186:
247:
438:
455:
283:
hyperplane section will be smooth). A Lefschetz pencil restricts the nature of the acquired singularities, so that the topology may be analysed by the
374:
463:
500:
408:
325:
403:
143:
362:
345:(Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)). Extra Volume II: 309–314.
68:
56:
480:
272:
398:
216:
313:
291:
287:
method. The fibres with singularities are required to have a unique quadratic singularity, only.
98:
90:
33:
25:
276:
370:
358:
37:
29:
303:
208:
476:
350:
472:
346:
338:
309:
299:
284:
64:
295:
79:
494:
451:
426:
430:
204:
17:
261:
252:
which is in fact well-defined only outside the points on the intersection of
271:
The second point is that the fibers may themselves 'degenerate' and acquire
75:
294:. They apply in ways similar to, but more complicated than,
164:
369:. Wiley Classics Library. Wiley Interscience. p. 509.
302:. It has also been shown that Lefschetz pencils exist in
341:(1998). "Lefschetz fibrations in symplectic geometry".
219:
146:
109:′, spanning the pencil — in other words,
63:, namely a one-parameter family, parametrised by the
316:, leading to more recent research interest in them.
290:It has been shown that Lefschetz pencils exist in
241:
180:
82:; but with two qualifications about singularity.
133:′, and the general hyperplane section is
181:{\displaystyle \lambda L+\mu L^{\prime }=0.\ }
8:
439:Notices of the American Mathematical Society
85:The first point comes up if we assume that
312:has found a role for Lefschetz pencils in
230:
218:
163:
145:
74:, a Lefschetz pencil is something like a
390:
260:. To make a well-defined mapping, some
456:"The topology of symplectic manifolds"
7:
242:{\displaystyle V\rightarrow P^{1}\ }
67:. This means that in the case of a
14:
367:Principles of Algebraic Geometry
464:Turkish Journal of Mathematics
223:
1:
431:"What is a Lefschetz pencil?"
125:′= 0 for linear forms
101:. Suppose given hyperplanes
404:Encyclopedia of Mathematics
517:
399:"Monodromy transformation"
306:for the Ă©tale topology.
69:complex algebraic variety
57:linear system of divisors
55:is a particular kind of
326:Picard–Lefschetz theory
243:
191:Then the intersection
182:
93:, and the divisors on
32:, used to analyse the
343:Documenta Mathematica
244:
183:
24:is a construction in
501:Geometry of divisors
217:
144:
339:Donaldson, Simon K.
314:symplectic topology
292:characteristic zero
264:must be applied to
99:hyperplane sections
359:Griffiths, Phillip
239:
178:
91:projective variety
34:algebraic topology
26:algebraic geometry
238:
177:
137:intersected with
38:algebraic variety
30:Solomon Lefschetz
508:
487:
485:
479:. Archived from
460:
447:
435:
413:
412:
395:
380:
354:
304:characteristic p
300:smooth manifolds
248:
246:
245:
240:
236:
235:
234:
209:rational mapping
207:two. There is a
187:
185:
184:
179:
175:
168:
167:
22:Lefschetz pencil
516:
515:
511:
510:
509:
507:
506:
505:
491:
490:
483:
458:
450:
433:
425:
422:
417:
416:
397:
396:
392:
387:
377:
357:
337:
334:
322:
310:Simon Donaldson
296:Morse functions
285:vanishing cycle
277:Bertini's lemma
273:singular points
226:
215:
214:
159:
142:
141:
65:projective line
49:
12:
11:
5:
514:
512:
504:
503:
493:
492:
489:
488:
486:on 2022-02-06.
448:
421:
420:External links
418:
415:
414:
389:
388:
386:
383:
382:
381:
375:
355:
333:
330:
329:
328:
321:
318:
250:
249:
233:
229:
225:
222:
189:
188:
174:
171:
166:
162:
158:
155:
152:
149:
89:is given as a
80:Riemann sphere
48:
45:
28:considered by
13:
10:
9:
6:
4:
3:
2:
513:
502:
499:
498:
496:
482:
478:
474:
470:
466:
465:
457:
453:
452:Gompf, Robert
449:
445:
441:
440:
432:
428:
427:Gompf, Robert
424:
423:
419:
410:
406:
405:
400:
394:
391:
384:
378:
376:0-471-05059-8
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
335:
331:
327:
324:
323:
319:
317:
315:
311:
307:
305:
301:
297:
293:
288:
286:
282:
279:applies, the
278:
274:
269:
267:
263:
259:
255:
231:
227:
220:
213:
212:
211:
210:
206:
202:
198:
194:
172:
169:
160:
156:
153:
150:
147:
140:
139:
138:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
96:
92:
88:
83:
81:
77:
73:
70:
66:
62:
58:
54:
46:
44:
42:
39:
35:
31:
27:
23:
19:
481:the original
468:
462:
443:
437:
402:
393:
366:
342:
308:
289:
280:
270:
265:
257:
253:
251:
203:′ has
200:
196:
192:
190:
134:
130:
126:
122:
118:
114:
113:is given by
110:
106:
102:
94:
86:
84:
71:
60:
52:
50:
40:
21:
15:
363:Harris, Joe
205:codimension
121:′ by
47:Description
18:mathematics
332:References
262:blowing up
471:: 43–59.
409:EMS Press
224:→
165:′
157:μ
148:λ
78:over the
76:fibration
495:Category
454:(2001).
429:(2005).
365:(1994).
320:See also
117:= 0 and
477:1829078
411:, 2001
351:1648081
281:general
275:(where
475:
373:
349:
237:
176:
53:pencil
36:of an
484:(PDF)
459:(PDF)
434:(PDF)
385:Notes
256:with
199:with
446:(8).
371:ISBN
129:and
105:and
97:are
20:, a
298:on
195:of
59:on
43:.
16:In
497::
473:MR
469:25
467:.
461:.
444:52
442:.
436:.
407:,
401:,
361:;
347:MR
268:.
173:0.
51:A
379:.
353:.
266:V
258:V
254:J
232:1
228:P
221:V
201:H
197:H
193:J
170:=
161:L
154:+
151:L
135:V
131:L
127:L
123:L
119:H
115:L
111:H
107:H
103:H
95:V
87:V
72:V
61:V
41:V
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.