468:
2213:
1562:
Another convenient way to construct a non complete intersection, which can never be a local complete intersection, is by taking the union of two different varieties where their dimensions do not agree. For example, the union of a line and a plane intersecting at a point is a classic example of this
1408:
709:
762:: it is a smooth local complete intersection meaning in any chart it can be expressed as the vanishing locus of two polynomials, but globally it is expressed by the vanishing locus of more than two polynomials. We can construct it using the very ample line bundle
266:
473:
gives an example of a quintic threefold. It can be difficult to find explicit examples of complete intersections of higher dimensional varieties using two or more explicit examples (bestiary), but, there is an explicit example of a 3-fold of type
1096:
1993:
1552:
1649:
1210:
879:
159:
724:
One method for constructing local complete intersections is to take a projective complete intersection variety and embed it into a higher dimensional projective space. A classic example of this is the
1202:
1215:
512:
463:{\displaystyle \mathbb {V} (x_{0}^{5}+\cdots +x_{4}^{5})={\text{Proj}}\left({\frac {\mathbb {F} }{(x_{0}^{5}+\cdots +x_{4}^{5})}}\right){\xrightarrow {i}}\mathbb {P} _{\mathbb {F} }^{4}}
1985:
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2208:{\displaystyle \sum _{n=0}^{\infty }\chi (X_{n}(a_{1},\ldots ,a_{r}))z^{n}={\frac {a_{1}\cdots a_{r}}{(1-z)^{2}}}\prod _{i=1}^{r}{\frac {1}{(1+(a_{i}-1)z)}}}
260:
Easy examples of complete intersections are given by hypersurfaces which are defined by the vanishing locus of a single polynomial. For example,
1419:
1403:{\displaystyle {\begin{aligned}f_{1}&=x_{0}x_{3}-x_{1}x_{2}\\f_{2}&=x_{1}^{2}-x_{0}x_{2}\\f_{3}&=x_{2}^{2}-x_{1}x_{3}\end{aligned}}}
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For more refined questions, the nature of the intersection has to be addressed more closely. The hypersurfaces may be required to satisfy a
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830:
75:
2337:
2282:
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704:{\displaystyle \mathbb {V} (x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}x_{5},x_{4}^{4}+x_{5}^{4}-2x_{0}x_{1}x_{2}x_{3})}
2311:
51:
2255:
226:, with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension
1924:
1708:
1935:
Hirzebruch gave a generating function computing the dimension of all complete intersections of multi-degree
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2298:, London Mathematical Society Lecture Note Series, vol. 77, Cambridge: Cambridge University Press,
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17:
1927:. This implies that the middle homology group is determined by the Euler characteristic of the space.
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1675:
again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in
165:
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884:
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1091:{\displaystyle \Gamma ({\mathcal {O}}(3))={\text{Span}}_{R}\{s^{3},s^{2}t,st^{2},t^{3}\}}
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1671:) of the degrees of defining hypersurfaces. For example, taking quadrics in
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being in general position at intersection points). The intersection may be
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193:
1547:{\displaystyle {\text{Proj}}\left({\frac {R}{(f_{1},f_{2},f_{3})}}\right)}
1644:{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} }{(xz,yz)}}\right)}
434:
1668:
176:, which generate all other homogeneous polynomials that vanish on
222:. The question is essentially, can we get the dimension down to
252:
is automatically a hypersurface and there is nothing to prove.
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1664:
1835:
are the intersection of hyperplane sections, we can use the
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of complex smooth complete intersections were worked out by
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874:{\displaystyle \mathbb {P} _{R}^{1}\to \mathbb {P} _{R}^{3}}
16:
For complete intersection rings in commutative algebra, see
154:{\displaystyle F_{i}(X_{0},\cdots ,X_{n}),1\leq i\leq n-m,}
2332:, vol. 22, Fields Institute Monographs, p. 194,
1752:, the complete intersection condition is translated into
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1197:{\displaystyle \mathbb {P} _{R}^{3}={\text{Proj}}(R)}
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196:; the intersection of these hypersurfaces should be
2296:Isolated singular points on complete intersections
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210:hypersurfaces will always have dimension at least
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1413:Hence the twisted cubic is the projective scheme
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2348:Euler Characteristics of Complete Intersections
2275:Calabi-Yau Manifolds, A Bestiary for Physicists
1740:) may be required to be the defining ideal of
1204:the embedding gives the following relations:
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1027:
214:, assuming that the field of scalars is an
1915:. In addition, it can be checked that the
1777:Since complete intersections of dimension
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1764:an ideal has defining regular sequences.
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1558:Union of varieties differing in dimension
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1563:phenomenon. It is given by the scheme
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1980:{\displaystyle (a_{1},\ldots ,a_{r})}
1879:{\displaystyle H^{j}(X)=\mathbb {Z} }
755:{\displaystyle \mathbb {P} _{R}^{3}}
1828:{\displaystyle \mathbb {CP} ^{n+m}}
1694:
13:
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2251:Algebraic Geometry, A First Course
2013:
1756:terms, allowing the definition of
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987:
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14:
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2366:
2277:, World Scientific, p. 380,
788:{\displaystyle {\mathcal {O}}(3)}
1744:, and not just have the correct
817:{\displaystyle \mathbb {P} ^{1}}
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1659:A complete intersection has a
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57:and lies in projective space
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2330:Modular Calabi-Yau Threefolds
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1925:universal coefficient theorem
1837:Lefschetz hyperplane theorem
7:
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1758:local complete intersection
255:
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2234:, p. 136, Definition.
1711:, in other words here the
216:algebraically closed field
18:complete intersection ring
15:
2328:Meyer, Christian (2005),
69:homogeneous polynomials:
2304:10.1017/CBO9780511662720
970:{\displaystyle \mapsto }
42:is generated by exactly
166:homogeneous coordinates
2373:Complete intersections
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2158:
2017:
1981:
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1908:{\displaystyle j<n}
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1703:condition (like their
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200:. The intersection of
155:
46:elements. That is, if
2375:at the Manifold Atlas
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824:giving the embedding
819:
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501:
499:{\displaystyle (2,4)}
465:
156:
61:, there should exist
36:complete intersection
2248:Harris, Joe (1992).
1994:
1939:
1931:Euler characteristic
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183:Geometrically, each
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2394:Commutative algebra
2292:Looijenga, E. J. N.
1750:commutative algebra
1667:(properly though a
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23:In mathematics, an
2389:Algebraic geometry
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2273:Hübsch, Tristan,
2265:978-0-387-97716-4
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1790:{\displaystyle n}
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1713:homogeneous ideal
1663:, written as the
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1695:General position
1689:Kunihiko Kodaira
1677:general position
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150:
147:
144:
141:
138:
135:
132:
129:
126:
123:
118:
114:
110:
107:
104:
99:
95:
91:
86:
82:
9:
6:
4:
3:
2:
2406:
2395:
2392:
2390:
2387:
2386:
2384:
2374:
2371:
2370:
2361:on 2017-08-15
2357:
2350:
2349:
2344:
2341:
2335:
2331:
2326:
2323:
2319:
2315:
2313:0-521-28674-3
2309:
2305:
2301:
2297:
2293:
2289:
2286:
2280:
2276:
2271:
2267:
2261:
2257:
2253:
2252:
2246:
2245:
2233:
2228:
2224:
2196:
2190:
2187:
2182:
2178:
2171:
2168:
2161:
2154:
2149:
2146:
2143:
2139:
2130:
2122:
2119:
2116:
2106:
2102:
2098:
2093:
2089:
2082:
2077:
2073:
2061:
2057:
2053:
2050:
2047:
2042:
2038:
2029:
2025:
2018:
2008:
2005:
2002:
1998:
1990:
1989:
1988:
1969:
1965:
1961:
1958:
1955:
1950:
1946:
1928:
1926:
1922:
1918:
1902:
1899:
1896:
1868:
1862:
1854:
1850:
1842:
1841:
1840:
1838:
1820:
1817:
1814:
1784:
1765:
1763:
1759:
1755:
1751:
1747:
1743:
1738:
1734:
1727:
1722:
1718:
1714:
1710:
1706:
1702:
1692:
1690:
1686:
1685:Hodge numbers
1682:
1678:
1674:
1670:
1666:
1662:
1637:
1628:
1625:
1622:
1619:
1616:
1605:
1602:
1599:
1596:
1593:
1579:
1566:
1565:
1564:
1540:
1529:
1525:
1521:
1516:
1512:
1508:
1503:
1499:
1485:
1481:
1477:
1472:
1468:
1464:
1459:
1455:
1451:
1446:
1442:
1435:
1429:
1416:
1415:
1414:
1391:
1387:
1381:
1377:
1373:
1368:
1363:
1359:
1355:
1353:
1346:
1342:
1332:
1328:
1322:
1318:
1314:
1309:
1304:
1300:
1296:
1294:
1287:
1283:
1273:
1269:
1263:
1259:
1255:
1250:
1246:
1240:
1236:
1232:
1230:
1223:
1219:
1207:
1206:
1205:
1183:
1179:
1175:
1170:
1166:
1162:
1157:
1153:
1149:
1144:
1140:
1133:
1122:
1117:
1112:
1080:
1076:
1072:
1067:
1063:
1059:
1056:
1053:
1048:
1044:
1040:
1035:
1031:
1022:
1012:
1003:
959:
955:
951:
946:
942:
938:
935:
932:
927:
923:
919:
914:
910:
897:
894:
891:
866:
861:
846:
841:
827:
826:
825:
809:
779:
747:
742:
727:
726:twisted cubic
720:Twisted cubic
693:
689:
683:
679:
673:
669:
663:
659:
655:
652:
647:
642:
638:
634:
629:
624:
620:
616:
611:
607:
601:
597:
593:
588:
583:
579:
575:
570:
565:
561:
557:
552:
547:
543:
539:
534:
529:
525:
509:
508:
507:
490:
487:
484:
455:
435:
431:
425:
414:
409:
405:
401:
398:
395:
390:
385:
381:
367:
363:
359:
356:
353:
348:
344:
329:
320:
312:
307:
303:
299:
296:
293:
288:
283:
279:
263:
262:
261:
253:
251:
245:
241:
234:
230:
225:
221:
217:
213:
208:
204:
199:
195:
190:
186:
181:
179:
174:
170:
167:
148:
145:
142:
139:
136:
133:
130:
127:
124:
116:
112:
108:
105:
102:
97:
93:
84:
80:
72:
71:
70:
68:
64:
60:
56:
53:
49:
45:
41:
37:
33:
29:
26:
19:
2356:the original
2347:
2329:
2295:
2274:
2250:
2227:
1934:
1921:torsion-free
1888:
1776:
1762:localization
1741:
1736:
1732:
1725:
1720:
1716:
1698:
1672:
1660:
1658:
1561:
1412:
1098:. If we let
979:
723:
715:Non-examples
472:
259:
249:
243:
239:
232:
228:
223:
218:such as the
211:
206:
202:
197:
194:hypersurface
188:
184:
182:
177:
172:
168:
163:
66:
62:
58:
54:
47:
43:
39:
35:
27:
22:
2232:Harris 1992
1987:. It reads
1919:are always
1661:multidegree
1655:Multidegree
2383:Categories
2242:References
1923:using the
980:Note that
506:given by
192:defines a
2188:−
2140:∏
2120:−
2099:⋯
2051:…
2019:χ
2014:∞
1999:∑
1959:…
1374:−
1315:−
1256:−
988:Γ
904:↦
852:→
653:−
399:⋯
357:…
297:⋯
143:−
137:≤
131:≤
106:⋯
52:dimension
2294:(1984),
2219:Citation
1773:Homology
1768:Topology
1669:multiset
432:→
256:Examples
2322:0747303
1746:radical
1731:, ...,
237:. When
164:in the
44:codim V
2336:
2320:
2310:
2281:
2262:
1683:. The
1679:is an
2359:(PDF)
2352:(PDF)
1748:. In
1665:tuple
795:over
248:then
34:is a
2334:ISBN
2308:ISBN
2279:ISBN
2260:ISBN
1900:<
1889:for
1575:Spec
1425:Proj
1127:Proj
1018:Span
325:Proj
50:has
2300:doi
1797:in
881:by
728:in
246:= 1
235:≥ 2
30:in
2385::
2318:MR
2316:,
2306:,
2258:.
2254:.
1691:.
242:−
231:−
205:−
180:.
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2200:)
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