22:
1186:
619:
916:
416:
162:
1085:
1246:
467:
330:
1364:
1287:
660:
1426:
524:
1454:
1014:
969:
209:
493:
989:
824:
792:
768:
684:
233:
737:
1493:
1498:
421:
That is not enough to complete the fiber as non-singular (smooth and proper), and then glue it to infinity by a change of classical maps.
1579:
1553:
1527:
58:
40:
32:
1571:
1545:
1519:
251:. But many problems of contemporary mathematics remain open, notably, for those examples which are not rational, the question of
1603:
1598:
1096:
529:
168:
349:
93:
1022:
844:
340:
273:
663:
427:
1324:
1375:
771:
240:
172:
84:
72:
1200:
1251:
624:
1575:
1549:
1523:
1488:
244:
212:
80:
941:
1511:
336:
248:
182:
498:
472:
1434:
994:
1563:
974:
809:
777:
753:
669:
264:
252:
218:
689:
339:
to complete the surface at infinity, which may be achieved by writing the equation in
1592:
176:
1537:
1192:
239:. In practice, it is more commonly observed as a surface with a well-understood
236:
803:
267:
on the left side. This can be achieved through a transformation, such as:
263:
In order to properly express a conic bundle, one must first simplify the
834: − 1, without multiple roots. Consider the scalar
15:
1568:
Commutative
Algebra with a View Toward Algebraic Geometry
971:
is the surface obtained as "gluing" of the two surfaces
1437:
1378:
1327:
1254:
1203:
1099:
1025:
997:
977:
944:
847:
812:
780:
756:
692:
672:
627:
532:
501:
475:
430:
352:
276:
221:
185:
96:
343:
and expressing the first visible part of the fiber:
1448:
1420:
1358:
1318:smooth and proper surface, the mapping defined by
1281:
1240:
1180:
1079:
1008:
983:
963:
910:
818:
786:
762:
731:
678:
654:
613:
518:
487:
461:
410:
324:
227:
203:
156:
1181:{\displaystyle X'^{2}-aY'^{2}=P^{*}(T')Z'^{2}}
750:For the sake of simplicity, suppose the field
614:{\displaystyle X'^{2}-aY'^{2}=P^{*}(T')Z'^{2}}
424:Seen from infinity, (i.e. through the change
8:
167:Conic bundles can be considered as either a
1436:
1377:
1344:
1326:
1273:
1253:
1202:
1171:
1143:
1129:
1108:
1098:
1071:
1046:
1030:
1024:
996:
976:
949:
943:
897:
879:
852:
846:
841:One defines the reciprocal polynomial by
811:
779:
755:
691:
686:. Details are below about the map-change
671:
632:
626:
604:
576:
562:
541:
531:
500:
474:
451:
429:
411:{\displaystyle X^{2}-aY^{2}=P(T)Z^{2}.\,}
407:
398:
373:
357:
351:
321:
297:
281:
275:
220:
184:
153:
129:
101:
95:
59:Learn how and when to remove this message
1494:Intersection number (algebraic geometry)
157:{\displaystyle X^{2}+aXY+bY^{2}=P(T).\,}
1168:
1126:
1105:
601:
559:
538:
469:), the same fiber (excepted the fibers
179:. It can be associated with the symbol
1499:List of complex and algebraic surfaces
1080:{\displaystyle X^{2}-aY^{2}=P(T)Z^{2}}
911:{\displaystyle P^{*}(T')=T^{2m}P(1/T)}
175:. This can be a double covering of a
7:
325:{\displaystyle X^{2}-aY^{2}=P(T).\,}
243:, and the simplest cases share with
1431:and the same definition applied to
526:), written as the set of solutions
1540:; John Little; Don O'Shea (1997).
31:tone or style may not reflect the
14:
1542:Ideals, Varieties, and Algorithms
1469:a structure of conic bundle over
335:This is followed by placement in
1293:One shows the following result:
83:that appears as a solution to a
41:guide to writing better articles
20:
806:with coefficients in the field
794:any nonzero integer. Denote by
462:{\displaystyle T\mapsto T'=1/T}
1412:
1409:
1400:
1382:
1379:
1359:{\displaystyle p:U\to P_{1,k}}
1337:
1160:
1149:
1064:
1058:
905:
891:
869:
858:
726:
693:
649:
638:
593:
582:
434:
391:
385:
315:
309:
198:
186:
147:
141:
1:
1421:{\displaystyle (,t)\mapsto t}
1620:
1241:{\displaystyle x'=x,y'=y,}
1282:{\displaystyle z'=zt^{m}}
662:appears naturally as the
655:{\displaystyle P^{*}(T')}
247:the property of being a
1191:along the open sets by
964:{\displaystyle F_{a,P}}
918:, and the conic bundle
341:homogeneous coordinates
1450:
1422:
1360:
1283:
1242:
1182:
1081:
1010:
985:
965:
912:
820:
788:
764:
733:
680:
656:
615:
520:
489:
463:
412:
326:
229:
205:
158:
1451:
1423:
1361:
1284:
1243:
1183:
1082:
1011:
986:
966:
913:
821:
789:
765:
734:
681:
664:reciprocal polynomial
657:
616:
521:
490:
464:
413:
327:
230:
206:
204:{\displaystyle (a,P)}
159:
1435:
1376:
1325:
1297:Fundamental property
1252:
1201:
1097:
1023:
995:
975:
942:
845:
810:
778:
754:
690:
670:
625:
530:
519:{\displaystyle T'=0}
499:
473:
428:
350:
274:
219:
183:
94:
1604:Algebraic varieties
1544:(second ed.).
772:characteristic zero
488:{\displaystyle T=0}
241:divisor class group
1599:Algebraic geometry
1516:Algebraic Geometry
1449:{\displaystyle U'}
1446:
1418:
1356:
1279:
1238:
1178:
1077:
1009:{\displaystyle U'}
1006:
981:
961:
908:
816:
784:
760:
729:
676:
652:
611:
516:
485:
459:
408:
322:
245:Del Pezzo surfaces
225:
201:
154:
85:Cartesian equation
73:algebraic geometry
1489:Algebraic surface
984:{\displaystyle U}
819:{\displaystyle k}
787:{\displaystyle m}
763:{\displaystyle k}
679:{\displaystyle P}
228:{\displaystyle k}
213:Galois cohomology
81:algebraic variety
69:
68:
61:
35:used on Knowledge
33:encyclopedic tone
1611:
1585:
1559:
1533:
1512:Robin Hartshorne
1455:
1453:
1452:
1447:
1445:
1427:
1425:
1424:
1419:
1365:
1363:
1362:
1357:
1355:
1354:
1288:
1286:
1285:
1280:
1278:
1277:
1262:
1247:
1245:
1244:
1239:
1228:
1211:
1187:
1185:
1184:
1179:
1177:
1176:
1175:
1159:
1148:
1147:
1135:
1134:
1133:
1114:
1113:
1112:
1086:
1084:
1083:
1078:
1076:
1075:
1051:
1050:
1035:
1034:
1015:
1013:
1012:
1007:
1005:
990:
988:
987:
982:
970:
968:
967:
962:
960:
959:
917:
915:
914:
909:
901:
887:
886:
868:
857:
856:
825:
823:
822:
817:
793:
791:
790:
785:
769:
767:
766:
761:
738:
736:
735:
732:{\displaystyle }
730:
725:
714:
703:
685:
683:
682:
677:
661:
659:
658:
653:
648:
637:
636:
620:
618:
617:
612:
610:
609:
608:
592:
581:
580:
568:
567:
566:
547:
546:
545:
525:
523:
522:
517:
509:
494:
492:
491:
486:
468:
466:
465:
460:
455:
444:
417:
415:
414:
409:
403:
402:
378:
377:
362:
361:
337:projective space
331:
329:
328:
323:
302:
301:
286:
285:
249:rational surface
234:
232:
231:
226:
210:
208:
207:
202:
173:Châtelet surface
163:
161:
160:
155:
134:
133:
106:
105:
64:
57:
53:
50:
44:
43:for suggestions.
39:See Knowledge's
24:
23:
16:
1619:
1618:
1614:
1613:
1612:
1610:
1609:
1608:
1589:
1588:
1582:
1572:Springer-Verlag
1562:
1556:
1546:Springer-Verlag
1536:
1530:
1520:Springer-Verlag
1510:
1507:
1485:
1478:
1468:
1438:
1433:
1432:
1374:
1373:
1340:
1323:
1322:
1313:
1299:
1269:
1255:
1250:
1249:
1221:
1204:
1199:
1198:
1167:
1163:
1152:
1139:
1125:
1121:
1104:
1100:
1095:
1094:
1067:
1042:
1026:
1021:
1020:
998:
993:
992:
973:
972:
945:
940:
939:
937:
930:
875:
861:
848:
843:
842:
808:
807:
776:
775:
752:
751:
748:
718:
707:
696:
688:
687:
668:
667:
641:
628:
623:
622:
600:
596:
585:
572:
558:
554:
537:
533:
528:
527:
502:
497:
496:
471:
470:
437:
426:
425:
394:
369:
353:
348:
347:
293:
277:
272:
271:
261:
217:
216:
181:
180:
125:
97:
92:
91:
65:
54:
48:
45:
38:
29:This article's
25:
21:
12:
11:
5:
1617:
1615:
1607:
1606:
1601:
1591:
1590:
1587:
1586:
1580:
1564:David Eisenbud
1560:
1554:
1534:
1528:
1506:
1503:
1502:
1501:
1496:
1491:
1484:
1481:
1473:
1460:
1444:
1441:
1429:
1428:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1367:
1366:
1353:
1350:
1347:
1343:
1339:
1336:
1333:
1330:
1305:
1298:
1295:
1291:
1290:
1276:
1272:
1268:
1265:
1261:
1258:
1237:
1234:
1231:
1227:
1224:
1220:
1217:
1214:
1210:
1207:
1189:
1188:
1174:
1170:
1166:
1162:
1158:
1155:
1151:
1146:
1142:
1138:
1132:
1128:
1124:
1120:
1117:
1111:
1107:
1103:
1088:
1087:
1074:
1070:
1066:
1063:
1060:
1057:
1054:
1049:
1045:
1041:
1038:
1033:
1029:
1004:
1001:
980:
958:
955:
952:
948:
936:
933:
922:
907:
904:
900:
896:
893:
890:
885:
882:
878:
874:
871:
867:
864:
860:
855:
851:
815:
783:
774:and denote by
759:
747:
741:
728:
724:
721:
717:
713:
710:
706:
702:
699:
695:
675:
651:
647:
644:
640:
635:
631:
607:
603:
599:
595:
591:
588:
584:
579:
575:
571:
565:
561:
557:
553:
550:
544:
540:
536:
515:
512:
508:
505:
484:
481:
478:
458:
454:
450:
447:
443:
440:
436:
433:
419:
418:
406:
401:
397:
393:
390:
387:
384:
381:
376:
372:
368:
365:
360:
356:
333:
332:
320:
317:
314:
311:
308:
305:
300:
296:
292:
289:
284:
280:
265:quadratic form
260:
257:
253:unirationality
224:
211:in the second
200:
197:
194:
191:
188:
165:
164:
152:
149:
146:
143:
140:
137:
132:
128:
124:
121:
118:
115:
112:
109:
104:
100:
67:
66:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
1616:
1605:
1602:
1600:
1597:
1596:
1594:
1583:
1581:0-387-94269-6
1577:
1573:
1569:
1565:
1561:
1557:
1555:0-387-94680-2
1551:
1547:
1543:
1539:
1535:
1531:
1529:0-387-90244-9
1525:
1521:
1517:
1513:
1509:
1508:
1504:
1500:
1497:
1495:
1492:
1490:
1487:
1486:
1482:
1480:
1477:
1472:
1467:
1463:
1459:
1442:
1439:
1415:
1406:
1403:
1397:
1394:
1391:
1388:
1385:
1372:
1371:
1370:
1351:
1348:
1345:
1341:
1334:
1331:
1328:
1321:
1320:
1319:
1317:
1312:
1308:
1304:
1296:
1294:
1274:
1270:
1266:
1263:
1259:
1256:
1235:
1232:
1229:
1225:
1222:
1218:
1215:
1212:
1208:
1205:
1197:
1196:
1195:
1194:
1172:
1164:
1156:
1153:
1144:
1140:
1136:
1130:
1122:
1118:
1115:
1109:
1101:
1093:
1092:
1091:
1072:
1068:
1061:
1055:
1052:
1047:
1043:
1039:
1036:
1031:
1027:
1019:
1018:
1017:
1016:of equations
1002:
999:
978:
956:
953:
950:
946:
934:
932:
929:
925:
921:
902:
898:
894:
888:
883:
880:
876:
872:
865:
862:
853:
849:
839:
837:
833:
829:
826:, of degree 2
813:
805:
801:
797:
781:
773:
757:
746:
742:
740:
722:
719:
715:
711:
708:
704:
700:
697:
673:
665:
645:
642:
633:
629:
605:
597:
589:
586:
577:
573:
569:
563:
555:
551:
548:
542:
534:
513:
510:
506:
503:
482:
479:
476:
456:
452:
448:
445:
441:
438:
431:
422:
404:
399:
395:
388:
382:
379:
374:
370:
366:
363:
358:
354:
346:
345:
344:
342:
338:
318:
312:
306:
303:
298:
294:
290:
287:
282:
278:
270:
269:
268:
266:
258:
256:
254:
250:
246:
242:
238:
222:
215:of the field
214:
195:
192:
189:
178:
177:ruled surface
174:
170:
169:Severi–Brauer
150:
144:
138:
135:
130:
126:
122:
119:
116:
113:
110:
107:
102:
98:
90:
89:
88:
87:of the form:
86:
82:
78:
74:
63:
60:
52:
42:
36:
34:
27:
18:
17:
1567:
1541:
1515:
1475:
1470:
1465:
1461:
1457:
1430:
1368:
1315:
1310:
1306:
1302:
1301:The surface
1300:
1292:
1190:
1089:
938:
931:as follows:
927:
923:
919:
840:
835:
831:
827:
799:
795:
749:
744:
423:
420:
334:
262:
166:
77:conic bundle
76:
70:
55:
46:
30:
1193:isomorphism
237:isomorphism
235:through an
1593:Categories
1505:References
935:Definition
804:polynomial
743:The fiber
259:Expression
1538:David Cox
1456:gives to
1413:↦
1338:→
1145:∗
1116:−
1037:−
854:∗
634:∗
578:∗
549:−
435:↦
364:−
288:−
49:June 2009
1566:(1999).
1514:(1977).
1483:See also
1443:′
1260:′
1226:′
1209:′
1169:′
1157:′
1127:′
1106:′
1003:′
866:′
723:′
712:′
701:′
646:′
602:′
590:′
560:′
539:′
507:′
442:′
1578:
1552:
1526:
770:is of
621:where
79:is an
1314:is a
991:and
1576:ISBN
1550:ISBN
1524:ISBN
1248:and
1090:and
830:or 2
802:) a
495:and
75:, a
1369:by
666:of
171:or
71:In
1595::
1574:.
1570:.
1548:.
1522:.
1518:.
1479:.
1474:1,
838:.
739:.
255:.
1584:.
1558:.
1532:.
1476:k
1471:P
1466:P
1464:,
1462:a
1458:F
1440:U
1416:t
1410:)
1407:t
1404:,
1401:]
1398:z
1395::
1392:y
1389::
1386:x
1383:[
1380:(
1352:k
1349:,
1346:1
1342:P
1335:U
1332::
1329:p
1316:k
1311:P
1309:,
1307:a
1303:F
1289:.
1275:m
1271:t
1267:z
1264:=
1257:z
1236:,
1233:y
1230:=
1223:y
1219:,
1216:x
1213:=
1206:x
1173:2
1165:Z
1161:)
1154:T
1150:(
1141:P
1137:=
1131:2
1123:Y
1119:a
1110:2
1102:X
1073:2
1069:Z
1065:)
1062:T
1059:(
1056:P
1053:=
1048:2
1044:Y
1040:a
1032:2
1028:X
1000:U
979:U
957:P
954:,
951:a
947:F
928:P
926:,
924:a
920:F
906:)
903:T
899:/
895:1
892:(
889:P
884:m
881:2
877:T
873:=
870:)
863:T
859:(
850:P
836:a
832:m
828:m
814:k
800:T
798:(
796:P
782:m
758:k
745:c
727:]
720:z
716::
709:y
705::
698:x
694:[
674:P
650:)
643:T
639:(
630:P
606:2
598:Z
594:)
587:T
583:(
574:P
570:=
564:2
556:Y
552:a
543:2
535:X
514:0
511:=
504:T
483:0
480:=
477:T
457:T
453:/
449:1
446:=
439:T
432:T
405:.
400:2
396:Z
392:)
389:T
386:(
383:P
380:=
375:2
371:Y
367:a
359:2
355:X
319:.
316:)
313:T
310:(
307:P
304:=
299:2
295:Y
291:a
283:2
279:X
223:k
199:)
196:P
193:,
190:a
187:(
151:.
148:)
145:T
142:(
139:P
136:=
131:2
127:Y
123:b
120:+
117:Y
114:X
111:a
108:+
103:2
99:X
62:)
56:(
51:)
47:(
37:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.