158:
1131:-dimensional real space are to say how many regions there are, or how many faces of dimension 4, or how many bounded regions. These questions can be answered just from the intersection semilattice. For instance, two basic theorems, from Zaslavsky (1975), are that the number of regions of an affine arrangement equals (−1)
1589:
contains the degenerate hyperplane, then it has no regions because the complement is empty. However, it still has flats, an intersection semilattice, and faces. The preceding discussion assumes the degenerate hyperplane is not in the arrangement.
627:
444:
except, in the affine case, subsets whose intersection is empty. (The dimension of the empty set is defined to be −1.) This polynomial helps to solve some basic questions; see below. Another polynomial associated with
1112:
As an example, if the arrangement consists of three parallel lines, the intersection semilattice consists of the plane and the three lines, but not the empty set. There are four regions, none of them bounded.
1544:
1115:
If we add a line crossing the three parallels, then the intersection semilattice consists of the plane, the four lines, and the three points of intersection. There are eight regions, still none of them
431:
925:
1446:
affine space (which is hard to visualize because even the complex affine plane has four real dimensions), the complement is connected (all one piece) with holes where the hyperplanes were removed.
75:), which is the set that remains when the hyperplanes are removed from the whole space. One may ask how these properties are related to the arrangement and its intersection semilattice. The
1017:
262:, which is a generalization of a matroid (and has the same relationship to the intersection semilattice as does the matroid to the lattice in the lattice case), but is not a matroid if
826:
191:. (This is why the semilattice must be ordered by reverse inclusion—rather than by inclusion, which might seem more natural but would not yield a geometric (semi)lattice.)
971:
1665:
1593:
Sometimes one wants to allow repeated hyperplanes in the arrangement. We did not consider this possibility in the preceding discussion, but it makes no material difference.
1428:
1399:
for every hyperplane H that separates these two regions. If these variables are specialized to be all value q, then this is called the q-matrix (over the
Euclidean domain
765:
857:
1567:
1483:
The isomorphism can be described explicitly and gives a presentation of the cohomology in terms of generators and relations, where generators are represented (in the
1397:
1370:
1343:
184:
473:
1669:
102:
itself, all the individual hyperplanes, all intersections of pairs of hyperplanes, etc. (excluding, in the affine case, the empty set). These
259:
1496:
187:. If the arrangement is linear or projective, or if the intersection of all hyperplanes is nonempty, the intersection lattice is a
756:). (The empty set is excluded from the semilattice in the affine case specifically so that these relationships will be valid.)
330:
1771:
870:
1801:
976:
1620:
785:
1823:
1630:
32:
1183:
designed a fast algorithm to determine the face of an arrangement of hyperplanes containing an input point.
1077:. The regions are faces because the whole space is a flat. The faces of codimension 1 may be called the
1625:
1069:
is also divided into pieces by the hyperplanes that do not contain the flat; these pieces are called the
930:
1866:
1856:
1641:
1602:
1683:
1464:) is completely determined by the intersection semilattice. To be precise, the cohomology ring of
1119:
If we add one more line, parallel to the last, then there are 12 regions, of which two are bounded
1402:
1814:(1975), "Facing up to arrangements: face-count formulas for partitions of space by hyperplanes",
764:
The intersection semilattice determines another combinatorial invariant of the arrangement, the
1861:
1678:
1220:, a different one for each region. This poset is formed by choosing an arbitrary base region,
842:
1552:
1207:
1839:
1781:
1739:
1708:
1375:
1348:
1321:
682:
142:
1758:, Grundlehren der Mathematischen Wissenschaften , vol. 300, Berlin: Springer-Verlag,
146:. Historically, real arrangements of lines were the first arrangements investigated. If
8:
1315:
1101:
864:
52:
1789:
1696:
1484:
622:{\displaystyle w_{A}(x,y):=\sum _{B}x^{n-\dim f(B)}\sum _{C}(-1)^{|C-B|}y^{\dim f(C)},}
137:
95:
1186:
Another question about an arrangement in real space is to decide how many regions are
1797:
1767:
1488:
1431:
1304:
188:
98:
that are obtained by intersecting some of the hyperplanes; among these subspaces are
723:
is the smallest dimension of any flat, except that in the projective case it equals
1827:
1811:
1759:
1725:
1688:
1607:
1203:
773:
1449:
A typical problem about an arrangement in complex space is to describe the holes.
973:
have empty intersection, and by boundaries of elements of the same form for which
1835:
1777:
1735:
1704:
48:
1443:
1062:
1055:
1763:
1318:
introduced a matrix indexed by the regions. The matrix element for the region
1206:
asks for the smallest arrangement of a given dimension in general position in
1202:). This cannot be answered based solely on the intersection semilattice. The
1108:(if it is bounded) or a convex polygonal region which goes off to infinity.
1850:
1751:
1120:
832:
24:
1730:
1300:
1039:
44:
1150:-dimensional faces or bounded faces can be read off as the coefficient of
1042:, the complement is disconnected: it is made up of separate pieces called
258:) is a semilattice, there is an analogous matroid-like structure called a
1747:
1292:
1199:
1036:
1020:
157:
78:
1831:
1716:
Meiser, Stefan (1993), "Point location in arrangements of hyperplanes",
1700:
1473:
1453:
1296:
63:
generally concern geometrical, topological, or other properties of the
40:
1692:
1569:
any linear form defining the generic hyperplane of the arrangement.
1195:
1058:
20:
1210:
for which there does not exist a cell touched by all hyperplanes.
1187:
1105:
1093:. Adding an extra top element to the face semilattice gives the
203:
1539:{\displaystyle {\frac {1}{2\pi i}}{\frac {d\alpha }{\alpha }}.}
1430:) for the arrangement and much information is contained in its
1213:
A real linear arrangement has, besides its face semilattice, a
1140:(−1) and the number of bounded regions equals (−1)p
170:
1215:
1638:
Edelman, Paul H. (1984), "A partial order on the regions of
426:{\displaystyle p_{A}(y):=\sum _{B}(-1)^{|B|}y^{\dim f(B)},}
1796:. Vol. 1 (2nd ed.). Cambridge University Press.
1291:). In the special case when the hyperplanes arise from a
1452:
The basic theorem about complex arrangements is that the
1299:
with the weak order. In general, the poset of regions is
920:{\displaystyle e_{H_{1}}\wedge \cdots \wedge e_{H_{p}}}
1089:
of an arrangement is the set of all faces, ordered by
1644:
1555:
1499:
1405:
1378:
1351:
1324:
1054:, each of which is either a bounded region that is a
979:
933:
873:
845:
788:
476:
333:
859:. The Orlik–Solomon algebra is then the quotient of
1372:is given by the product of indeterminate variables
183:) is a meet semilattice and more specifically is a
1659:
1561:
1538:
1422:
1391:
1364:
1337:
1065:region which goes off to infinity. Each flat of
1011:
965:
919:
851:
820:
621:
425:
1670:Transactions of the American Mathematical Society
294:) := the intersection of the hyperplanes in
1848:
1303:by the number of separating hyperplanes and its
1239:) consisting of the hyperplanes that separate
59:. Questions about a hyperplane arrangement
1816:Memoirs of the American Mathematical Society
1247:. The regions are partially ordered so that
1100:In two dimensions (i.e., in the real affine
759:
171:The intersection semilattice and the matroid
1295:, the resulting poset is the corresponding
1012:{\displaystyle H_{1}\cap \cdots \cap H_{p}}
1746:
1061:, or an unbounded region that is a convex
768:. To define it, fix a commutative subring
655:Being a geometric lattice or semilattice,
246:is the intersection of the hyperplanes in
1810:
1729:
1682:
1647:
1407:
1127:Typical problems about an arrangement in
821:{\displaystyle \bigoplus _{H\in A}Ke_{H}}
222:for its ground set and has rank function
140:; such an arrangement is often called an
1792:(2011). "3.11 Hyperplane Arrangements".
1577:Sometimes it is convenient to allow the
156:
1788:
1637:
1437:
1308:
1849:
1715:
1180:
744:). The Whitney-number polynomial of
681:), which has an extensive theory (see
136:is 2-dimensional, the hyperplanes are
1030:
1585:, to belong to an arrangement. If
1227:, and associating with each region
663:) has a characteristic polynomial,
16:Partition of space by a hyperplanes
13:
966:{\displaystyle H_{1},\dots ,H_{p}}
867:generated by elements of the form
846:
14:
1878:
1572:
1472:) (with integer coefficients) is
839:with the usual boundary operator
685:). Thus it is good to know that
165:
161:A hyperplane arrangement in space
1660:{\displaystyle \mathbb {R} ^{n}}
1476:to the Orlik–Solomon algebra on
831:generated by the hyperplanes. A
748:is similarly related to that of
117:. The intersection semilattice
1194:-dimensional generalization of
1146:(1). Similarly, the number of
772:of the base field and form the
1417:
1411:
611:
605:
585:
571:
566:
556:
541:
535:
499:
487:
415:
409:
389:
381:
376:
366:
350:
344:
273:
1:
1824:American Mathematical Society
1613:
175:The intersection semilattice
1621:"Arrangement of hyperplanes"
1423:{\displaystyle \mathbb {Q} }
308:characteristic polynomial of
150:is 3-dimensional one has an
7:
1756:Arrangements of Hyperplanes
1718:Information and Computation
1667:dissected by hyperplanes",
1626:Encyclopedia of Mathematics
1596:
1581:, which is the whole space
10:
1883:
1104:) each region is a convex
125:) is partially ordered by
29:arrangement of hyperplanes
1822:(154), Providence, R.I.:
1794:Enumerative Combinatorics
1764:10.1007/978-3-662-02772-1
1603:Supersolvable arrangement
1164:, −1) or (−1)
852:{\displaystyle \partial }
760:The Orlik–Solomon algebra
451:Whitney-number polynomial
835:structure is defined on
436:summed over all subsets
1562:{\displaystyle \alpha }
1731:10.1006/inco.1993.1057
1661:
1563:
1540:
1424:
1393:
1366:
1339:
1013:
967:
921:
853:
822:
623:
427:
324:), can be defined by
162:
104:intersection subspaces
1662:
1579:degenerate hyperplane
1564:
1541:
1425:
1394:
1392:{\displaystyle a_{H}}
1367:
1365:{\displaystyle R_{j}}
1340:
1338:{\displaystyle R_{i}}
1314:Vadim Schechtman and
1208:real projective space
1014:
968:
922:
854:
823:
766:Orlik–Solomon algebra
624:
428:
185:geometric semilattice
160:
152:arrangement of planes
94:), is the set of all
1642:
1553:
1497:
1438:Complex arrangements
1403:
1376:
1349:
1322:
977:
931:
871:
843:
786:
779:of the vector space
474:
331:
270:) is not a lattice.
250:. In general, when
202:) is a lattice, the
143:arrangement of lines
110:are also called the
1316:Alexander Varchenko
1307:has been computed (
132:If the whole space
1657:
1559:
1536:
1489:differential forms
1485:de Rham cohomology
1456:of the complement
1420:
1389:
1362:
1335:
1009:
963:
917:
849:
818:
804:
619:
555:
514:
423:
365:
163:
1867:Oriented matroids
1857:Discrete geometry
1832:10.1090/memo/0154
1812:Zaslavsky, Thomas
1773:978-3-642-08137-8
1531:
1516:
1487:) as logarithmic
1432:Smith normal form
1031:Real arrangements
789:
546:
505:
356:
238:is any subset of
189:geometric lattice
127:reverse inclusion
1874:
1842:
1807:
1790:Stanley, Richard
1784:
1742:
1733:
1711:
1686:
1666:
1664:
1663:
1658:
1656:
1655:
1650:
1634:
1608:Oriented matroid
1568:
1566:
1565:
1560:
1545:
1543:
1542:
1537:
1532:
1527:
1519:
1517:
1515:
1501:
1429:
1427:
1426:
1421:
1410:
1398:
1396:
1395:
1390:
1388:
1387:
1371:
1369:
1368:
1363:
1361:
1360:
1344:
1342:
1341:
1336:
1334:
1333:
1204:McMullen problem
1087:face semilattice
1018:
1016:
1015:
1010:
1008:
1007:
989:
988:
972:
970:
969:
964:
962:
961:
943:
942:
926:
924:
923:
918:
916:
915:
914:
913:
890:
889:
888:
887:
858:
856:
855:
850:
827:
825:
824:
819:
817:
816:
803:
774:exterior algebra
628:
626:
625:
620:
615:
614:
590:
589:
588:
574:
554:
545:
544:
513:
486:
485:
432:
430:
429:
424:
419:
418:
394:
393:
392:
384:
364:
343:
342:
306:is empty. The
286:, let us define
230:) := codim(
35:of a finite set
1882:
1881:
1877:
1876:
1875:
1873:
1872:
1871:
1847:
1846:
1804:
1774:
1693:10.2307/1999150
1651:
1646:
1645:
1643:
1640:
1639:
1619:
1616:
1599:
1575:
1554:
1551:
1550:
1520:
1518:
1505:
1500:
1498:
1495:
1494:
1440:
1406:
1404:
1401:
1400:
1383:
1379:
1377:
1374:
1373:
1356:
1352:
1350:
1347:
1346:
1329:
1325:
1323:
1320:
1319:
1305:Möbius function
1286:
1271:
1260:
1253:
1226:
1172:
1159:
1154:in (−1) w
1145:
1139:
1033:
1003:
999:
984:
980:
978:
975:
974:
957:
953:
938:
934:
932:
929:
928:
909:
905:
904:
900:
883:
879:
878:
874:
872:
869:
868:
844:
841:
840:
812:
808:
793:
787:
784:
783:
762:
739:
714:
693:
676:
652:) is nonempty.
595:
591:
584:
570:
569:
565:
550:
519:
515:
509:
481:
477:
475:
472:
471:
457:
399:
395:
388:
380:
379:
375:
360:
338:
334:
332:
329:
328:
318:
276:
173:
168:
17:
12:
11:
5:
1880:
1870:
1869:
1864:
1859:
1845:
1844:
1808:
1803:978-1107602625
1802:
1786:
1772:
1752:Terao, Hiroaki
1744:
1724:(2): 286–303,
1713:
1684:10.1.1.308.820
1677:(2): 617–631,
1654:
1649:
1635:
1615:
1612:
1611:
1610:
1605:
1598:
1595:
1574:
1573:Technicalities
1571:
1558:
1547:
1546:
1535:
1530:
1526:
1523:
1514:
1511:
1508:
1504:
1439:
1436:
1419:
1416:
1413:
1409:
1386:
1382:
1359:
1355:
1332:
1328:
1284:
1269:
1258:
1251:
1224:
1168:
1155:
1141:
1135:
1125:
1124:
1121:parallelograms
1117:
1113:
1032:
1029:
1006:
1002:
998:
995:
992:
987:
983:
960:
956:
952:
949:
946:
941:
937:
912:
908:
903:
899:
896:
893:
886:
882:
877:
848:
829:
828:
815:
811:
807:
802:
799:
796:
792:
761:
758:
730:
705:
689:
667:
630:
629:
618:
613:
610:
607:
604:
601:
598:
594:
587:
583:
580:
577:
573:
568:
564:
561:
558:
553:
549:
543:
540:
537:
534:
531:
528:
525:
522:
518:
512:
508:
504:
501:
498:
495:
492:
489:
484:
480:
467:), defined by
455:
434:
433:
422:
417:
414:
411:
408:
405:
402:
398:
391:
387:
383:
378:
374:
371:
368:
363:
359:
355:
352:
349:
346:
341:
337:
316:
275:
272:
172:
169:
167:
166:General theory
164:
15:
9:
6:
4:
3:
2:
1879:
1868:
1865:
1863:
1862:Combinatorics
1860:
1858:
1855:
1854:
1852:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1732:
1727:
1723:
1719:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1685:
1680:
1676:
1672:
1671:
1652:
1636:
1632:
1628:
1627:
1622:
1618:
1617:
1609:
1606:
1604:
1601:
1600:
1594:
1591:
1588:
1584:
1580:
1570:
1556:
1533:
1528:
1524:
1521:
1512:
1509:
1506:
1502:
1493:
1492:
1491:
1490:
1486:
1481:
1479:
1475:
1471:
1467:
1463:
1459:
1455:
1450:
1447:
1445:
1435:
1433:
1414:
1384:
1380:
1357:
1353:
1330:
1326:
1317:
1312:
1310:
1306:
1302:
1298:
1294:
1290:
1283:
1279:
1275:
1268:
1264:
1257:
1250:
1246:
1242:
1238:
1234:
1230:
1223:
1219:
1217:
1211:
1209:
1205:
1201:
1197:
1193:
1189:
1184:
1182:
1181:Meiser (1993)
1178:
1176:
1171:
1167:
1163:
1158:
1153:
1149:
1144:
1138:
1134:
1130:
1122:
1118:
1114:
1111:
1110:
1109:
1107:
1103:
1098:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1057:
1053:
1049:
1045:
1041:
1038:
1028:
1026:
1022:
1004:
1000:
996:
993:
990:
985:
981:
958:
954:
950:
947:
944:
939:
935:
910:
906:
901:
897:
894:
891:
884:
880:
875:
866:
862:
838:
834:
833:chain complex
813:
809:
805:
800:
797:
794:
790:
782:
781:
780:
778:
775:
771:
767:
757:
755:
751:
747:
743:
737:
733:
729:
726:
722:
718:
712:
708:
704:
701:
697:
692:
688:
684:
680:
674:
670:
666:
662:
658:
653:
651:
647:
643:
639:
635:
616:
608:
602:
599:
596:
592:
581:
578:
575:
562:
559:
551:
547:
538:
532:
529:
526:
523:
520:
516:
510:
506:
502:
496:
493:
490:
482:
478:
470:
469:
468:
466:
462:
458:
452:
448:
443:
439:
420:
412:
406:
403:
400:
396:
385:
372:
369:
361:
357:
353:
347:
339:
335:
327:
326:
325:
323:
319:
312:
309:
305:
301:
297:
293:
289:
285:
281:
278:For a subset
271:
269:
265:
261:
257:
253:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
197:
192:
190:
186:
182:
178:
159:
155:
153:
149:
145:
144:
139:
135:
130:
128:
124:
120:
116:
113:
109:
105:
101:
97:
93:
89:
85:
81:
80:
77:intersection
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
25:combinatorics
22:
1819:
1815:
1793:
1755:
1748:Orlik, Peter
1721:
1717:
1674:
1668:
1624:
1592:
1586:
1582:
1578:
1576:
1548:
1482:
1477:
1469:
1465:
1461:
1457:
1451:
1448:
1441:
1313:
1309:Edelman 1984
1288:
1281:
1277:
1273:
1266:
1262:
1255:
1248:
1244:
1240:
1236:
1232:
1228:
1221:
1214:
1212:
1191:
1185:
1179:
1174:
1169:
1165:
1161:
1156:
1151:
1147:
1142:
1136:
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1095:face lattice
1094:
1090:
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1066:
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1293:root system
1276:) contains
1021:codimension
274:Polynomials
260:semimatroid
79:semilattice
41:hyperplanes
33:arrangement
1851:Categories
1614:References
1474:isomorphic
1454:cohomology
1297:Weyl group
1218:of regions
1200:tetrahedra
1063:polyhedral
1023:less than
927:for which
644:such that
313:, written
298:; this is
210:, written
86:, written
65:complement
53:projective
1679:CiteSeerX
1631:EMS Press
1557:α
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1510:π
1196:triangles
1188:simplices
1091:inclusion
997:∩
994:⋯
991:∩
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898:∧
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892:∧
847:∂
798:∈
791:⨁
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600:
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560:−
548:∑
530:
524:−
507:∑
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96:subspaces
1754:(1992),
1597:See also
1231:the set
1173:(−
1160:(−
1116:bounded.
1085:. The
1059:polytope
1052:chambers
112:flats of
21:geometry
1840:0357135
1782:1217488
1740:1241314
1709:0737888
1701:1999150
1633:, 2001
1444:complex
1106:polygon
1048:regions
863:by the
683:matroid
449:is the
218:), has
204:matroid
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1301:ranked
1177:, 1).
1079:facets
1056:convex
55:space
49:affine
45:linear
31:is an
1697:JSTOR
1549:with
1243:from
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1190:(the
1102:plane
1071:faces
1044:cells
865:ideal
194:When
138:lines
51:, or
43:in a
27:, an
1798:ISBN
1768:ISBN
1345:and
1198:and
1037:real
1019:has
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242:and
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