(2); … … 794 799 multipleCover(4); 795 800 multipleCover(5); 801 multipleCover(6); 796 802 } 797 803 … … 812 818 def F = fixedPoints(M); 813 819 int i; 814 polyr = 0;820 number r = 0; 815 821 for (i=1;i<=size(F);i++) 816 822 { … … 826 832 { 827 833 "EXAMPLE:"; echo=2; 828 ring r = 0, (x),dp;834 ring r = 0,x,dp; 829 835 linesHypersurface(2); 830 836 linesHypersurface(3); … … 874 880 { 875 881 "EXAMPLE:"; echo=2; 876 ring r = 0, (x),dp;882 ring r = 0,x,dp; 877 883 rationalCurve(1); 878 884 /* … … 880 886 rationalCurve(3); 881 887 rationalCurve(4); 882 rationalCurve(5);883 888 rationalCurve(1,list(4,2)); 884 889 rationalCurve(1,list(3,3)); … … 897 902 rationalCurve(4,list(3,2,2)); 898 903 rationalCurve(4,list(2,2,2,2)); 899 rationalCurve(5,list(4,2));900 rationalCurve(5,list(3,3));901 rationalCurve(5,list(3,2,2));902 rationalCurve(5,list(2,2,2,2));903 */904 904 } 905 905 … … 921 921 { 922 922 "EXAMPLE:"; echo=2; 923 ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp;923 ring r = 0,x,dp; 924 924 graph G = makeGraph(list(list(0,1,list(0,1,2)),list(1,1,list(1,0,2))), 925 925 list(list(0,1,2))); … … 943 943 { 944 944 "EXAMPLE:"; echo=2; 945 ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp;945 ring r = 0,x,dp; 946 946 graph G = makeGraph(list(list(0,1,list(0,1,2)),list(1,1,list(1,0,2))), 947 947 list(list(0,1,2))); … … 1373 1373 list v3 = k,3,f4,f5,f7; 1374 1374 list v4 = h,2,f6,f9; 1375 list v5 = m, 1,f8,f11;1375 list v5 = m,2,f8,f11; 1376 1376 list v6 = n,1,f10; 1377 1377 list v7 = p,1,f12; … … 1516 1516 } 1517 1517 1518 ////////////////////////////////////////////////////////////////////////////////1519 /// Auxilary Static Procedures in this Library /////////////////////////////////1520 ////////////////////////////////////////////////////////////////////////////////1521 1522 1518 proc part(poly f, int n) 1523 1519 "USAGE: part(f,n); f poly, n int … … 1646 1642 { 1647 1643 "EXAMPLE:"; echo=2; 1648 ring r = 0, (x),dp;1644 ring r = 0,x,dp; 1649 1645 poly f = 3+x; 1650 1646 expp(f,3); … … 1718 1714 { 1719 1715 "EXAMPLE:"; echo=2; 1720 ring r = 0, (h,e),wp(1,1);1716 ring r = 0,(h,e),wp(1,1); 1721 1717 ideal rels = he,h2+e2; 1722 1718 variety V = makeVariety(2,rels); … … 1744 1740 { 1745 1741 "EXAMPLE:"; echo=2; 1746 ring r = 0, (h,e),wp(1,1);1742 ring r = 0,(h,e),wp(1,1); 1747 1743 ideal rels = he,h2+e2; 1748 1744 variety V = makeVariety(2,rels); … … 1769 1765 { 1770 1766 "EXAMPLE:"; echo=2; 1771 ring r = 0, (h,e),wp(1,1);1767 ring r = 0,(h,e),wp(1,1); 1772 1768 ideal rels = he,h2+e2; 1773 1769 int d = 2; … … 2086 2082 2087 2083 //////////////////////////////////////////////////////////////////////////////// 2088 ////////// Procedures concerned with sheaves ///////////////////////////////////2084 ////////// Procedures concerned with abstract sheaves /////////////////////////////////// 2089 2085 //////////////////////////////////////////////////////////////////////////////// 2090 2086
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