1 | //////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="version schubert.lib 4.0.0.0 Jun_2013 "; // $Id$ |
---|
3 | category="Algebraic Geometry"; |
---|
4 | info=" |
---|
5 | LIBRARY: Schubert.lib Proceduces for Intersection Theory |
---|
6 | |
---|
7 | AUTHOR: Hiep Dang, email: hiep@mathematik.uni-kl.de |
---|
8 | |
---|
9 | OVERVIEW: |
---|
10 | |
---|
11 | We implement new classes (variety, sheaf, stack, graph) and methods for |
---|
12 | computing with them. Here a variety is represented by a nonnegative integer |
---|
13 | which is its dimension and a graded ring which is its Chow ring. A sheaf is |
---|
14 | represented by a variety and a polynomial which is its Chern character. |
---|
15 | In particular, we implement the concrete varieties such as projective spaces |
---|
16 | , Grassmannians, and projective bundles. |
---|
17 | |
---|
18 | An important task of this library is related to the computation of |
---|
19 | Gromov-Witten invariants. In particular, we implement new tools for the |
---|
20 | computation in equivariant intersection theory. These tools are based on the |
---|
21 | localization of moduli spaces of stable maps and Bott's formula. They are |
---|
22 | useful for the computation of Gromov-Witten invariants. In order to do this, |
---|
23 | we have to deal with moduli spaces of stable maps, which were introduced by |
---|
24 | Kontsevich, and the graphs corresponding to the fixed point components of a |
---|
25 | torus action on the moduli spaces of stable maps. |
---|
26 | |
---|
27 | REFERENCES: |
---|
28 | |
---|
29 | Hiep Dang, Intersection theory with applications to the computation of |
---|
30 | Gromov-Witten invariants, Ph.D thesis, TU Kaiserslautern, 2013. |
---|
31 | |
---|
32 | Sheldon Katz and Stein A. Stromme, Schubert-A Maple package for intersection |
---|
33 | theory and enumerative geometry, 1992. |
---|
34 | |
---|
35 | Daniel R. Grayson, Michael E. Stillman, Stein A. Stromme, David Eisenbud and |
---|
36 | Charley Crissman, Schubert2-A Macaulay2 package for computation in |
---|
37 | intersection theory, 2010. |
---|
38 | |
---|
39 | Maxim Kontsevich, Enumeration of rational curves via torus actions, 1995. |
---|
40 | |
---|
41 | PROCEDURES: |
---|
42 | mod_init() create new objects in this library |
---|
43 | makeVariety(int,ideal) create a variety |
---|
44 | printVariety(variety) print procedure for a variety |
---|
45 | productVariety(variety,variety) make the product of two varieties |
---|
46 | ChowRing(variety) create the Chow ring of a variety |
---|
47 | Grassmannian(int,int) create a Grassmannian as a variety |
---|
48 | projectiveSpace(int) create a projective space as a variety |
---|
49 | projectiveBundle(sheaf) create a projective bundle as a variety |
---|
50 | integral(variety,poly) degree of a 0-cycle on a variety |
---|
51 | makeSheaf(variety,poly) create a sheaf |
---|
52 | printSheaf(sheaf) print procedure for sheaves |
---|
53 | rankSheaf(sheaf) return the rank of a sheaf |
---|
54 | totalChernClass(sheaf) compute the total Chern class of a sheaf |
---|
55 | ChernClass(sheaf,int) compute the k-th Chern class of a sheaf |
---|
56 | topChernClass(sheaf) compute the top Chern class of a sheaf |
---|
57 | totalSegreClass(sheaf) compute the total Segre class of a sheaf |
---|
58 | dualSheaf(sheaf) make the dual of a sheaf |
---|
59 | tensorSheaf(sheaf,sheaf) make the tensor of two sheaves |
---|
60 | symmetricPowerSheaf(sheaf,int) make the k-th symmetric power of a sheaf |
---|
61 | quotSheaf(sheaf,sheaf) make the quotient of two sheaves |
---|
62 | addSheaf(sheaf,sheaf) make the direct sum of two sheaves |
---|
63 | makeGraph(list,list) create a graph from a list of vertices |
---|
64 | and a list of edges |
---|
65 | printGraph(graph) print procedure for graphs |
---|
66 | moduliSpace(variety,int) create a moduli space of stable maps as |
---|
67 | an algebraic stack |
---|
68 | printStack(stack) print procedure for stacks |
---|
69 | dimStack(stack) compute the dimension of a stack |
---|
70 | fixedPoints(stack) compute the list of graphs corresponding |
---|
71 | the fixed point components of a torus |
---|
72 | action on the stack |
---|
73 | contributionBundle(stack,graph) compute the contribution bundle on a |
---|
74 | stack at a graph |
---|
75 | normalBundle(stack,graph) compute the normal bundle on a stack at |
---|
76 | a graph |
---|
77 | multipleCover(int) compute the contribution of multiple |
---|
78 | covers of a smooth rational curve as a |
---|
79 | Gromov-Witten invariant |
---|
80 | linesHypersurface(int) compute the number of lines on a general |
---|
81 | hypersurface |
---|
82 | rationalCurve(int,list) compute the Gromov-Witten invariant |
---|
83 | corresponding the number of rational |
---|
84 | curves on a general Calabi-Yau threefold |
---|
85 | part(poly,int) compute a homogeneous component of a |
---|
86 | polynomial. |
---|
87 | parts(poly,int,int) compute the sum of homogeneous |
---|
88 | components of a polynomial |
---|
89 | logg(poly,int) compute Chern characters from total |
---|
90 | Chern classes. |
---|
91 | expp(poly,int) compute total Chern classes from Chern |
---|
92 | characters |
---|
93 | SchubertClass(list) compute the Schubert classes on a |
---|
94 | Grassmannian |
---|
95 | dualPartition(list) compute the dual of a partition |
---|
96 | |
---|
97 | KEYWORDS: Intersection Theory; Enumerative Geometry; Schubert Calculus; |
---|
98 | Bott's formula. |
---|
99 | |
---|
100 | "; |
---|
101 | |
---|
102 | //////////////////////////////////////////////////////////////////////////////// |
---|
103 | |
---|
104 | LIB "general.lib"; |
---|
105 | LIB "homolog.lib"; |
---|
106 | |
---|
107 | //////////////////////////////////////////////////////////////////////////////// |
---|
108 | /////////// create new objects in this library //////////////////////////////// |
---|
109 | //////////////////////////////////////////////////////////////////////////////// |
---|
110 | |
---|
111 | proc mod_init() |
---|
112 | "USAGE: mod_init(); |
---|
113 | THEORY: This is to create new objects in this library such as variety, |
---|
114 | sheaf, stack, and graph. |
---|
115 | KEYWORDS: variety, sheaf, stack, graph |
---|
116 | EXAMPLE: example mod_init(); shows an example |
---|
117 | " |
---|
118 | { |
---|
119 | newstruct("variety","int dimension, ring baseRing, ideal relations"); |
---|
120 | newstruct("sheaf","variety currentVariety, poly ChernCharacter"); |
---|
121 | newstruct("graph","list vertices, list edges"); |
---|
122 | newstruct("stack","variety currentVariety, int degreeCurve"); |
---|
123 | |
---|
124 | system("install","variety","print",printVariety,1); |
---|
125 | system("install","variety","*",productVariety,2); |
---|
126 | system("install","sheaf","print",printSheaf,1); |
---|
127 | system("install","sheaf","*",tensorSheaf,2); |
---|
128 | system("install","sheaf","+",addSheaf,2); |
---|
129 | system("install","sheaf","-",quotSheaf,2); |
---|
130 | system("install","sheaf","^",symmetricPowerSheaf,2); |
---|
131 | system("install","graph","print",printGraph,1); |
---|
132 | system("install","stack","print",printStack,1); |
---|
133 | } |
---|
134 | example |
---|
135 | { |
---|
136 | "EXAMPLE:"; echo=2; |
---|
137 | mod_init(); |
---|
138 | } |
---|
139 | |
---|
140 | //////////////////////////////////////////////////////////////////////////////// |
---|
141 | //////// Procedures concerned with moduli spaces of stable maps //////////////// |
---|
142 | //////////////////////////////////////////////////////////////////////////////// |
---|
143 | |
---|
144 | proc printStack(stack M) |
---|
145 | "USAGE: printStack(M); M stack |
---|
146 | ASSUME: M is a moduli space of stable maps. |
---|
147 | THEORY: This is the print function used by Singular to print a stack. |
---|
148 | KEYWORDS: stack, moduli space of stable maps |
---|
149 | EXAMPLE: example printStack; shows an example |
---|
150 | " |
---|
151 | { |
---|
152 | "A moduli space of dimension", dimStack(M); |
---|
153 | } |
---|
154 | example |
---|
155 | { |
---|
156 | "EXAMPLE:"; echo=2; |
---|
157 | ring r = 0,(x),dp; |
---|
158 | variety P = projectiveSpace(4); |
---|
159 | stack M = moduliSpace(P,2); |
---|
160 | M; |
---|
161 | } |
---|
162 | |
---|
163 | proc moduliSpace(variety V, int d) |
---|
164 | "USAGE: moduliSpace(V,d); V variety, d int |
---|
165 | ASSUME: V is a projective space and d is a positive integer. |
---|
166 | THEORY: This is the function used by Singular to create a moduli space of |
---|
167 | stable maps from a genus zero curve to a projective space. |
---|
168 | KEYWORDS: stack, moduli space of stable maps, rational curves |
---|
169 | EXAMPLE: example moduliSpace; shows an example |
---|
170 | " |
---|
171 | { |
---|
172 | stack M; |
---|
173 | M.currentVariety = V; |
---|
174 | M.degreeCurve = d; |
---|
175 | return(M); |
---|
176 | } |
---|
177 | example |
---|
178 | { |
---|
179 | "EXAMPLE:"; echo=2; |
---|
180 | ring r = 0,(x),dp; |
---|
181 | variety P = projectiveSpace(4); |
---|
182 | stack M = moduliSpace(P,2); |
---|
183 | M; |
---|
184 | } |
---|
185 | |
---|
186 | proc dimStack(stack M) |
---|
187 | "USAGE: dimStack(M); M stack |
---|
188 | RETURN: int |
---|
189 | INPUT: M is a moduli space of stable maps. |
---|
190 | OUTPUT: the dimension of moduli space of stable maps. |
---|
191 | KEYWORDS: dimension, moduli space of stable maps, rational curves |
---|
192 | EXAMPLE: example dimStack; shows an example |
---|
193 | " |
---|
194 | { |
---|
195 | variety V = M.currentVariety; |
---|
196 | int n = V.dimension; |
---|
197 | int d = M.degreeCurve; |
---|
198 | return (n*d+n+d-3); |
---|
199 | } |
---|
200 | example |
---|
201 | { |
---|
202 | "EXAMPLE:"; echo=2; |
---|
203 | ring r = 0,(x),dp; |
---|
204 | variety P = projectiveSpace(4); |
---|
205 | stack M = moduliSpace(P,2); |
---|
206 | dimStack(M); |
---|
207 | } |
---|
208 | |
---|
209 | proc fixedPoints(stack M) |
---|
210 | "USAGE: fixedPoints(M); M stack |
---|
211 | RETURN: list |
---|
212 | INPUT: M is a moduli space of stable maps. |
---|
213 | OUTPUT: a list of graphs corresponding the fixed point components of a torus |
---|
214 | action on a moduli space of stable maps. |
---|
215 | KEYWORDS: fixed points, moduli space of stable maps, graph |
---|
216 | EXAMPLE: example fixedPoints; shows an example |
---|
217 | " |
---|
218 | { |
---|
219 | int i,j,k,h,m,n,p; |
---|
220 | list l; |
---|
221 | int d = M.degreeCurve; |
---|
222 | variety V = M.currentVariety; |
---|
223 | int r = V.dimension; |
---|
224 | for (i=0;i<=r;i++) |
---|
225 | { |
---|
226 | for (j=0;j<=r;j++) |
---|
227 | { |
---|
228 | if (i <> j) |
---|
229 | { |
---|
230 | l = insert(l,list(graph1(d,i,j),2*d),size(l)); |
---|
231 | } |
---|
232 | } |
---|
233 | } |
---|
234 | if (d == 2) |
---|
235 | { |
---|
236 | for (i=0;i<=r;i++) |
---|
237 | { |
---|
238 | for (j=0;j<=r;j++) |
---|
239 | { |
---|
240 | for (k=0;k<=r;k++) |
---|
241 | { |
---|
242 | if (i <> j and j <> k) |
---|
243 | { |
---|
244 | l = insert(l,list(graph2(list(1,1),i,j,k),2),size(l)); |
---|
245 | } |
---|
246 | } |
---|
247 | } |
---|
248 | } |
---|
249 | } |
---|
250 | if (d == 3) |
---|
251 | { |
---|
252 | for (i=0;i<=r;i++) |
---|
253 | { |
---|
254 | for (j=0;j<=r;j++) |
---|
255 | { |
---|
256 | for (k=0;k<=r;k++) |
---|
257 | { |
---|
258 | if (i <> j and j <> k) |
---|
259 | { |
---|
260 | l = insert(l,list(graph2(list(2,1),i,j,k),2),size(l)); |
---|
261 | for (h=0;h<=r;h++) |
---|
262 | { |
---|
263 | if (h <> k) |
---|
264 | { |
---|
265 | l = insert(l,list(graph31(list(1,1,1),i,j,k,h),2),size(l)); |
---|
266 | } |
---|
267 | if (h <> j) |
---|
268 | { |
---|
269 | l = insert(l,list(graph32(list(1,1,1),i,j,k,h),6),size(l)); |
---|
270 | } |
---|
271 | } |
---|
272 | } |
---|
273 | } |
---|
274 | } |
---|
275 | } |
---|
276 | } |
---|
277 | if (d == 4) |
---|
278 | { |
---|
279 | for (i=0;i<=r;i++) |
---|
280 | { |
---|
281 | for (j=0;j<=r;j++) |
---|
282 | { |
---|
283 | for (k=0;k<=r;k++) |
---|
284 | { |
---|
285 | if (i <> j and j <> k) |
---|
286 | { |
---|
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)); |
---|
289 | for (h=0;h<=r;h++) |
---|
290 | { |
---|
291 | if (h <> k) |
---|
292 | { |
---|
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)); |
---|
295 | } |
---|
296 | if (h <> j) |
---|
297 | { |
---|
298 | l = insert(l,list(graph32(list(2,1,1),i,j,k,h),4),size(l)); |
---|
299 | } |
---|
300 | for (m=0;m<=r;m++) |
---|
301 | { |
---|
302 | if (k <> h and m <> h) |
---|
303 | { |
---|
304 | l = insert(l,list(graph41(list(1,1,1,1),i,j,k,h,m),2),size(l)); |
---|
305 | } |
---|
306 | if (k <> h and m <> k) |
---|
307 | { |
---|
308 | l = insert(l,list(graph42(list(1,1,1,1),i,j,k,h,m),2),size(l)); |
---|
309 | } |
---|
310 | if (h <> j and m <> j) |
---|
311 | { |
---|
312 | l = insert(l,list(graph43(list(1,1,1,1),i,j,k,h,m),24),size(l)); |
---|
313 | } |
---|
314 | } |
---|
315 | } |
---|
316 | } |
---|
317 | } |
---|
318 | } |
---|
319 | } |
---|
320 | } |
---|
321 | if (d == 5) |
---|
322 | { |
---|
323 | for (i=0;i<=r;i++) |
---|
324 | { |
---|
325 | for (j=0;j<=r;j++) |
---|
326 | { |
---|
327 | for (k=0;k<=r;k++) |
---|
328 | { |
---|
329 | if (i <> j and j <> k) |
---|
330 | { |
---|
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)); |
---|
333 | for (h=0;h<=r;h++) |
---|
334 | { |
---|
335 | if (k <> h) |
---|
336 | { |
---|
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)); |
---|
341 | } |
---|
342 | if (j <> h) |
---|
343 | { |
---|
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)); |
---|
346 | } |
---|
347 | for (m=0;m<=r;m++) |
---|
348 | { |
---|
349 | if (k <> h and h <> m) |
---|
350 | { |
---|
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)); |
---|
353 | } |
---|
354 | if (k <> h and k <> m) |
---|
355 | { |
---|
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)); |
---|
359 | } |
---|
360 | if (j <> h and j <> m) |
---|
361 | { |
---|
362 | l = insert(l,list(graph43(list(2,1,1,1),i,j,k,h,m),12),size(l)); |
---|
363 | } |
---|
364 | for (n=0;n<=r;n++) |
---|
365 | { |
---|
366 | if (k <> h and h <> m and m <> n) |
---|
367 | { |
---|
368 | l = insert(l,list(graph51(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l)); |
---|
369 | } |
---|
370 | if (k <> h and h <> m and h <> n) |
---|
371 | { |
---|
372 | l = insert(l,list(graph52(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l)); |
---|
373 | } |
---|
374 | if (k <> h and k <> m and k <> n) |
---|
375 | { |
---|
376 | l = insert(l,list(graph53(list(1,1,1,1,1),i,j,k,h,m,n),6),size(l)); |
---|
377 | } |
---|
378 | if (j <> h and h <> m and h <> n) |
---|
379 | { |
---|
380 | l = insert(l,list(graph54(list(1,1,1,1,1),i,j,k,h,m,n),8),size(l)); |
---|
381 | } |
---|
382 | if (k <> h and k <> m and h <> n) |
---|
383 | { |
---|
384 | l = insert(l,list(graph55(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l)); |
---|
385 | } |
---|
386 | if (j <> h and j <> m and j <> n) |
---|
387 | { |
---|
388 | l = insert(l,list(graph56(list(1,1,1,1,1),i,j,k,h,m,n),120),size(l)); |
---|
389 | } |
---|
390 | } |
---|
391 | } |
---|
392 | } |
---|
393 | } |
---|
394 | } |
---|
395 | } |
---|
396 | } |
---|
397 | } |
---|
398 | if (d == 6) |
---|
399 | { |
---|
400 | for (i=0;i<=r;i++) |
---|
401 | { |
---|
402 | for (j=0;j<=r;j++) |
---|
403 | { |
---|
404 | for (k=0;k<=r;k++) |
---|
405 | { |
---|
406 | if (i <> j and j <> k) |
---|
407 | { |
---|
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)); |
---|
411 | for (h=0;h<=r;h++) |
---|
412 | { |
---|
413 | if (k <> h) |
---|
414 | { |
---|
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)); |
---|
421 | } |
---|
422 | if (j <> h) |
---|
423 | { |
---|
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)); |
---|
427 | } |
---|
428 | for (m=0;m<=r;m++) |
---|
429 | { |
---|
430 | if (k <> h and h <> m) |
---|
431 | { |
---|
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)); |
---|
438 | } |
---|
439 | if (k <> h and k <> m) |
---|
440 | { |
---|
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)); |
---|
448 | } |
---|
449 | if (j <> h and j <> m) |
---|
450 | { |
---|
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)); |
---|
453 | } |
---|
454 | for (n=0;n<=r;n++) |
---|
455 | { |
---|
456 | if (k <> h and h <> m and m <> n) |
---|
457 | { |
---|
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)); |
---|
461 | } |
---|
462 | if (k <> h and h <> m and h <> n) |
---|
463 | { |
---|
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)); |
---|
468 | } |
---|
469 | if (k <> h and k <> m and k <> n) |
---|
470 | { |
---|
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)); |
---|
474 | } |
---|
475 | if (j <> h and h <> m and h <> n) |
---|
476 | { |
---|
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)); |
---|
479 | } |
---|
480 | if (k <> h and k <> m and h <> n) |
---|
481 | { |
---|
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)); |
---|
485 | } |
---|
486 | if (j <> h and j <> m and j <> n) |
---|
487 | { |
---|
488 | l = insert(l,list(graph56(list(2,1,1,1,1),i,j,k,h,m,n),48),size(l)); |
---|
489 | } |
---|
490 | for (p=0;p<=r;p++) |
---|
491 | { |
---|
492 | if (k <> h and h <> m and m <> n and n <> p) |
---|
493 | { |
---|
494 | l = insert(l,list(graph61(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l)); |
---|
495 | } |
---|
496 | if (k <> h and h <> m and m <> n and m <> p) |
---|
497 | { |
---|
498 | l = insert(l,list(graph62(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l)); |
---|
499 | } |
---|
500 | if (k <> h and h <> m and h <> n and n <> p) |
---|
501 | { |
---|
502 | l = insert(l,list(graph63(list(1,1,1,1,1,1),i,j,k,h,m,n,p),1),size(l)); |
---|
503 | } |
---|
504 | if (k <> h and h <> m and h <> n and h <> p) |
---|
505 | { |
---|
506 | l = insert(l,list(graph64(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6),size(l)); |
---|
507 | } |
---|
508 | if (k <> h and k <> m and k <> n and n <> p) |
---|
509 | { |
---|
510 | l = insert(l,list(graph65(list(1,1,1,1,1,1),i,j,k,h,m,n,p),4),size(l)); |
---|
511 | } |
---|
512 | if (k <> h and k <> m and m <> p and h <> n) |
---|
513 | { |
---|
514 | l = insert(l,list(graph66(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6),size(l)); |
---|
515 | } |
---|
516 | if (j <> h and h <> m and m <> n and m <> p) |
---|
517 | { |
---|
518 | l = insert(l,list(graph67(list(1,1,1,1,1,1),i,j,k,h,m,n,p),8),size(l)); |
---|
519 | } |
---|
520 | if (j <> h and h <> m and h <> n and h <> p) |
---|
521 | { |
---|
522 | l = insert(l,list(graph68(list(1,1,1,1,1,1),i,j,k,h,m,n,p),12),size(l)); |
---|
523 | } |
---|
524 | if (j <> h and h <> m and h <> n and n <> p) |
---|
525 | { |
---|
526 | l = insert(l,list(graph69(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l)); |
---|
527 | } |
---|
528 | if (k <> h and k <> m and k <> n and k <> p) |
---|
529 | { |
---|
530 | l = insert(l,list(graph610(list(1,1,1,1,1,1),i,j,k,h,m,n,p),24),size(l)); |
---|
531 | } |
---|
532 | if (j <> h and j <> m and j <> n and j <> p) |
---|
533 | { |
---|
534 | l = insert(l,list(graph611(list(1,1,1,1,1,1),i,j,k,h,m,n,p),720),size(l)); |
---|
535 | } |
---|
536 | } |
---|
537 | } |
---|
538 | } |
---|
539 | } |
---|
540 | } |
---|
541 | } |
---|
542 | } |
---|
543 | } |
---|
544 | } |
---|
545 | return (l); |
---|
546 | } |
---|
547 | example |
---|
548 | { |
---|
549 | "EXAMPLE:"; echo=2; |
---|
550 | ring r = 0,(x),dp; |
---|
551 | variety P = projectiveSpace(4); |
---|
552 | stack M = moduliSpace(P,2); |
---|
553 | def F = fixedPoints(M); |
---|
554 | size(F); |
---|
555 | typeof(F[1]) == "list"; |
---|
556 | typeof(F[1][1]) == "graph"; |
---|
557 | typeof(F[1][2]) == "int"; |
---|
558 | } |
---|
559 | |
---|
560 | static proc torusList(variety P) |
---|
561 | "USAGE: torusList(P); P variety |
---|
562 | RETURN: list |
---|
563 | INPUT: P is a projective space |
---|
564 | OUTPUT: a list of numbers |
---|
565 | THEORY: This is a procedure concerning the enumeration of rational curves. |
---|
566 | KEYWORDS: torus action |
---|
567 | EXAMPLE: example torusList; shows an example |
---|
568 | " |
---|
569 | { |
---|
570 | int i; |
---|
571 | int n = P.dimension; |
---|
572 | list l; |
---|
573 | for (i=0;i<=n;i++) |
---|
574 | { |
---|
575 | l = insert(l,number(5^i),size(l)); |
---|
576 | } |
---|
577 | return (l); |
---|
578 | } |
---|
579 | example |
---|
580 | { |
---|
581 | "EXAMPLE:"; echo=2; |
---|
582 | ring r = 0,(x),dp; |
---|
583 | variety P = projectiveSpace(4); |
---|
584 | def L = torusList(P); |
---|
585 | L; |
---|
586 | } |
---|
587 | |
---|
588 | proc contributionBundle(stack M, graph G, list #) |
---|
589 | "USAGE: contributionBundle(M,G,#); M stack, G graph, # list |
---|
590 | RETURN: number |
---|
591 | INPUT: M is a moduli space of stable maps, G is a graph, # is a list. |
---|
592 | OUTPUT: a number corresponding to the contribution bundle on a moduli space |
---|
593 | of stable maps at a fixed point component (graph) |
---|
594 | KEYWORDS: contribution bundle, graph, multiple cover, rational curve, |
---|
595 | SEE ALSO: normalBundle |
---|
596 | EXAMPLE: example contributionBundle; shows an example |
---|
597 | " |
---|
598 | { |
---|
599 | def R = basering; |
---|
600 | setring R; |
---|
601 | int i,j,a; |
---|
602 | variety P = M.currentVariety; |
---|
603 | def L = torusList(P); |
---|
604 | int r = P.dimension; |
---|
605 | int d; |
---|
606 | if (size(#)==0) {d = 2*r - 3;} |
---|
607 | else |
---|
608 | { |
---|
609 | if (typeof(#[1]) == "int") {d = #[1];} |
---|
610 | else {Error("invalid optional argument");} |
---|
611 | } |
---|
612 | list e = G.edges; |
---|
613 | list v = G.vertices; |
---|
614 | number E = 1; |
---|
615 | number V = 1; |
---|
616 | if (r == 1) |
---|
617 | { |
---|
618 | for (i=1;i<=size(v);i++) |
---|
619 | { |
---|
620 | V = V*(-L[v[i][1]+1])^(v[i][2]-1); |
---|
621 | } |
---|
622 | for (j=1;j<=size(e);j++) |
---|
623 | { |
---|
624 | number f = 1; |
---|
625 | if (e[j][3]<>1) |
---|
626 | { |
---|
627 | for (a=1;a<e[j][3];a++) |
---|
628 | { |
---|
629 | f=f*(-a*L[e[j][1]+1]-(e[j][3]-a)*L[e[j][2]+1])/e[j][3]; |
---|
630 | } |
---|
631 | } |
---|
632 | E = E*f; |
---|
633 | kill f; |
---|
634 | } |
---|
635 | return ((E*V)^2); |
---|
636 | } |
---|
637 | else |
---|
638 | { |
---|
639 | for (i=1;i<=size(v);i++) |
---|
640 | { |
---|
641 | V = V*((d*L[v[i][1]+1])^(v[i][2]-1)); |
---|
642 | } |
---|
643 | for (j=1;j<=size(e);j++) |
---|
644 | { |
---|
645 | number f = 1; |
---|
646 | for (a=0;a<=d*e[j][3];a++) |
---|
647 | { |
---|
648 | f = f*((a*L[e[j][1]+1]+(d*e[j][3]-a)*L[e[j][2]+1])/e[j][3]); |
---|
649 | } |
---|
650 | E = E*f; |
---|
651 | kill f; |
---|
652 | } |
---|
653 | return (E/V); |
---|
654 | } |
---|
655 | } |
---|
656 | example |
---|
657 | { |
---|
658 | "EXAMPLE:"; echo=2; |
---|
659 | ring r = 0,(x),dp; |
---|
660 | variety P = projectiveSpace(4); |
---|
661 | stack M = moduliSpace(P,2); |
---|
662 | def F = fixedPoints(M); |
---|
663 | graph G = F[1][1]; |
---|
664 | number f = contributionBundle(M,G); |
---|
665 | number g = contributionBundle(M,G,5); |
---|
666 | f == g; |
---|
667 | } |
---|
668 | |
---|
669 | proc normalBundle(stack M, graph G) |
---|
670 | "USAGE: normalBundle(M,G); M stack, G graph |
---|
671 | RETURN: number |
---|
672 | INPUT: M is a moduli space of stable maps, G is a graph |
---|
673 | OUTPUT: a number corresponding to the normal bundle on a moduli space of |
---|
674 | stable maps at a graph |
---|
675 | KEYWORDS: normal bundle, graph, rational curves, mutiple covers, lines on |
---|
676 | hypersurfaces |
---|
677 | SEE ALSO: contributionBundle |
---|
678 | EXAMPLE: example normalBundle; shows an example |
---|
679 | { |
---|
680 | def R = basering; |
---|
681 | setring R; |
---|
682 | variety P = M.currentVariety; |
---|
683 | def L = torusList(P); |
---|
684 | int n = P.dimension; |
---|
685 | list e = G.edges; |
---|
686 | list v = G.vertices; |
---|
687 | int i,j,k,h,b,m,a; |
---|
688 | number N = 1; |
---|
689 | for (j=1;j<=size(e);j++) |
---|
690 | { |
---|
691 | int d = e[j][3]; |
---|
692 | 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)); |
---|
694 | for (k=0;k<=n;k++) |
---|
695 | { |
---|
696 | if (k <> e[j][1] and k <> e[j][2]) |
---|
697 | { |
---|
698 | for (a=0;a<=d;a++) |
---|
699 | { |
---|
700 | y=y*((a*L[e[j][1]+1]+(d-a)*L[e[j][2]+1])/d - L[k+1]); |
---|
701 | } |
---|
702 | } |
---|
703 | } |
---|
704 | N = N*y; |
---|
705 | kill y,d,c; |
---|
706 | } |
---|
707 | for (i=1;i<=size(v);i++) |
---|
708 | { |
---|
709 | number F = 1; |
---|
710 | for (h=3;h<=size(v[i]);h++) |
---|
711 | { |
---|
712 | F = F*(L[v[i][h][1]+1]-L[v[i][h][2]+1])/v[i][h][3]; |
---|
713 | } |
---|
714 | if (v[i][2] == 1) |
---|
715 | { |
---|
716 | N = N/F; |
---|
717 | kill F; |
---|
718 | } |
---|
719 | else |
---|
720 | { |
---|
721 | number z = 1; |
---|
722 | for (m=0;m<=n;m++) |
---|
723 | { |
---|
724 | if (m<>v[i][1]) |
---|
725 | { |
---|
726 | z = z*(L[v[i][1]+1]-L[m+1]); |
---|
727 | } |
---|
728 | } |
---|
729 | if (v[i][2] == 3) |
---|
730 | { |
---|
731 | N = N*F/z^2; |
---|
732 | kill F,z; |
---|
733 | } |
---|
734 | else |
---|
735 | { |
---|
736 | number g = 0; |
---|
737 | for (b=3;b<=size(v[i]);b++) |
---|
738 | { |
---|
739 | g = g + v[i][b][3]/(L[v[i][b][1]+1]-L[v[i][b][2]+1]); |
---|
740 | } |
---|
741 | N = N*F*g^(3-v[i][2])/(z^(v[i][2]-1)); |
---|
742 | kill g,F,z; |
---|
743 | } |
---|
744 | } |
---|
745 | } |
---|
746 | return (N); |
---|
747 | } |
---|
748 | example |
---|
749 | { |
---|
750 | "EXAMPLE:"; echo=2; |
---|
751 | ring r = 0,(x),dp; |
---|
752 | variety P = projectiveSpace(4); |
---|
753 | stack M = moduliSpace(P,2); |
---|
754 | def F = fixedPoints(M); |
---|
755 | graph G = F[1][1]; |
---|
756 | number f = normalBundle(M,G); |
---|
757 | f <> 0; |
---|
758 | } |
---|
759 | |
---|
760 | proc multipleCover(int d) |
---|
761 | "USAGE: multipleCover(d); d int |
---|
762 | RETURN: number |
---|
763 | THEORY: This is the contribution of degree d multiple covers of a smooth |
---|
764 | rational curve as a Gromov-Witten invariant. |
---|
765 | KEYWORDS: Gromov-Witten invariants, multiple covers |
---|
766 | SEE ALSO: rationalCurve, linesHypersurface |
---|
767 | EXAMPLE: example multipleCover; shows an example |
---|
768 | " |
---|
769 | { |
---|
770 | def R = basering; |
---|
771 | setring R; |
---|
772 | variety P = projectiveSpace(1); |
---|
773 | stack M = moduliSpace(P,d); |
---|
774 | def F = fixedPoints(M); |
---|
775 | int i; |
---|
776 | number r = 0; |
---|
777 | for (i=1;i<=size(F);i++) |
---|
778 | { |
---|
779 | graph G = F[i][1]; |
---|
780 | number s = contributionBundle(M,G); |
---|
781 | number t = F[i][2]*normalBundle(M,G); |
---|
782 | r = r + s/t; |
---|
783 | kill s,t,G; |
---|
784 | } |
---|
785 | return (r); |
---|
786 | } |
---|
787 | example |
---|
788 | { |
---|
789 | "EXAMPLE:"; echo=2; |
---|
790 | ring r = 0,(x),dp; |
---|
791 | multipleCover(1); |
---|
792 | multipleCover(2); |
---|
793 | multipleCover(3); |
---|
794 | multipleCover(4); |
---|
795 | multipleCover(5); |
---|
796 | } |
---|
797 | |
---|
798 | proc linesHypersurface(int n) |
---|
799 | "USAGE: linesHypersurface(n); n int |
---|
800 | RETURN: number |
---|
801 | THEORY: This is the number of lines on a general hypersurface of degree |
---|
802 | d = 2n-3 in an n-dimensional projective space. |
---|
803 | KEYWORDS: Gromov-Witten invariants, lines on hypersurfaces |
---|
804 | SEE ALSO: linesHypersurface, multipleCover |
---|
805 | EXAMPLE: example linesHypersurface; shows an example |
---|
806 | " |
---|
807 | { |
---|
808 | def R = basering; |
---|
809 | setring R; |
---|
810 | variety P = projectiveSpace(n); |
---|
811 | stack M = moduliSpace(P,1); |
---|
812 | def F = fixedPoints(M); |
---|
813 | int i; |
---|
814 | poly r = 0; |
---|
815 | for (i=1;i<=size(F);i++) |
---|
816 | { |
---|
817 | graph G = F[i][1]; |
---|
818 | number s = contributionBundle(M,G); |
---|
819 | number t = F[i][2]*normalBundle(M,G); |
---|
820 | r = r + s/t; |
---|
821 | kill s,t,G; |
---|
822 | } |
---|
823 | return (r); |
---|
824 | } |
---|
825 | example |
---|
826 | { |
---|
827 | "EXAMPLE:"; echo=2; |
---|
828 | ring r = 0,(x),dp; |
---|
829 | linesHypersurface(2); |
---|
830 | linesHypersurface(3); |
---|
831 | linesHypersurface(4); |
---|
832 | linesHypersurface(5); |
---|
833 | linesHypersurface(6); |
---|
834 | linesHypersurface(7); |
---|
835 | linesHypersurface(8); |
---|
836 | linesHypersurface(9); |
---|
837 | linesHypersurface(10); |
---|
838 | } |
---|
839 | |
---|
840 | proc rationalCurve(int d, list #) |
---|
841 | "USAGE: rationalCurve(d,#); d int, # list |
---|
842 | RETURN: number |
---|
843 | THEORY: This is the Gromov-Witten invariant corresponding the number of |
---|
844 | rational curves on a general Calabi-Yau threefold. |
---|
845 | KEYWORDS: Gromov-Witten invariants, rational curves on Calabi-Yau threefolds |
---|
846 | SEE ALSO: linesHypersurface, multipleCover |
---|
847 | EXAMPLE: example rationalCurve; shows an example |
---|
848 | " |
---|
849 | { |
---|
850 | def R = basering; |
---|
851 | setring R; |
---|
852 | int n,i,j; |
---|
853 | if (size(#) == 0) {n = 4; list l = 5;} |
---|
854 | else {n = size(#)+3; list l = #;} |
---|
855 | variety P = projectiveSpace(n); |
---|
856 | stack M = moduliSpace(P,d); |
---|
857 | def F = fixedPoints(M); |
---|
858 | number r = 0; |
---|
859 | for (i=1;i<=size(F);i++) |
---|
860 | { |
---|
861 | graph G = F[i][1]; |
---|
862 | number s = 1; |
---|
863 | for (j=1;j<=size(l);j++) |
---|
864 | { |
---|
865 | s = s*contributionBundle(M,G,list(l[j])); |
---|
866 | } |
---|
867 | number t = F[i][2]*normalBundle(M,G); |
---|
868 | r = r + s/t; |
---|
869 | kill s,t,G; |
---|
870 | } |
---|
871 | return (r); |
---|
872 | } |
---|
873 | example |
---|
874 | { |
---|
875 | "EXAMPLE:"; echo=2; |
---|
876 | ring r = 0,(x),dp; |
---|
877 | rationalCurve(1); |
---|
878 | /* |
---|
879 | rationalCurve(2); |
---|
880 | rationalCurve(3); |
---|
881 | rationalCurve(4); |
---|
882 | rationalCurve(5); |
---|
883 | rationalCurve(1,list(4,2)); |
---|
884 | rationalCurve(1,list(3,3)); |
---|
885 | rationalCurve(1,list(3,2,2)); |
---|
886 | rationalCurve(1,list(2,2,2,2)); |
---|
887 | rationalCurve(2,list(4,2)); |
---|
888 | rationalCurve(2,list(3,3)); |
---|
889 | rationalCurve(2,list(3,2,2)); |
---|
890 | rationalCurve(2,list(2,2,2,2)); |
---|
891 | rationalCurve(3,list(4,2)); |
---|
892 | rationalCurve(3,list(3,3)); |
---|
893 | rationalCurve(3,list(3,2,2)); |
---|
894 | rationalCurve(3,list(2,2,2,2)); |
---|
895 | rationalCurve(4,list(4,2)); |
---|
896 | rationalCurve(4,list(3,3)); |
---|
897 | rationalCurve(4,list(3,2,2)); |
---|
898 | 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 | } |
---|
905 | |
---|
906 | //////////////////////////////////////////////////////////////////////////////// |
---|
907 | /////////// Procedures concerned with graphs /////////////////////////////////// |
---|
908 | //////////////////////////////////////////////////////////////////////////////// |
---|
909 | |
---|
910 | proc printGraph(graph G) |
---|
911 | "USAGE: printGraph(G); G graph |
---|
912 | ASSUME: G is a graph. |
---|
913 | THEORY: This is the print function used by Singular to print a graph. |
---|
914 | KEYWORDS: graph |
---|
915 | EXAMPLE: example printGraph; shows an example |
---|
916 | " |
---|
917 | { |
---|
918 | "A graph with", size(G.vertices), "vertices and", size(G.edges), "edges"; |
---|
919 | } |
---|
920 | example |
---|
921 | { |
---|
922 | "EXAMPLE:"; echo=2; |
---|
923 | ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp; |
---|
924 | graph G = makeGraph(list(list(0,1,list(0,1,2)),list(1,1,list(1,0,2))), |
---|
925 | list(list(0,1,2))); |
---|
926 | G; |
---|
927 | } |
---|
928 | |
---|
929 | proc makeGraph(list v, list e) |
---|
930 | "USAGE: makeGraph(v,e); v list, e list |
---|
931 | ASSUME: v is a list of vertices, e is a list of edges. |
---|
932 | RETURN: graph with vertices v and edges e. |
---|
933 | THEORY: Creates a graph from a list of vertices and edges. |
---|
934 | KEYWORDS: graph |
---|
935 | EXAMPLE: example makeGraph; shows an example |
---|
936 | { |
---|
937 | graph G; |
---|
938 | G.vertices = v; |
---|
939 | G.edges = e; |
---|
940 | return(G); |
---|
941 | } |
---|
942 | example |
---|
943 | { |
---|
944 | "EXAMPLE:"; echo=2; |
---|
945 | ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp; |
---|
946 | graph G = makeGraph(list(list(0,1,list(0,1,2)),list(1,1,list(1,0,2))), |
---|
947 | list(list(0,1,2))); |
---|
948 | G; |
---|
949 | } |
---|
950 | |
---|
951 | static proc graph1(int d, int i, int j) |
---|
952 | { |
---|
953 | graph G; |
---|
954 | list f1 = i,j,d; |
---|
955 | list f2 = j,i,d; |
---|
956 | list v1 = i,1,f1; |
---|
957 | list v2 = j,1,f2; |
---|
958 | G.vertices = v1,v2; |
---|
959 | G.edges = list(f1); |
---|
960 | return (G); |
---|
961 | } |
---|
962 | |
---|
963 | static proc graph2(list d, int i, int j, int k) |
---|
964 | { |
---|
965 | graph G; |
---|
966 | list f1 = i,j,d[1]; |
---|
967 | list f2 = j,i,d[1]; |
---|
968 | list f3 = j,k,d[2]; |
---|
969 | list f4 = k,j,d[2]; |
---|
970 | list v1 = i,1,f1; |
---|
971 | list v2 = j,2,f2,f3; |
---|
972 | list v3 = k,1,f4; |
---|
973 | G.vertices = v1,v2,v3; |
---|
974 | G.edges = f1,f3; |
---|
975 | return (G); |
---|
976 | } |
---|
977 | |
---|
978 | static proc graph31(list d, int i, int j, int k, int h) |
---|
979 | { |
---|
980 | graph G; |
---|
981 | list f1 = i,j,d[1]; |
---|
982 | list f2 = j,i,d[1]; |
---|
983 | list f3 = j,k,d[2]; |
---|
984 | list f4 = k,j,d[2]; |
---|
985 | list f5 = k,h,d[3]; |
---|
986 | list f6 = h,k,d[3]; |
---|
987 | list v1 = i,1,f1; |
---|
988 | list v2 = j,2,f2,f3; |
---|
989 | list v3 = k,2,f4,f5; |
---|
990 | list v4 = h,1,f6; |
---|
991 | G.vertices = v1,v2,v3,v4; |
---|
992 | G.edges = f1,f3,f5; |
---|
993 | return (G); |
---|
994 | } |
---|
995 | |
---|
996 | static proc graph32(list d, int i, int j, int k, int h) |
---|
997 | { |
---|
998 | graph G; |
---|
999 | list f1 = i,j,d[1]; |
---|
1000 | list f2 = j,i,d[1]; |
---|
1001 | list f3 = j,k,d[2]; |
---|
1002 | list f4 = j,h,d[3]; |
---|
1003 | list f5 = k,j,d[2]; |
---|
1004 | list f6 = h,j,d[3]; |
---|
1005 | list v1 = i,1,f1; |
---|
1006 | list v2 = j,3,f2,f3,f4; |
---|
1007 | list v3 = k,1,f5; |
---|
1008 | list v4 = h,1,f6; |
---|
1009 | G.vertices = v1,v2,v3,v4; |
---|
1010 | G.edges = f1,f3,f4; |
---|
1011 | return (G); |
---|
1012 | } |
---|
1013 | |
---|
1014 | static proc graph41(list d, int i, int j, int k, int h, int l) |
---|
1015 | { |
---|
1016 | graph G; |
---|
1017 | list f1 = i,j,d[1]; |
---|
1018 | list f2 = j,i,d[1]; |
---|
1019 | list f3 = j,k,d[2]; |
---|
1020 | list f4 = k,j,d[2]; |
---|
1021 | list f5 = k,h,d[3]; |
---|
1022 | list f6 = h,k,d[3]; |
---|
1023 | list f7 = h,l,d[4]; |
---|
1024 | list f8 = l,h,d[4]; |
---|
1025 | list v1 = i,1,f1; |
---|
1026 | list v2 = j,2,f2,f3; |
---|
1027 | list v3 = k,2,f4,f5; |
---|
1028 | list v4 = h,2,f6,f7; |
---|
1029 | list v5 = l,1,f8; |
---|
1030 | G.vertices = v1,v2,v3,v4,v5; |
---|
1031 | G.edges = f1,f3,f5,f7; |
---|
1032 | return (G); |
---|
1033 | } |
---|
1034 | |
---|
1035 | static proc graph42(list d, int i, int j, int k, int h, int l) |
---|
1036 | { |
---|
1037 | graph G; |
---|
1038 | list f1 = i,j,d[1]; |
---|
1039 | list f2 = j,i,d[1]; |
---|
1040 | list f3 = j,k,d[2]; |
---|
1041 | list f4 = k,j,d[2]; |
---|
1042 | list f5 = k,h,d[3]; |
---|
1043 | list f6 = k,l,d[4]; |
---|
1044 | list f7 = h,k,d[3]; |
---|
1045 | list f8 = l,k,d[4]; |
---|
1046 | list v1 = i,1,f1; |
---|
1047 | list v2 = j,2,f2,f3; |
---|
1048 | list v3 = k,3,f4,f5,f6; |
---|
1049 | list v4 = h,1,f7; |
---|
1050 | list v5 = l,1,f8; |
---|
1051 | G.vertices = v1,v2,v3,v4,v5; |
---|
1052 | G.edges = f1,f3,f5,f6; |
---|
1053 | return (G); |
---|
1054 | } |
---|
1055 | |
---|
1056 | static proc graph43(list d, int i, int j, int k, int h, int l) |
---|
1057 | { |
---|
1058 | graph G; |
---|
1059 | list f1 = i,j,d[1]; |
---|
1060 | list f2 = j,i,d[1]; |
---|
1061 | list f3 = j,k,d[2]; |
---|
1062 | list f4 = j,h,d[3]; |
---|
1063 | list f5 = j,l,d[4]; |
---|
1064 | list f6 = k,j,d[2]; |
---|
1065 | list f7 = h,j,d[3]; |
---|
1066 | list f8 = l,j,d[4]; |
---|
1067 | list v1 = i,1,f1; |
---|
1068 | list v2 = j,4,f2,f3,f4,f5; |
---|
1069 | list v3 = k,1,f6; |
---|
1070 | list v4 = h,1,f7; |
---|
1071 | list v5 = l,1,f8; |
---|
1072 | G.vertices = v1,v2,v3,v4,v5; |
---|
1073 | G.edges = f1,f3,f4,f5; |
---|
1074 | return (G); |
---|
1075 | } |
---|
1076 | |
---|
1077 | static proc graph51(list d, int i, int j, int k, int h, int m, int n) |
---|
1078 | { |
---|
1079 | graph G; |
---|
1080 | list f1 = i,j,d[1]; |
---|
1081 | list f2 = j,i,d[1]; |
---|
1082 | list f3 = j,k,d[2]; |
---|
1083 | list f4 = k,j,d[2]; |
---|
1084 | list f5 = k,h,d[3]; |
---|
1085 | list f6 = h,k,d[3]; |
---|
1086 | list f7 = h,m,d[4]; |
---|
1087 | list f8 = m,h,d[4]; |
---|
1088 | list f9 = m,n,d[5]; |
---|
1089 | list f10 = n,m,d[5]; |
---|
1090 | list v1 = i,1,f1; |
---|
1091 | list v2 = j,2,f2,f3; |
---|
1092 | list v3 = k,2,f4,f5; |
---|
1093 | list v4 = h,2,f6,f7; |
---|
1094 | list v5 = m,2,f8,f9; |
---|
1095 | list v6 = n,1,f10; |
---|
1096 | G.vertices = v1,v2,v3,v4,v5,v6; |
---|
1097 | G.edges = f1,f3,f5,f7,f9; |
---|
1098 | return (G); |
---|
1099 | } |
---|
1100 | |
---|
1101 | static proc graph52(list d, int i, int j, int k, int h, int m, int n) |
---|
1102 | { |
---|
1103 | graph G; |
---|
1104 | list f1 = i,j,d[1]; |
---|
1105 | list f2 = j,i,d[1]; |
---|
1106 | list f3 = j,k,d[2]; |
---|
1107 | list f4 = k,j,d[2]; |
---|
1108 | list f5 = k,h,d[3]; |
---|
1109 | list f6 = h,k,d[3]; |
---|
1110 | list f7 = h,m,d[4]; |
---|
1111 | list f8 = m,h,d[4]; |
---|
1112 | list f9 = h,n,d[5]; |
---|
1113 | list f10 = n,h,d[5]; |
---|
1114 | list v1 = i,1,f1; |
---|
1115 | list v2 = j,2,f2,f3; |
---|
1116 | list v3 = k,2,f4,f5; |
---|
1117 | list v4 = h,3,f6,f7,f9; |
---|
1118 | list v5 = m,1,f8; |
---|
1119 | list v6 = n,1,f10; |
---|
1120 | G.vertices = v1,v2,v3,v4,v5,v6; |
---|
1121 | G.edges = f1,f3,f5,f7,f9; |
---|
1122 | return (G); |
---|
1123 | } |
---|
1124 | |
---|
1125 | static proc graph53(list d, int i, int j, int k, int h, int m, int n) |
---|
1126 | { |
---|
1127 | graph G; |
---|
1128 | list f1 = i,j,d[1]; |
---|
1129 | list f2 = j,i,d[1]; |
---|
1130 | list f3 = j,k,d[2]; |
---|
1131 | list f4 = k,j,d[2]; |
---|
1132 | list f5 = k,h,d[3]; |
---|
1133 | list f6 = h,k,d[3]; |
---|
1134 | list f7 = k,m,d[4]; |
---|
1135 | list f8 = m,k,d[4]; |
---|
1136 | list f9 = k,n,d[5]; |
---|
1137 | list f10 = n,k,d[5]; |
---|
1138 | list v1 = i,1,f1; |
---|
1139 | list v2 = j,2,f2,f3; |
---|
1140 | list v3 = k,4,f4,f5,f7,f9; |
---|
1141 | list v4 = h,1,f6; |
---|
1142 | list v5 = m,1,f8; |
---|
1143 | list v6 = n,1,f10; |
---|
1144 | G.vertices = v1,v2,v3,v4,v5,v6; |
---|
1145 | G.edges = f1,f3,f5,f7,f9; |
---|
1146 | return (G); |
---|
1147 | } |
---|
1148 | |
---|
1149 | static proc graph54(list d, int i, int j, int k, int h, int m, int n) |
---|
1150 | { |
---|
1151 | graph G; |
---|
1152 | list f1 = i,j,d[1]; |
---|
1153 | list f2 = j,i,d[1]; |
---|
1154 | list f3 = j,k,d[2]; |
---|
1155 | list f4 = k,j,d[2]; |
---|
1156 | list f5 = j,h,d[3]; |
---|
1157 | list f6 = h,j,d[3]; |
---|
1158 | list f7 = h,m,d[4]; |
---|
1159 | list f8 = m,h,d[4]; |
---|
1160 | list f9 = h,n,d[5]; |
---|
1161 | list f10 = n,h,d[5]; |
---|
1162 | list v1 = i,1,f1; |
---|
1163 | list v2 = j,3,f2,f3,f5; |
---|
1164 | list v3 = k,1,f4; |
---|
1165 | list v4 = h,3,f6,f7,f9; |
---|
1166 | list v5 = m,1,f8; |
---|
1167 | list v6 = n,1,f10; |
---|
1168 | G.vertices = v1,v2,v3,v4,v5,v6; |
---|
1169 | G.edges = f1,f3,f5,f7,f9; |
---|
1170 | return (G); |
---|
1171 | } |
---|
1172 | |
---|
1173 | static proc graph55(list d, int i, int j, int k, int h, int m, int n) |
---|
1174 | { |
---|
1175 | graph G; |
---|
1176 | list f1 = i,j,d[1]; |
---|
1177 | list f2 = j,i,d[1]; |
---|
1178 | list f3 = j,k,d[2]; |
---|
1179 | list f4 = k,j,d[2]; |
---|
1180 | list f5 = k,h,d[3]; |
---|
1181 | list f6 = h,k,d[3]; |
---|
1182 | list f7 = k,m,d[4]; |
---|
1183 | list f8 = m,k,d[4]; |
---|
1184 | list f9 = h,n,d[5]; |
---|
1185 | list f10 = n,h,d[5]; |
---|
1186 | list v1 = i,1,f1; |
---|
1187 | list v2 = j,2,f2,f3; |
---|
1188 | list v3 = k,3,f4,f5,f7; |
---|
1189 | list v4 = h,2,f6,f9; |
---|
1190 | list v5 = m,1,f8; |
---|
1191 | list v6 = n,1,f10; |
---|
1192 | G.vertices = v1,v2,v3,v4,v5,v6; |
---|
1193 | G.edges = f1,f3,f5,f7,f9; |
---|
1194 | return (G); |
---|
1195 | } |
---|
1196 | |
---|
1197 | static proc graph56(list d, int i, int j, int k, int h, int m, int n) |
---|
1198 | { |
---|
1199 | graph G; |
---|
1200 | list f1 = i,j,d[1]; |
---|
1201 | list f2 = j,i,d[1]; |
---|
1202 | list f3 = j,k,d[2]; |
---|
1203 | list f4 = k,j,d[2]; |
---|
1204 | list f5 = j,h,d[3]; |
---|
1205 | list f6 = h,j,d[3]; |
---|
1206 | list f7 = j,m,d[4]; |
---|
1207 | list f8 = m,j,d[4]; |
---|
1208 | list f9 = j,n,d[5]; |
---|
1209 | list f10 = n,j,d[5]; |
---|
1210 | list v1 = i,1,f1; |
---|
1211 | list v2 = j,5,f2,f3,f5,f7,f9; |
---|
1212 | list v3 = k,1,f4; |
---|
1213 | list v4 = h,1,f6; |
---|
1214 | list v5 = m,1,f8; |
---|
1215 | list v6 = n,1,f10; |
---|
1216 | G.vertices = v1,v2,v3,v4,v5,v6; |
---|
1217 | G.edges = f1,f3,f5,f7,f9; |
---|
1218 | return (G); |
---|
1219 | } |
---|
1220 | |
---|
1221 | static proc graph61(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1222 | { |
---|
1223 | graph G; |
---|
1224 | list f1 = i,j,d[1]; |
---|
1225 | list f2 = j,i,d[1]; |
---|
1226 | list f3 = j,k,d[2]; |
---|
1227 | list f4 = k,j,d[2]; |
---|
1228 | list f5 = k,h,d[3]; |
---|
1229 | list f6 = h,k,d[3]; |
---|
1230 | list f7 = h,m,d[4]; |
---|
1231 | list f8 = m,h,d[4]; |
---|
1232 | list f9 = m,n,d[5]; |
---|
1233 | list f10 = n,m,d[5]; |
---|
1234 | list f11 = n,p,d[6]; |
---|
1235 | list f12 = p,n,d[6]; |
---|
1236 | list v1 = i,1,f1; |
---|
1237 | list v2 = j,2,f2,f3; |
---|
1238 | list v3 = k,2,f4,f5; |
---|
1239 | list v4 = h,2,f6,f7; |
---|
1240 | list v5 = m,2,f8,f9; |
---|
1241 | list v6 = n,2,f10,f11; |
---|
1242 | list v7 = p,1,f12; |
---|
1243 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1244 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1245 | return (G); |
---|
1246 | } |
---|
1247 | |
---|
1248 | static proc graph62(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1249 | { |
---|
1250 | graph G; |
---|
1251 | list f1 = i,j,d[1]; |
---|
1252 | list f2 = j,i,d[1]; |
---|
1253 | list f3 = j,k,d[2]; |
---|
1254 | list f4 = k,j,d[2]; |
---|
1255 | list f5 = k,h,d[3]; |
---|
1256 | list f6 = h,k,d[3]; |
---|
1257 | list f7 = h,m,d[4]; |
---|
1258 | list f8 = m,h,d[4]; |
---|
1259 | list f9 = m,n,d[5]; |
---|
1260 | list f10 = n,m,d[5]; |
---|
1261 | list f11 = m,p,d[6]; |
---|
1262 | list f12 = p,m,d[6]; |
---|
1263 | list v1 = i,1,f1; |
---|
1264 | list v2 = j,2,f2,f3; |
---|
1265 | list v3 = k,2,f4,f5; |
---|
1266 | list v4 = h,2,f6,f7; |
---|
1267 | list v5 = m,3,f8,f9,f11; |
---|
1268 | list v6 = n,1,f10; |
---|
1269 | list v7 = p,1,f12; |
---|
1270 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1271 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1272 | return (G); |
---|
1273 | } |
---|
1274 | |
---|
1275 | static proc graph63(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1276 | { |
---|
1277 | graph G; |
---|
1278 | list f1 = i,j,d[1]; |
---|
1279 | list f2 = j,i,d[1]; |
---|
1280 | list f3 = j,k,d[2]; |
---|
1281 | list f4 = k,j,d[2]; |
---|
1282 | list f5 = k,h,d[3]; |
---|
1283 | list f6 = h,k,d[3]; |
---|
1284 | list f7 = h,m,d[4]; |
---|
1285 | list f8 = m,h,d[4]; |
---|
1286 | list f9 = h,n,d[5]; |
---|
1287 | list f10 = n,h,d[5]; |
---|
1288 | list f11 = n,p,d[6]; |
---|
1289 | list f12 = p,n,d[6]; |
---|
1290 | list v1 = i,1,f1; |
---|
1291 | list v2 = j,2,f2,f3; |
---|
1292 | list v3 = k,2,f4,f5; |
---|
1293 | list v4 = h,3,f6,f7,f9; |
---|
1294 | list v5 = m,1,f8; |
---|
1295 | list v6 = n,2,f10,f11; |
---|
1296 | list v7 = p,1,f12; |
---|
1297 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1298 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1299 | return (G); |
---|
1300 | } |
---|
1301 | |
---|
1302 | static proc graph64(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1303 | { |
---|
1304 | graph G; |
---|
1305 | list f1 = i,j,d[1]; |
---|
1306 | list f2 = j,i,d[1]; |
---|
1307 | list f3 = j,k,d[2]; |
---|
1308 | list f4 = k,j,d[2]; |
---|
1309 | list f5 = k,h,d[3]; |
---|
1310 | list f6 = h,k,d[3]; |
---|
1311 | list f7 = h,m,d[4]; |
---|
1312 | list f8 = m,h,d[4]; |
---|
1313 | list f9 = h,n,d[5]; |
---|
1314 | list f10 = n,h,d[5]; |
---|
1315 | list f11 = h,p,d[6]; |
---|
1316 | list f12 = p,h,d[6]; |
---|
1317 | list v1 = i,1,f1; |
---|
1318 | list v2 = j,2,f2,f3; |
---|
1319 | list v3 = k,2,f4,f5; |
---|
1320 | list v4 = h,4,f6,f7,f9,f11; |
---|
1321 | list v5 = m,1,f8; |
---|
1322 | list v6 = n,1,f10; |
---|
1323 | list v7 = p,1,f12; |
---|
1324 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1325 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1326 | return (G); |
---|
1327 | } |
---|
1328 | |
---|
1329 | static proc graph65(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1330 | { |
---|
1331 | graph G; |
---|
1332 | list f1 = i,j,d[1]; |
---|
1333 | list f2 = j,i,d[1]; |
---|
1334 | list f3 = j,k,d[2]; |
---|
1335 | list f4 = k,j,d[2]; |
---|
1336 | list f5 = k,h,d[3]; |
---|
1337 | list f6 = h,k,d[3]; |
---|
1338 | list f7 = k,m,d[4]; |
---|
1339 | list f8 = m,k,d[4]; |
---|
1340 | list f9 = k,n,d[5]; |
---|
1341 | list f10 = n,k,d[5]; |
---|
1342 | list f11 = n,p,d[6]; |
---|
1343 | list f12 = p,n,d[6]; |
---|
1344 | list v1 = i,1,f1; |
---|
1345 | list v2 = j,2,f2,f3; |
---|
1346 | list v3 = k,4,f4,f5,f7,f9; |
---|
1347 | list v4 = h,1,f6; |
---|
1348 | list v5 = m,1,f8; |
---|
1349 | list v6 = n,2,f10,f11; |
---|
1350 | list v7 = p,1,f12; |
---|
1351 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1352 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1353 | return (G); |
---|
1354 | } |
---|
1355 | |
---|
1356 | static proc graph66(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1357 | { |
---|
1358 | graph G; |
---|
1359 | list f1 = i,j,d[1]; |
---|
1360 | list f2 = j,i,d[1]; |
---|
1361 | list f3 = j,k,d[2]; |
---|
1362 | list f4 = k,j,d[2]; |
---|
1363 | list f5 = k,h,d[3]; |
---|
1364 | list f6 = h,k,d[3]; |
---|
1365 | list f7 = k,m,d[4]; |
---|
1366 | list f8 = m,k,d[4]; |
---|
1367 | list f9 = h,n,d[5]; |
---|
1368 | list f10 = n,h,d[5]; |
---|
1369 | list f11 = m,p,d[6]; |
---|
1370 | list f12 = p,m,d[6]; |
---|
1371 | list v1 = i,1,f1; |
---|
1372 | list v2 = j,2,f2,f3; |
---|
1373 | list v3 = k,3,f4,f5,f7; |
---|
1374 | list v4 = h,2,f6,f9; |
---|
1375 | list v5 = m,1,f8,f11; |
---|
1376 | list v6 = n,1,f10; |
---|
1377 | list v7 = p,1,f12; |
---|
1378 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1379 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1380 | return (G); |
---|
1381 | } |
---|
1382 | |
---|
1383 | static proc graph67(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1384 | { |
---|
1385 | graph G; |
---|
1386 | list f1 = i,j,d[1]; |
---|
1387 | list f2 = j,i,d[1]; |
---|
1388 | list f3 = j,k,d[2]; |
---|
1389 | list f4 = k,j,d[2]; |
---|
1390 | list f5 = j,h,d[3]; |
---|
1391 | list f6 = h,j,d[3]; |
---|
1392 | list f7 = h,m,d[4]; |
---|
1393 | list f8 = m,h,d[4]; |
---|
1394 | list f9 = m,n,d[5]; |
---|
1395 | list f10 = n,m,d[5]; |
---|
1396 | list f11 = m,p,d[6]; |
---|
1397 | list f12 = p,m,d[6]; |
---|
1398 | list v1 = i,1,f1; |
---|
1399 | list v2 = j,3,f2,f3,f5; |
---|
1400 | list v3 = k,1,f4; |
---|
1401 | list v4 = h,2,f6,f7; |
---|
1402 | list v5 = m,3,f8,f9,f11; |
---|
1403 | list v6 = n,1,f10; |
---|
1404 | list v7 = p,1,f12; |
---|
1405 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1406 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1407 | return (G); |
---|
1408 | } |
---|
1409 | |
---|
1410 | static proc graph68(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1411 | { |
---|
1412 | graph G; |
---|
1413 | list f1 = i,j,d[1]; |
---|
1414 | list f2 = j,i,d[1]; |
---|
1415 | list f3 = j,k,d[2]; |
---|
1416 | list f4 = k,j,d[2]; |
---|
1417 | list f5 = j,h,d[3]; |
---|
1418 | list f6 = h,j,d[3]; |
---|
1419 | list f7 = h,m,d[4]; |
---|
1420 | list f8 = m,h,d[4]; |
---|
1421 | list f9 = h,n,d[5]; |
---|
1422 | list f10 = n,h,d[5]; |
---|
1423 | list f11 = h,p,d[6]; |
---|
1424 | list f12 = p,h,d[6]; |
---|
1425 | list v1 = i,1,f1; |
---|
1426 | list v2 = j,3,f2,f3,f5; |
---|
1427 | list v3 = k,1,f4; |
---|
1428 | list v4 = h,4,f6,f7,f9,f11; |
---|
1429 | list v5 = m,1,f8; |
---|
1430 | list v6 = n,1,f10; |
---|
1431 | list v7 = p,1,f12; |
---|
1432 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1433 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1434 | return (G); |
---|
1435 | } |
---|
1436 | |
---|
1437 | static proc graph69(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1438 | { |
---|
1439 | graph G; |
---|
1440 | list f1 = i,j,d[1]; |
---|
1441 | list f2 = j,i,d[1]; |
---|
1442 | list f3 = j,k,d[2]; |
---|
1443 | list f4 = k,j,d[2]; |
---|
1444 | list f5 = j,h,d[3]; |
---|
1445 | list f6 = h,j,d[3]; |
---|
1446 | list f7 = h,m,d[4]; |
---|
1447 | list f8 = m,h,d[4]; |
---|
1448 | list f9 = h,n,d[5]; |
---|
1449 | list f10 = n,h,d[5]; |
---|
1450 | list f11 = n,p,d[6]; |
---|
1451 | list f12 = p,n,d[6]; |
---|
1452 | list v1 = i,1,f1; |
---|
1453 | list v2 = j,3,f2,f3,f5; |
---|
1454 | list v3 = k,1,f4; |
---|
1455 | list v4 = h,3,f6,f7,f9; |
---|
1456 | list v5 = m,1,f8; |
---|
1457 | list v6 = n,2,f10,f11; |
---|
1458 | list v7 = p,1,f12; |
---|
1459 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1460 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1461 | return (G); |
---|
1462 | } |
---|
1463 | |
---|
1464 | static proc graph610(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1465 | { |
---|
1466 | graph G; |
---|
1467 | list f1 = i,j,d[1]; |
---|
1468 | list f2 = j,i,d[1]; |
---|
1469 | list f3 = j,k,d[2]; |
---|
1470 | list f4 = k,j,d[2]; |
---|
1471 | list f5 = k,h,d[3]; |
---|
1472 | list f6 = h,k,d[3]; |
---|
1473 | list f7 = k,m,d[4]; |
---|
1474 | list f8 = m,k,d[4]; |
---|
1475 | list f9 = k,n,d[5]; |
---|
1476 | list f10 = n,k,d[5]; |
---|
1477 | list f11 = k,p,d[6]; |
---|
1478 | list f12 = p,k,d[6]; |
---|
1479 | list v1 = i,1,f1; |
---|
1480 | list v2 = j,2,f2,f3; |
---|
1481 | list v3 = k,5,f4,f5,f7,f9,f11; |
---|
1482 | list v4 = h,1,f6; |
---|
1483 | list v5 = m,1,f8; |
---|
1484 | list v6 = n,1,f10; |
---|
1485 | list v7 = p,1,f12; |
---|
1486 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1487 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1488 | return (G); |
---|
1489 | } |
---|
1490 | |
---|
1491 | static proc graph611(list d, int i, int j, int k, int h, int m, int n, int p) |
---|
1492 | { |
---|
1493 | graph G; |
---|
1494 | list f1 = i,j,d[1]; |
---|
1495 | list f2 = j,i,d[1]; |
---|
1496 | list f3 = j,k,d[2]; |
---|
1497 | list f4 = k,j,d[2]; |
---|
1498 | list f5 = j,h,d[3]; |
---|
1499 | list f6 = h,j,d[3]; |
---|
1500 | list f7 = j,m,d[4]; |
---|
1501 | list f8 = m,j,d[4]; |
---|
1502 | list f9 = j,n,d[5]; |
---|
1503 | list f10 = n,j,d[5]; |
---|
1504 | list f11 = j,p,d[6]; |
---|
1505 | list f12 = p,j,d[6]; |
---|
1506 | list v1 = i,1,f1; |
---|
1507 | list v2 = j,6,f2,f3,f5,f7,f9,f11; |
---|
1508 | list v3 = k,1,f4; |
---|
1509 | list v4 = h,1,f6; |
---|
1510 | list v5 = m,1,f8; |
---|
1511 | list v6 = n,1,f10; |
---|
1512 | list v7 = p,1,f12; |
---|
1513 | G.vertices = v1,v2,v3,v4,v5,v6,v7; |
---|
1514 | G.edges = f1,f3,f5,f7,f9,f11; |
---|
1515 | return (G); |
---|
1516 | } |
---|
1517 | |
---|
1518 | //////////////////////////////////////////////////////////////////////////////// |
---|
1519 | /// Auxilary Static Procedures in this Library ///////////////////////////////// |
---|
1520 | //////////////////////////////////////////////////////////////////////////////// |
---|
1521 | |
---|
1522 | proc part(poly f, int n) |
---|
1523 | "USAGE: part(f,n); f poly, n int |
---|
1524 | RETURN: poly |
---|
1525 | PURPOSE: computing the homogeneous component of a polynomial. |
---|
1526 | EXAMPLE: example part; shows examples |
---|
1527 | " |
---|
1528 | { |
---|
1529 | int i; |
---|
1530 | poly p; |
---|
1531 | for (i=1;i<=size(f);i++) |
---|
1532 | { |
---|
1533 | if (deg(f[i])==n) {p=p+f[i];} |
---|
1534 | } |
---|
1535 | return (p); |
---|
1536 | } |
---|
1537 | example |
---|
1538 | { |
---|
1539 | "EXAMPLE:"; echo=2; |
---|
1540 | ring r = 0,(x,y,z),wp(1,2,3); |
---|
1541 | poly f = 1+x+x2+x3+x4+y+y2+y3+z+z2+xy+xz+yz+xyz; |
---|
1542 | part(f,0); |
---|
1543 | part(f,1); |
---|
1544 | part(f,2); |
---|
1545 | part(f,3); |
---|
1546 | part(f,4); |
---|
1547 | part(f,5); |
---|
1548 | part(f,6); |
---|
1549 | } |
---|
1550 | |
---|
1551 | proc parts(poly f, int i, int j) |
---|
1552 | "USAGE: parts(f,i,j); f poly, i int, j int |
---|
1553 | RETURN: poly |
---|
1554 | THEORY: computing a polynomial which is the sum of the homogeneous |
---|
1555 | components of a polynomial. |
---|
1556 | EXAMPLE: example parts; shows examples |
---|
1557 | " |
---|
1558 | { |
---|
1559 | int k; |
---|
1560 | poly p; |
---|
1561 | for (k=i;k<=j;k++) |
---|
1562 | { |
---|
1563 | p=p+part(f,k); |
---|
1564 | } |
---|
1565 | return (p); |
---|
1566 | } |
---|
1567 | example |
---|
1568 | { |
---|
1569 | "EXAMPLE:"; echo=2; |
---|
1570 | ring r = 0,(x,y,z),wp(1,2,3); |
---|
1571 | poly f = 1+x+x2+x3+x4+y+y2+y3+z+z2+xy+xz+yz+xyz; |
---|
1572 | parts(f,2,4); |
---|
1573 | } |
---|
1574 | |
---|
1575 | proc logg(poly f, int n) |
---|
1576 | "USAGE: logg(f,n); f poly, n int |
---|
1577 | RETURN: poly |
---|
1578 | THEORY: computing Chern characters from total Chern classes. |
---|
1579 | EXAMPLE: example logg; shows examples |
---|
1580 | " |
---|
1581 | { |
---|
1582 | poly p; |
---|
1583 | int i,j,k,m; |
---|
1584 | if (n==0) {p=0;} |
---|
1585 | if (n==1) {p=part(f,1);} |
---|
1586 | else |
---|
1587 | { |
---|
1588 | list l=-part(f,1); |
---|
1589 | for (j=2;j<=n;j++) |
---|
1590 | { |
---|
1591 | poly q; |
---|
1592 | for (k=1;k<j;k++) |
---|
1593 | { |
---|
1594 | q=q+part(f,k)*l[j-k]; |
---|
1595 | } |
---|
1596 | q=-j*part(f,j)-q; |
---|
1597 | l=insert(l,q,size(l)); |
---|
1598 | kill q; |
---|
1599 | } |
---|
1600 | for (m=1;m<=n;m++) |
---|
1601 | { |
---|
1602 | p=p+1/factorial(m)*(-1)^m*l[m]; |
---|
1603 | } |
---|
1604 | } |
---|
1605 | return (p); |
---|
1606 | } |
---|
1607 | example |
---|
1608 | { |
---|
1609 | "EXAMPLE:"; echo=2; |
---|
1610 | ring r = 0,(x,y),wp(1,2); |
---|
1611 | poly f = 1+x+y; |
---|
1612 | logg(f,4); |
---|
1613 | } |
---|
1614 | |
---|
1615 | proc expp(poly f, int n) |
---|
1616 | "USAGE: expp(f,n); f poly, n int |
---|
1617 | RETURN: poly |
---|
1618 | PURPOSE: computing total Chern classes from Chern characters. |
---|
1619 | EXAMPLE: example expp; shows examples |
---|
1620 | " |
---|
1621 | { |
---|
1622 | poly p; |
---|
1623 | int i,j,k; |
---|
1624 | if (deg(f)==0) {p=1;} |
---|
1625 | else |
---|
1626 | { |
---|
1627 | list l=1; |
---|
1628 | for (i=1;i<=n;i++) |
---|
1629 | { |
---|
1630 | poly q; |
---|
1631 | for (j=1;j<=i;j++) |
---|
1632 | { |
---|
1633 | q=q+factorial(j)*(-1)^(j-1)*l[i-j+1]*part(f,j)/i; |
---|
1634 | } |
---|
1635 | l=insert(l,q,size(l)); |
---|
1636 | kill q; |
---|
1637 | } |
---|
1638 | for (k=1;k<=size(l);k++) |
---|
1639 | { |
---|
1640 | p=p+l[k]; |
---|
1641 | } |
---|
1642 | } |
---|
1643 | return (p); |
---|
1644 | } |
---|
1645 | example |
---|
1646 | { |
---|
1647 | "EXAMPLE:"; echo=2; |
---|
1648 | ring r = 0,(x),dp; |
---|
1649 | poly f = 3+x; |
---|
1650 | expp(f,3); |
---|
1651 | } |
---|
1652 | |
---|
1653 | static proc adams(poly f, int n) |
---|
1654 | { |
---|
1655 | poly p; |
---|
1656 | int i; |
---|
1657 | for (i=0;i<=deg(f);i++) |
---|
1658 | { |
---|
1659 | p=p+n^i*part(f,i); |
---|
1660 | } |
---|
1661 | return (p); |
---|
1662 | } |
---|
1663 | |
---|
1664 | static proc wedges(int n, poly f, int d) |
---|
1665 | { |
---|
1666 | int i,j; |
---|
1667 | list l; |
---|
1668 | if (n==0) {l=1;} |
---|
1669 | if (n==1) {l=1,f;} |
---|
1670 | else |
---|
1671 | { |
---|
1672 | l=1,f; |
---|
1673 | for (i=2;i<=n;i++) |
---|
1674 | { |
---|
1675 | poly q; |
---|
1676 | for (j=1;j<=i;j++) |
---|
1677 | { |
---|
1678 | q=q+((-1)^(i-j))*parts(l[j]*adams(f,i-j+1),0,d)/i; |
---|
1679 | } |
---|
1680 | l=insert(l,q,size(l)); |
---|
1681 | kill q; |
---|
1682 | } |
---|
1683 | } |
---|
1684 | return (l); |
---|
1685 | } |
---|
1686 | |
---|
1687 | static proc schur(list p, poly f) |
---|
1688 | { |
---|
1689 | int i,j; |
---|
1690 | int n = size(p); |
---|
1691 | matrix M[n][n]; |
---|
1692 | for (i=1;i<=n;i++) |
---|
1693 | { |
---|
1694 | for (j=1;j<=n;j++) |
---|
1695 | { |
---|
1696 | M[i,j] = part(f,p[i]+j-i); |
---|
1697 | } |
---|
1698 | } |
---|
1699 | return (det(M)); |
---|
1700 | } |
---|
1701 | |
---|
1702 | //////////////////////////////////////////////////////////////////////////////// |
---|
1703 | //////// Procedures concerned with abstract varieties ////////////////////////// |
---|
1704 | //////////////////////////////////////////////////////////////////////////////// |
---|
1705 | |
---|
1706 | proc printVariety(variety V) |
---|
1707 | "USAGE: printVariety(V); V variety |
---|
1708 | ASSUME: V is an abstract variety |
---|
1709 | THEORY: This is the print function used by Singular to print an abstract |
---|
1710 | variety. |
---|
1711 | KEYWORDS: abstract variety, projective space, Grassmannian |
---|
1712 | EXAMPLE: example printVariety; shows an example |
---|
1713 | " |
---|
1714 | { |
---|
1715 | "A variety of dimension", V.dimension; |
---|
1716 | } |
---|
1717 | example |
---|
1718 | { |
---|
1719 | "EXAMPLE:"; echo=2; |
---|
1720 | ring r = 0, (h,e), wp(1,1); |
---|
1721 | ideal rels = he,h2+e2; |
---|
1722 | variety V = makeVariety(2,rels); |
---|
1723 | V; |
---|
1724 | } |
---|
1725 | |
---|
1726 | proc makeVariety(int d, ideal i) |
---|
1727 | "USAGE: makeVariety(d,i); d int, i ideal |
---|
1728 | ASSUME: d is a nonnegative integer, i is an ideal |
---|
1729 | RETURN: variety |
---|
1730 | THEORY: create an abstract variety which has dimension d, and its Chow ring |
---|
1731 | should be a quotient ring |
---|
1732 | KEYWORDS: abstract variety, projective space, Grassmannian |
---|
1733 | EXAMPLE: example makeVariety; shows an example |
---|
1734 | " |
---|
1735 | { |
---|
1736 | def R = basering; |
---|
1737 | variety V; |
---|
1738 | V.dimension = d; |
---|
1739 | V.baseRing = R; |
---|
1740 | V.relations = i; |
---|
1741 | return(V); |
---|
1742 | } |
---|
1743 | example |
---|
1744 | { |
---|
1745 | "EXAMPLE:"; echo=2; |
---|
1746 | ring r = 0, (h,e), wp(1,1); |
---|
1747 | ideal rels = he,h2+e2; |
---|
1748 | variety V = makeVariety(2,rels); |
---|
1749 | V; |
---|
1750 | V.dimension; |
---|
1751 | V.relations; |
---|
1752 | } |
---|
1753 | |
---|
1754 | proc ChowRing(variety V) |
---|
1755 | "USAGE: ChowRing(V); V variety |
---|
1756 | ASSUME: V is an abstract variety |
---|
1757 | RETURN: qring |
---|
1758 | KEYWORDS: Chow ring, abstract variety, projective space, Grassmannian |
---|
1759 | EXAMPLE: example makeVariety; shows an example |
---|
1760 | " |
---|
1761 | { |
---|
1762 | def R = V.baseRing; |
---|
1763 | setring R; |
---|
1764 | ideal rels = V.relations; |
---|
1765 | qring CR = std(rels); |
---|
1766 | return (CR); |
---|
1767 | } |
---|
1768 | example |
---|
1769 | { |
---|
1770 | "EXAMPLE:"; echo=2; |
---|
1771 | ring r = 0, (h,e), wp(1,1); |
---|
1772 | ideal rels = he,h2+e2; |
---|
1773 | int d = 2; |
---|
1774 | variety V = makeVariety(2,rels); |
---|
1775 | ChowRing(V); |
---|
1776 | } |
---|
1777 | |
---|
1778 | //////////////////////////////////////////////////////////////////////////////// |
---|
1779 | |
---|
1780 | proc Grassmannian(int k, int n, list #) |
---|
1781 | "USAGE: Grassmannian(k,n); k int, n int |
---|
1782 | RETURN: variety |
---|
1783 | THEORY: create a Grassmannian G(k,n) as an abstract variety. This abstract |
---|
1784 | variety has diemnsion k(n-k) and its Chow ring is the quotient ring |
---|
1785 | of a polynomial ring in n-k variables q(1),...,q(n-k), which are the |
---|
1786 | Chern classes of tautological quotient bundle on G(k,n), modulo some |
---|
1787 | ideal generated by n-k polynomials which come from the Giambelli |
---|
1788 | formula. The monomial ordering of this Chow ring is 'wp' with vector |
---|
1789 | (1..k,1..n-k). Moreover, we export the Chern characters of |
---|
1790 | tautological subbundle and quotient bundle on G(k,n) |
---|
1791 | (say 'subBundle' and 'quotientBundle'). |
---|
1792 | KEYWORDS: Grassmannian, abstract variety, Schubert calculus |
---|
1793 | SEE ALSO: projectiveSpace, projectiveBundle |
---|
1794 | EXAMPLE: example Grassmannian; shows examples |
---|
1795 | " |
---|
1796 | { |
---|
1797 | string q; |
---|
1798 | if (size(#)==0) {q = "q";} |
---|
1799 | else |
---|
1800 | { |
---|
1801 | if (typeof(#[1]) == "string") {q = #[1];} |
---|
1802 | else {Error("invalid optional argument");} |
---|
1803 | } |
---|
1804 | variety G; |
---|
1805 | G.dimension = k*(n-k); |
---|
1806 | execute("ring r = 0,("+q+"(1..n-k)),wp(1..n-k);"); |
---|
1807 | setring r; |
---|
1808 | G.baseRing = r; |
---|
1809 | int i,j; |
---|
1810 | poly v = 1; |
---|
1811 | poly u = 1; |
---|
1812 | for (j=1;j<=n-k;j++) {v=v+q(j);} |
---|
1813 | list l; |
---|
1814 | for (i=1;i<=k;i++) |
---|
1815 | { |
---|
1816 | l=insert(l,1,size(l)); |
---|
1817 | u=u+(-1)^i*schur(l,v); |
---|
1818 | } |
---|
1819 | l=insert(l,1,size(l)); |
---|
1820 | ideal rels = schur(l,v); |
---|
1821 | int h = k+2; |
---|
1822 | while (h<=n) |
---|
1823 | { |
---|
1824 | l=insert(l,1,size(l)); |
---|
1825 | rels = rels,schur(l,v); |
---|
1826 | h++; |
---|
1827 | } |
---|
1828 | G.relations = rels; |
---|
1829 | int d = k*(n-k); |
---|
1830 | poly subBundle = reduce(logg(u,d)+k,std(rels)); |
---|
1831 | poly quotientBundle = reduce(logg(v,d)+n-k,std(rels)); |
---|
1832 | export (subBundle,quotientBundle); |
---|
1833 | kill u,v,d,l,rels; |
---|
1834 | return (G); |
---|
1835 | } |
---|
1836 | example |
---|
1837 | { |
---|
1838 | "EXAMPLE:"; echo=2; |
---|
1839 | variety G24 = Grassmannian(2,4); |
---|
1840 | G24; |
---|
1841 | def r = G24.baseRing; |
---|
1842 | setring r; |
---|
1843 | subBundle; |
---|
1844 | quotientBundle; |
---|
1845 | G24.dimension; |
---|
1846 | G24.relations; |
---|
1847 | ChowRing(G24); |
---|
1848 | } |
---|
1849 | |
---|
1850 | proc projectiveSpace(int n, list #) |
---|
1851 | "USAGE: projectiveSpace(n); n int |
---|
1852 | RETURN: variety |
---|
1853 | THEORY: create a projective space of dimension n as an abstract variety. Its |
---|
1854 | Chow ring is a quotient ring in one variable h modulo the ideal |
---|
1855 | generated by h^(n+1). |
---|
1856 | KEYWORDS: projective space, abstract variety |
---|
1857 | SEE ALSO: Grassmannian, projectiveBundle |
---|
1858 | EXAMPLE: example projectiveSpace; shows examples |
---|
1859 | " |
---|
1860 | { |
---|
1861 | string h; |
---|
1862 | if (size(#)==0) {h = "h";} |
---|
1863 | else |
---|
1864 | { |
---|
1865 | if (typeof(#[1]) == "string") {h = #[1];} |
---|
1866 | else {Error("invalid optional argument");} |
---|
1867 | } |
---|
1868 | variety P; |
---|
1869 | P.dimension = n; |
---|
1870 | execute("ring r = 0, ("+h+"), wp(1);"); |
---|
1871 | setring r; |
---|
1872 | P.baseRing = r; |
---|
1873 | ideal rels = var(1)^(n+1); |
---|
1874 | P.relations = rels; |
---|
1875 | poly u = 1; |
---|
1876 | poly v = 1 + var(1); |
---|
1877 | list l; |
---|
1878 | int i; |
---|
1879 | for (i=1;i<=n;i++) |
---|
1880 | { |
---|
1881 | l=insert(l,1,size(l)); |
---|
1882 | u=u+(-1)^i*schur(l,v); |
---|
1883 | } |
---|
1884 | poly subBundle = reduce(logg(u,n)+n,std(rels)); |
---|
1885 | poly quotientBundle = reduce(logg(v,n)+1,std(rels)); |
---|
1886 | export(subBundle,quotientBundle); |
---|
1887 | kill rels,u,v,l; |
---|
1888 | return (P); |
---|
1889 | } |
---|
1890 | example |
---|
1891 | { |
---|
1892 | "EXAMPLE:"; echo=2; |
---|
1893 | variety P = projectiveSpace(3); |
---|
1894 | P; |
---|
1895 | P.dimension; |
---|
1896 | def r = P.baseRing; |
---|
1897 | setring r; |
---|
1898 | P.relations; |
---|
1899 | ChowRing(P); |
---|
1900 | } |
---|
1901 | |
---|
1902 | proc projectiveBundle(sheaf S, list #) |
---|
1903 | "USAGE: projectiveBundle(S); S sheaf |
---|
1904 | INPUT: a sheaf on an abstract variety |
---|
1905 | RETURN: variety |
---|
1906 | THEORY: create a projective bundle as an abstract variety. This is related |
---|
1907 | to the enumeration of conics. |
---|
1908 | KEYWORDS: projective bundle, abstract variety, sheaf, enumeration of conics |
---|
1909 | SEE ALSO: projectiveSpace, Grassmannian |
---|
1910 | EXAMPLE: example projectiveBundle; shows examples |
---|
1911 | " |
---|
1912 | { |
---|
1913 | string z; |
---|
1914 | if (size(#)==0) {z = "z";} |
---|
1915 | else |
---|
1916 | { |
---|
1917 | if (typeof(#[1]) == "string") {z = #[1];} |
---|
1918 | else {Error("invalid optional argument");} |
---|
1919 | } |
---|
1920 | variety A; |
---|
1921 | def B = S.currentVariety; |
---|
1922 | def R = B.baseRing; |
---|
1923 | setring R; |
---|
1924 | ideal rels = B.relations; |
---|
1925 | int r = rankSheaf(S); |
---|
1926 | A.dimension = r - 1 + B.dimension; |
---|
1927 | poly c = totalChernClass(S); |
---|
1928 | execute("ring P = 0, ("+z+"), wp(1);"); |
---|
1929 | def CR = P + R; |
---|
1930 | setring CR; |
---|
1931 | A.baseRing = CR; |
---|
1932 | poly c = imap(R,c); |
---|
1933 | ideal rels = imap(R,rels); |
---|
1934 | poly g = var(1)^r; |
---|
1935 | int i; |
---|
1936 | for (i=1;i<=r;i++) {g=g+var(1)^(r-i)*part(c,i);} |
---|
1937 | A.relations = rels,g; |
---|
1938 | poly u = 1 + var(1); |
---|
1939 | poly f = logg(u,A.dimension)+1; |
---|
1940 | poly QuotientBundle = reduce(f,std(A.relations)); |
---|
1941 | export (QuotientBundle); |
---|
1942 | kill f,rels; |
---|
1943 | return (A); |
---|
1944 | } |
---|
1945 | example |
---|
1946 | { |
---|
1947 | "EXAMPLE:"; echo=2; |
---|
1948 | variety G = Grassmannian(3,5); |
---|
1949 | def r = G.baseRing; |
---|
1950 | setring r; |
---|
1951 | sheaf S = makeSheaf(G,subBundle); |
---|
1952 | sheaf B = dualSheaf(S)^2; |
---|
1953 | variety PB = projectiveBundle(B); |
---|
1954 | PB; |
---|
1955 | def R = PB.baseRing; |
---|
1956 | setring R; |
---|
1957 | QuotientBundle; |
---|
1958 | ChowRing(PB); |
---|
1959 | } |
---|
1960 | |
---|
1961 | proc productVariety(variety U, variety V) |
---|
1962 | "USAGE: productVariety(U,V); U variety, V variety |
---|
1963 | INPUT: two abstract varieties |
---|
1964 | OUTPUT: a product variety as an abstract variety |
---|
1965 | RETURN: variety |
---|
1966 | KEYWORDS: product variety, abstract variety |
---|
1967 | SEE ALSO: projectiveSpace, Grassmannian, projectiveBundle |
---|
1968 | EXAMPLE: example productVariety; shows examples |
---|
1969 | " |
---|
1970 | { |
---|
1971 | //def br = basering; |
---|
1972 | def ur = U.baseRing; setring ur; |
---|
1973 | ideal ii1 = U.relations; |
---|
1974 | def vr = V.baseRing; setring vr; |
---|
1975 | ideal ii2 = V.relations; |
---|
1976 | variety W; |
---|
1977 | W.dimension = U.dimension + V.dimension; |
---|
1978 | def temp = ringtensor(ur,vr); |
---|
1979 | setring temp; |
---|
1980 | W.baseRing = temp; |
---|
1981 | ideal i1 = imap(ur,ii1); |
---|
1982 | ideal i2 = imap(vr,ii2); |
---|
1983 | W.relations = i1 + i2; |
---|
1984 | setring ur; |
---|
1985 | kill ii1; |
---|
1986 | setring vr; |
---|
1987 | kill ii2; |
---|
1988 | //setring br; |
---|
1989 | return (W); |
---|
1990 | } |
---|
1991 | example |
---|
1992 | { |
---|
1993 | "EXAMPLE:"; echo=2; |
---|
1994 | variety P = projectiveSpace(3); |
---|
1995 | variety G = Grassmannian(2,4); |
---|
1996 | variety W = productVariety(P,G); |
---|
1997 | W; |
---|
1998 | W.dimension == P.dimension + G.dimension; |
---|
1999 | def r = W.baseRing; |
---|
2000 | setring r; |
---|
2001 | W.relations; |
---|
2002 | } |
---|
2003 | |
---|
2004 | //////////////////////////////////////////////////////////////////////////////// |
---|
2005 | |
---|
2006 | proc integral(variety V, poly f) |
---|
2007 | "USAGE: integral(V,f); V variety, f poly |
---|
2008 | INPUT: a abstract variety and a polynomial |
---|
2009 | RETURN: int |
---|
2010 | PURPOSE: computing intersection numbers. |
---|
2011 | EXAMPLE: example integral; shows an example |
---|
2012 | " |
---|
2013 | { |
---|
2014 | def R = V.baseRing; |
---|
2015 | setring R; |
---|
2016 | ideal rels = V.relations; |
---|
2017 | return (leadcoef(reduce(f,std(rels)))); |
---|
2018 | } |
---|
2019 | example |
---|
2020 | { |
---|
2021 | "EXAMPLE:"; echo=2; |
---|
2022 | variety G = Grassmannian(2,4); |
---|
2023 | def r = G.baseRing; |
---|
2024 | setring r; |
---|
2025 | integral(G,q(1)^4); |
---|
2026 | } |
---|
2027 | |
---|
2028 | proc SchubertClass(list p) |
---|
2029 | "USAGE: SchubertClass(p); p list |
---|
2030 | INPUT: a list of integers which is a partition |
---|
2031 | RETURN: poly |
---|
2032 | PURPOSE: compute the Schubert classes on a Grassmannian. |
---|
2033 | EXAMPLE: example SchubertClass; shows an example |
---|
2034 | " |
---|
2035 | { |
---|
2036 | def R = basering; |
---|
2037 | setring R; |
---|
2038 | poly f = 1; |
---|
2039 | if (size(p) == 0) {return (f);} |
---|
2040 | int i; |
---|
2041 | for (i=1;i<=nvars(R);i++) |
---|
2042 | { |
---|
2043 | f = f + var(i); |
---|
2044 | } |
---|
2045 | return (schur(p,f)); |
---|
2046 | } |
---|
2047 | example |
---|
2048 | { |
---|
2049 | "EXAMPLE:"; echo=2; |
---|
2050 | variety G = Grassmannian(2,4); |
---|
2051 | def r = G.baseRing; |
---|
2052 | setring r; |
---|
2053 | list p = 1,1; |
---|
2054 | SchubertClass(p); |
---|
2055 | } |
---|
2056 | |
---|
2057 | //////////////////////////////////////////////////////////////////////////////// |
---|
2058 | |
---|
2059 | proc dualPartition(int k, int n, list p) |
---|
2060 | "USAGE: dualPartition(k,n,p); k int, n int, p list |
---|
2061 | INPUT: two integers and a partition |
---|
2062 | RETURN: list |
---|
2063 | PURPOSE: compute the dual of a partition. |
---|
2064 | SEE ALSO: SchubertClass |
---|
2065 | EXAMPLE: example dualPartition; shows an example |
---|
2066 | " |
---|
2067 | { |
---|
2068 | while (size(p) < k) |
---|
2069 | { |
---|
2070 | p = insert(p,0,size(p)); |
---|
2071 | } |
---|
2072 | int i; |
---|
2073 | list l; |
---|
2074 | for (i=1;i<=size(p);i++) |
---|
2075 | { |
---|
2076 | l[i] = n-k-p[size(p)-i+1]; |
---|
2077 | } |
---|
2078 | return (l); |
---|
2079 | } |
---|
2080 | example |
---|
2081 | { |
---|
2082 | "EXAMPLE:"; echo=2; |
---|
2083 | ring r = 0,(x),dp; |
---|
2084 | dualPartition(2,4,list(2,1)); |
---|
2085 | } |
---|
2086 | |
---|
2087 | //////////////////////////////////////////////////////////////////////////////// |
---|
2088 | ////////// Procedures concerned with sheaves /////////////////////////////////// |
---|
2089 | //////////////////////////////////////////////////////////////////////////////// |
---|
2090 | |
---|
2091 | proc printSheaf(sheaf S) |
---|
2092 | "USAGE: printSheaf(S); S sheaf |
---|
2093 | RETURN: string |
---|
2094 | INPUT: a sheaf |
---|
2095 | THEORY: This is the print function used by Singular to print a sheaf. |
---|
2096 | SEE ALSO: makeSheaf, rankSheaf |
---|
2097 | EXAMPLE: example printSheaf; shows an example |
---|
2098 | " |
---|
2099 | { |
---|
2100 | "A sheaf of rank ", rankSheaf(S); |
---|
2101 | } |
---|
2102 | example |
---|
2103 | { |
---|
2104 | "EXAMPLE:"; echo=2; |
---|
2105 | variety X; |
---|
2106 | X.dimension = 4; |
---|
2107 | ring r = 0,(c(1..2),d(1..3)),wp(1..2,1..3); |
---|
2108 | setring r; |
---|
2109 | X.baseRing = r; |
---|
2110 | poly c = 1 + c(1) + c(2); |
---|
2111 | poly ch = 2 + logg(c,4); |
---|
2112 | sheaf S = makeSheaf(X,ch); |
---|
2113 | S; |
---|
2114 | } |
---|
2115 | |
---|
2116 | proc makeSheaf(variety V, poly ch) |
---|
2117 | "USAGE: makeSheaf(V,ch); V variety, ch poly |
---|
2118 | RETURN: sheaf |
---|
2119 | THEORY: create a sheaf on an abstract variety, and its Chern character is |
---|
2120 | the polynomial ch. |
---|
2121 | SEE ALSO: printSheaf, rankSheaf |
---|
2122 | EXAMPLE: example makeSheaf; shows an example |
---|
2123 | " |
---|
2124 | { |
---|
2125 | def R = basering; |
---|
2126 | sheaf S; |
---|
2127 | S.currentVariety = V; |
---|
2128 | S.ChernCharacter = ch; |
---|
2129 | return(S); |
---|
2130 | } |
---|
2131 | example |
---|
2132 | { |
---|
2133 | "EXAMPLE:"; echo=2; |
---|
2134 | variety X; |
---|
2135 | X.dimension = 4; |
---|
2136 | ring r = 0,(c(1..2),d(1..3)),wp(1..2,1..3); |
---|
2137 | setring r; |
---|
2138 | X.baseRing = r; |
---|
2139 | poly c = 1 + c(1) + c(2); |
---|
2140 | poly ch = 2 + logg(c,4); |
---|
2141 | sheaf S = makeSheaf(X,ch); |
---|
2142 | S; |
---|
2143 | } |
---|
2144 | |
---|
2145 | proc rankSheaf(sheaf S) |
---|
2146 | "USAGE: rankSheaf(S); S sheaf |
---|
2147 | RETURN: int |
---|
2148 | INPUT: S is a sheaf |
---|
2149 | OUTPUT: a positive integer which is the rank of a sheaf. |
---|
2150 | SEE ALSO: makeSheaf, printSheaf |
---|
2151 | EXAMPLE: example rankSheaf; shows an example |
---|
2152 | " |
---|
2153 | { |
---|
2154 | variety V = S.currentVariety; |
---|
2155 | def R = V.baseRing; |
---|
2156 | setring R; |
---|
2157 | poly f = S.ChernCharacter; |
---|
2158 | return (int(part(f,0))); |
---|
2159 | } |
---|
2160 | example |
---|
2161 | { |
---|
2162 | "EXAMPLE:"; echo=2; |
---|
2163 | variety G = Grassmannian(2,4); |
---|
2164 | def R = G.baseRing; |
---|
2165 | setring R; |
---|
2166 | sheaf S = makeSheaf(G,subBundle); |
---|
2167 | rankSheaf(S); |
---|
2168 | } |
---|
2169 | |
---|
2170 | proc totalChernClass(sheaf S) |
---|
2171 | "USAGE: totalChernClass(S); S sheaf |
---|
2172 | RETURN: poly |
---|
2173 | INPUT: S is a sheaf |
---|
2174 | OUTPUT: a polynomial which is the total Chern class of a sheaf |
---|
2175 | SEE ALSO: totalSegreClass, topChernClass, ChernClass |
---|
2176 | EXAMPLE: example totalChernClass; shows an example |
---|
2177 | " |
---|
2178 | { |
---|
2179 | variety V = S.currentVariety; |
---|
2180 | int d = V.dimension; |
---|
2181 | def R = V.baseRing; |
---|
2182 | setring R; |
---|
2183 | poly ch = S.ChernCharacter; |
---|
2184 | poly f = expp(ch,d); |
---|
2185 | ideal rels = std(V.relations); |
---|
2186 | return (reduce(f,rels)); |
---|
2187 | } |
---|
2188 | example |
---|
2189 | { |
---|
2190 | "EXAMPLE:"; echo=2; |
---|
2191 | variety X; |
---|
2192 | X.dimension = 4; |
---|
2193 | ring r = 0,(c(1..2),d(1..3)),wp(1..2,1..3); |
---|
2194 | setring r; |
---|
2195 | X.baseRing = r; |
---|
2196 | poly c = 1 + c(1) + c(2); |
---|
2197 | poly ch = 2 + logg(c,4); |
---|
2198 | sheaf E = makeSheaf(X,ch); |
---|
2199 | sheaf S = E^3; |
---|
2200 | totalChernClass(S); |
---|
2201 | } |
---|
2202 | |
---|
2203 | proc ChernClass(sheaf S, int i) |
---|
2204 | "USAGE: ChernClass(S,i); S sheaf, i int |
---|
2205 | INPUT: S is a sheaf, i is a nonnegative integer |
---|
2206 | RETURN: poly |
---|
2207 | THEORY: This is the i-th Chern class of a sheaf |
---|
2208 | SEE ALSO: topChernClass, totalChernClass |
---|
2209 | EXAMPLE: example ChernClass; shows an example |
---|
2210 | " |
---|
2211 | { |
---|
2212 | return (part(totalChernClass(S),i)); |
---|
2213 | } |
---|
2214 | example |
---|
2215 | { |
---|
2216 | "EXAMPLE:"; echo=2; |
---|
2217 | variety X; |
---|
2218 | X.dimension = 4; |
---|
2219 | ring r = 0,(c(1..2),d(1..3)),wp(1..2,1..3); |
---|
2220 | setring r; |
---|
2221 | X.baseRing = r; |
---|
2222 | poly c = 1 + c(1) + c(2); |
---|
2223 | poly ch = 2 + logg(c,4); |
---|
2224 | sheaf E = makeSheaf(X,ch); |
---|
2225 | sheaf S = E^3; |
---|
2226 | ChernClass(S,1); |
---|
2227 | ChernClass(S,2); |
---|
2228 | ChernClass(S,3); |
---|
2229 | ChernClass(S,4); |
---|
2230 | } |
---|
2231 | |
---|
2232 | proc topChernClass(sheaf S) |
---|
2233 | "USAGE: topChernClass(S); S sheaf |
---|
2234 | RETURN: poly |
---|
2235 | INPUT: S is a sheaf |
---|
2236 | THEORY: This is the top Chern class of a sheaf |
---|
2237 | SEE ALSO: ChernClass, totalChernClass |
---|
2238 | EXAMPLE: example topChernClass; shows an example |
---|
2239 | " |
---|
2240 | { |
---|
2241 | return (ChernClass(S,rankSheaf(S))); |
---|
2242 | } |
---|
2243 | example |
---|
2244 | { |
---|
2245 | "EXAMPLE:"; echo=2; |
---|
2246 | variety G = Grassmannian(2,4); |
---|
2247 | def R = G.baseRing; |
---|
2248 | setring R; |
---|
2249 | sheaf S = makeSheaf(G,quotientBundle); |
---|
2250 | sheaf B = S^3; |
---|
2251 | topChernClass(B); |
---|
2252 | } |
---|
2253 | |
---|
2254 | proc totalSegreClass(sheaf S) |
---|
2255 | "USAGE: totalSegreClass(S); S sheaf |
---|
2256 | RETURN: poly |
---|
2257 | INPUT: S is a sheaf |
---|
2258 | THEORY: This is the total Segre class of a sheaf. |
---|
2259 | SEE AlSO: totalChernClass |
---|
2260 | EXAMPLE: example totalSegreClass; shows an example |
---|
2261 | " |
---|
2262 | { |
---|
2263 | //def D = dualSheaf(S); |
---|
2264 | variety V = S.currentVariety; |
---|
2265 | def R = V.baseRing; |
---|
2266 | setring R; |
---|
2267 | poly f = totalChernClass(S); |
---|
2268 | poly g; |
---|
2269 | int d = V.dimension; |
---|
2270 | ideal rels = std(V.relations); |
---|
2271 | if (f == 1) {return (1);} |
---|
2272 | else |
---|
2273 | { |
---|
2274 | poly t,h; |
---|
2275 | int i,j; |
---|
2276 | for (i=0;i<=d;i++) {t = t + (1-f)^i;} |
---|
2277 | for (j=0;j<=d;j++) {h = h + part(t,j);} |
---|
2278 | return (reduce(h,rels)); |
---|
2279 | } |
---|
2280 | } |
---|
2281 | example |
---|
2282 | { |
---|
2283 | "EXAMPLE:"; echo=2; |
---|
2284 | variety G = Grassmannian(2,4); |
---|
2285 | def R = G.baseRing; |
---|
2286 | setring R; |
---|
2287 | sheaf S = makeSheaf(G,subBundle); |
---|
2288 | totalSegreClass(S); |
---|
2289 | } |
---|
2290 | |
---|
2291 | proc dualSheaf(sheaf S) |
---|
2292 | "USAGE: dualSheaf(S); S sheaf |
---|
2293 | RETURN: sheaf |
---|
2294 | THEORY: This is the dual of a sheaf |
---|
2295 | SEE ALSO: addSheaf, symmetricPowerSheaf, tensorSheaf, quotSheaf |
---|
2296 | EXAMPLE: example dualSheaf; shows examples |
---|
2297 | " |
---|
2298 | { |
---|
2299 | variety V = S.currentVariety; |
---|
2300 | int d = V.dimension; |
---|
2301 | def R = V.baseRing; |
---|
2302 | setring R; |
---|
2303 | poly ch = S.ChernCharacter; |
---|
2304 | poly f = adams(ch,-1); |
---|
2305 | sheaf D; |
---|
2306 | D.currentVariety = V; |
---|
2307 | D.ChernCharacter = f; |
---|
2308 | return (D); |
---|
2309 | } |
---|
2310 | example |
---|
2311 | { |
---|
2312 | "EXAMPLE:"; echo=2; |
---|
2313 | variety G = Grassmannian(2,4); |
---|
2314 | def R = G.baseRing; |
---|
2315 | setring R; |
---|
2316 | sheaf S = makeSheaf(G,subBundle); |
---|
2317 | sheaf D = dualSheaf(S); |
---|
2318 | D; |
---|
2319 | } |
---|
2320 | |
---|
2321 | proc tensorSheaf(sheaf A, sheaf B) |
---|
2322 | "USAGE: tensorSheaf(A,B); A sheaf, B sheaf |
---|
2323 | RETURN: sheaf |
---|
2324 | THEORY: This is the tensor product of two sheaves |
---|
2325 | SEE ALSO: addSheaf, symmetricPowerSheaf, quotSheaf, dualSheaf |
---|
2326 | EXAMPLE: example tensorSheaf; shows examples |
---|
2327 | " |
---|
2328 | { |
---|
2329 | sheaf S; |
---|
2330 | variety V1 = A.currentVariety; |
---|
2331 | variety V2 = B.currentVariety; |
---|
2332 | def R1 = V1.baseRing; |
---|
2333 | setring R1; |
---|
2334 | poly c1 = A.ChernCharacter; |
---|
2335 | def R2 = V2.baseRing; |
---|
2336 | setring R2; |
---|
2337 | poly c2 = B.ChernCharacter; |
---|
2338 | if (nvars(R1) < nvars(R2)) |
---|
2339 | { |
---|
2340 | S.currentVariety = V2; |
---|
2341 | poly c = imap(R1,c1); |
---|
2342 | poly f = parts(c*c2,0,V2.dimension); |
---|
2343 | S.ChernCharacter = f; |
---|
2344 | return (S); |
---|
2345 | } |
---|
2346 | else |
---|
2347 | { |
---|
2348 | setring R1; |
---|
2349 | S.currentVariety = V1; |
---|
2350 | poly c = imap(R2,c2); |
---|
2351 | poly f = parts(c1*c,0,V1.dimension); |
---|
2352 | S.ChernCharacter = f; |
---|
2353 | return (S); |
---|
2354 | } |
---|
2355 | } |
---|
2356 | example |
---|
2357 | { |
---|
2358 | "EXAMPLE:"; echo=2; |
---|
2359 | variety G = Grassmannian(3,4); |
---|
2360 | def R = G.baseRing; |
---|
2361 | setring R; |
---|
2362 | sheaf S = makeSheaf(G,subBundle); |
---|
2363 | sheaf Q = makeSheaf(G,quotientBundle); |
---|
2364 | sheaf T = S*Q; |
---|
2365 | T; |
---|
2366 | } |
---|
2367 | |
---|
2368 | proc symmetricPowerSheaf(sheaf S, int n) |
---|
2369 | "USAGE: symmetricPowerSheaf(S,n); S sheaf, n int |
---|
2370 | RETURN: sheaf |
---|
2371 | THEORY: This is the n-th symmetric power of a sheaf |
---|
2372 | SEE ALSO: quotSheaf, addSheaf, tensorSheaf, dualSheaf |
---|
2373 | EXAMPLE: example symmetricPowerSheaf; shows examples |
---|
2374 | " |
---|
2375 | { |
---|
2376 | variety V = S.currentVariety; |
---|
2377 | def R = V.baseRing; |
---|
2378 | setring R; |
---|
2379 | int r = rankSheaf(S); |
---|
2380 | int d = V.dimension; |
---|
2381 | int i,j,m; |
---|
2382 | poly f = S.ChernCharacter; |
---|
2383 | poly result; |
---|
2384 | list s,w; |
---|
2385 | if (n==0) {result=1;} |
---|
2386 | if (n==1) {result=f;} |
---|
2387 | else |
---|
2388 | { |
---|
2389 | s = 1,f; |
---|
2390 | w = wedges(n,f,d); |
---|
2391 | for (i=2;i<=n;i++) |
---|
2392 | { |
---|
2393 | if (i<=r) {m=i;} |
---|
2394 | else {m=r;} |
---|
2395 | poly q; |
---|
2396 | for (j=1;j<=m;j++) |
---|
2397 | { |
---|
2398 | q = q + ((-1)^(j+1))*parts(w[j+1]*s[i-j+1],0,d); |
---|
2399 | } |
---|
2400 | s = insert(s,q,size(s)); |
---|
2401 | kill q; |
---|
2402 | } |
---|
2403 | result = s[n+1]; |
---|
2404 | } |
---|
2405 | sheaf A; |
---|
2406 | A.currentVariety = V; |
---|
2407 | A.ChernCharacter = result; |
---|
2408 | return (A); |
---|
2409 | } |
---|
2410 | example |
---|
2411 | { |
---|
2412 | "EXAMPLE:"; echo=2; |
---|
2413 | variety G = Grassmannian(2,4); |
---|
2414 | def R = G.baseRing; |
---|
2415 | setring R; |
---|
2416 | sheaf S = makeSheaf(G,quotientBundle); |
---|
2417 | sheaf B = symmetricPowerSheaf(S,3); |
---|
2418 | B; |
---|
2419 | sheaf A = S^3; |
---|
2420 | A; |
---|
2421 | A.ChernCharacter == B.ChernCharacter; |
---|
2422 | } |
---|
2423 | |
---|
2424 | proc quotSheaf(sheaf A, sheaf B) |
---|
2425 | "USAGE: quotSheaf(A,B); A sheaf, B sheaf |
---|
2426 | RETURN: sheaf |
---|
2427 | THEORY: This is the quotient of two sheaves |
---|
2428 | SEE ALSO: addSheaf, symmetricPowerSheaf, tensorSheaf, dualSheaf |
---|
2429 | EXAMPLE: example quotSheaf; shows an example |
---|
2430 | " |
---|
2431 | { |
---|
2432 | sheaf S; |
---|
2433 | variety V1 = A.currentVariety; |
---|
2434 | variety V2 = B.currentVariety; |
---|
2435 | def R1 = V1.baseRing; |
---|
2436 | setring R1; |
---|
2437 | poly c1 = A.ChernCharacter; |
---|
2438 | def R2 = V2.baseRing; |
---|
2439 | setring R2; |
---|
2440 | poly c2 = B.ChernCharacter; |
---|
2441 | if (nvars(R1) < nvars(R2)) |
---|
2442 | { |
---|
2443 | S.currentVariety = V2; |
---|
2444 | poly c = imap(R1,c1); |
---|
2445 | S.ChernCharacter = c - c2; |
---|
2446 | return (S); |
---|
2447 | } |
---|
2448 | else |
---|
2449 | { |
---|
2450 | setring R1; |
---|
2451 | S.currentVariety = V1; |
---|
2452 | poly c = imap(R2,c2); |
---|
2453 | S.ChernCharacter = c1 - c; |
---|
2454 | return (S); |
---|
2455 | } |
---|
2456 | } |
---|
2457 | example |
---|
2458 | { |
---|
2459 | "EXAMPLE:"; echo=2; |
---|
2460 | variety G = Grassmannian(3,5); |
---|
2461 | def r = G.baseRing; |
---|
2462 | setring r; |
---|
2463 | sheaf S = makeSheaf(G,subBundle); |
---|
2464 | sheaf B = dualSheaf(S)^2; |
---|
2465 | sheaf B3 = dualSheaf(S)^3; |
---|
2466 | sheaf B5 = dualSheaf(S)^5; |
---|
2467 | variety PB = projectiveBundle(B); |
---|
2468 | def R = PB.baseRing; |
---|
2469 | setring R; |
---|
2470 | sheaf Q = makeSheaf(PB,QuotientBundle); |
---|
2471 | sheaf V = dualSheaf(Q)*B3; |
---|
2472 | sheaf A = B5 - V; |
---|
2473 | A; |
---|
2474 | } |
---|
2475 | |
---|
2476 | proc addSheaf(sheaf A, sheaf B) |
---|
2477 | "USAGE: addSheaf(A,B); A sheaf, B sheaf |
---|
2478 | RETURN: sheaf |
---|
2479 | THEORY: This is the direct sum of two sheaves. |
---|
2480 | SEE ALSO: quotSheaf, symmetricPowerSheaf, tensorSheaf, dualSheaf |
---|
2481 | EXAMPLE: example addSheaf; shows an example |
---|
2482 | " |
---|
2483 | { |
---|
2484 | sheaf S; |
---|
2485 | variety V1 = A.currentVariety; |
---|
2486 | variety V2 = B.currentVariety; |
---|
2487 | def R1 = V1.baseRing; |
---|
2488 | setring R1; |
---|
2489 | poly c1 = A.ChernCharacter; |
---|
2490 | def R2 = V2.baseRing; |
---|
2491 | setring R2; |
---|
2492 | poly c2 = B.ChernCharacter; |
---|
2493 | if (nvars(R1) < nvars(R2)) |
---|
2494 | { |
---|
2495 | S.currentVariety = V2; |
---|
2496 | poly c = imap(R1,c1); |
---|
2497 | S.ChernCharacter = c + c2; |
---|
2498 | return (S); |
---|
2499 | } |
---|
2500 | else |
---|
2501 | { |
---|
2502 | setring R1; |
---|
2503 | S.currentVariety = V1; |
---|
2504 | poly c = imap(R2,c2); |
---|
2505 | S.ChernCharacter = c1 + c; |
---|
2506 | return (S); |
---|
2507 | } |
---|
2508 | } |
---|
2509 | example |
---|
2510 | { |
---|
2511 | "EXAMPLE:"; echo=2; |
---|
2512 | variety G = Grassmannian(3,5); |
---|
2513 | def r = G.baseRing; |
---|
2514 | setring r; |
---|
2515 | sheaf S = makeSheaf(G,subBundle); |
---|
2516 | sheaf Q = makeSheaf(G,quotientBundle); |
---|
2517 | sheaf D = S + Q; |
---|
2518 | D; |
---|
2519 | D.ChernCharacter == rankSheaf(D); |
---|
2520 | totalChernClass(D) == 1; |
---|
2521 | } |
---|