32:
463:
3942:
2090:
The space of 2-planes in a split quadric 5-fold is isomorphic to a split quadric 6-fold. Likewise, both components of the space of 3-planes in a split quadric 6-fold are isomorphic to a split quadric 6-fold. (This is related to the phenomenon of
3546:
3649:
3684:
1285:
4884:
4639:
2882:
4995:
4745:
4473:
1410:
2724:
896:
3431:
1750:
453:
3248:
3147:
2050:
1900:
677:
729:
282:
3047:
936:
1134:
1085:
2519:
1013:
982:
772:
214:
138:
3085: + 2). (The numbering refers to the dimensions of the corresponding vector spaces. In the case of middle-dimensional linear subspaces of a quadric of even dimension 2
2423:
2377:
2083:
1991:
1949:
1816:
1779:
1693:
1656:
1623:
1584:
2472:
2348:
2235:
4326:
4180:
3451:
3010:
5089:
5049:
4404:
4359:
4266:
4221:
3293:
2963:
2261:
1527:
1039:
332:
4009:
2607:
2577:
4036:
3975:
3569:
3356:
104:
3937:{\displaystyle CH^{*}\operatorname {OGr} (m,2m+1)\cong \mathbb {Z} /(e_{j}^{2}-2e_{j-1}e_{j+1}+2e_{j-2}e_{j+2}-\cdots +(-1)^{j}e_{2j}=0{\text{ for all }}j),}
953:
A central part of the geometry of quadrics is the study of the linear spaces that they contain. (In the context of projective geometry, a linear subspace of
3149:
for one of the two connected components.) As a result, the isotropic
Grassmannians of a split quadric over a field also have algebraic cell decompositions.
2306:, except in the middle dimension of an even-dimensional quadric, where there are two cells. The corresponding cell closures (Schubert varieties) are:
1156:
4768:
4528:
1041:.) A key point is that every linear space contained in a smooth quadric has dimension at most half the dimension of the quadric. Moreover, when
5600:
5384:
5351:
5277:
5242:
2754:
5490:
4899:
4654:
5413:
4428:
1308:
2624:
159:
815:
5368:
3384:
2132:
1711:
372:
4062:
on a quadric, analogous to the maximal linear subspaces among all subvarieties of a quadric. To describe these bundles, let
1425:. In particular, over an algebraically closed field, there is only one smooth quadric of each dimension, up to isomorphism.
5335:
3678:+1). These descriptions can be used to compute the cohomology ring (or equivalently the Chow ring) of the spinor variety:
735:, not all zero (hence corresponding to a point in projective space), then there is a one-to-one correspondence defined by
5595:
3193:
3092:
2965:.) This calculation shows the importance of the linear subspaces of a quadric: the Chow ring of all algebraic cycles on
2000:
1850:
5590:
567:
5401:
4123:
474:
can be written (after some linear change of coordinates) as a polynomial in a proper subset of the variables, then
3434:
2136:
2116:
791:
491:
624:
4059:
1828:
682:
235:
4507:
3015:
1840:
1662:. (For example, ellipses, parabolas and hyperbolas are different kinds of conics in the affine plane over
904:
339:
225:
2173:.) Cellular varieties are very special among all algebraic varieties. For example, a cellular variety is
1105:
1056:
2489:
2144:
987:
956:
746:
579:
184:
112:
2399:
2353:
2059:
1967:
1925:
1792:
1755:
1669:
1632:
1599:
1560:
5442:
5013:
2155:. (In particular, this applies to every smooth quadric over an algebraically closed field.) That is,
2152:
366:
354:
2437:
2313:
5339:
3541:{\displaystyle \operatorname {SO} (n+2)/(\operatorname {U} (r+1)\times \operatorname {SO} (n-2r)),}
3324:
346:
60:
2192:
1045:
is algebraically closed, this is an optimal bound, meaning that every smooth quadric of dimension
5557:
5466:
5234:
4759:
4291:
4145:
3069:-dimensional quadric (like the quadric itself) is a projective homogeneous variety, known as the
2272:
2169:
1659:
152:
64:
17:
2545:
over a field, as follows. When the base field is the complex numbers, this is also the integral
2976:
2159:
can be written as a finite union of disjoint subsets that are isomorphic to affine spaces over
5585:
5486:
5409:
5380:
5347:
5273:
5238:
5058:
5018:
3644:{\displaystyle \operatorname {SO} (n+2)/(\operatorname {U} (1)\times \operatorname {SO} (n)).}
3438:
2276:
736:
4483:
viewed as the isotropic
Grassmannian of 2-planes in a 4-dimensional symplectic vector space.
4376:
4331:
4238:
4193:
3265:
2935:
2429:, one from each of the two families of middle-dimensional linear spaces (as described above).
2240:
1964:
Gr(2,4), the space of 2-planes in a 4-dimensional vector space (or equivalently, of lines in
1506:
1018:
311:
5549:
5516:
5450:
5430:
5265:
4495:
4491:
4487:
3984:
3308:
2582:
2552:
2174:
1832:
598:
148:
68:
5569:
5530:
5500:
5462:
5423:
5394:
5361:
5252:
4014:
3953:
3334:
3299: + 1) on this projective space does not come from a linear representation of SO(2
1993:). (This is related to the exceptional isomorphism of linear algebraic groups between SO(6,
1918:
on which the form restricts to zero. Furthermore, the space of lines in the quadric 3-fold
77:
5565:
5526:
5496:
5458:
5419:
5390:
5376:
5357:
5248:
1907:
1782:
1449:
479:
1460:
has even dimension. That is, there are two different "types" of maximal linear spaces in
5446:
3563:= 0, the isotropic Grassmannian is the quadric itself, which can therefore be viewed as
482:
over a lower-dimensional quadric. It is reasonable to focus attention on the case where
4511:
4115:
2107:
always has two connected components, each isomorphic to the isotropic
Grassmannian of (
614:
515:
293:
163:
141:
5521:
1140:. Thus every smooth quadric over an algebraically closed field is split. If a quadric
369:, one can also think of a projective variety in a more elementary way, as a subset of
5579:
5470:
3556:
503:
5433:(1988), "On the derived categories of coherent sheaves on some homogeneous spaces",
5537:
5478:
4755:
2186:
2163:
of various dimensions. (For projective homogeneous varieties, the cells are called
1961:
1501:
2525:-dimensional quadric. It is the iterated cone over a smooth quadric of dimension 2
1549:
can be any smooth quadric over an algebraically closed field.) In low dimensions,
5228:
4475:
are the line bundles O(−1,0) and O(0,−1). The spinor bundle on a quadric 3-fold
4418:
even, any reflection in the orthogonal group switches the two spinor bundles on
3948:
3255:
3174:
939:
523:
40:
21:
3445: + 2). From the latter point of view, this isotropic Grassmannian is
1468:
has even dimension. The two families can be described by: for a smooth quadric
794:. In particular, every quadric over an algebraically closed field is rational.
4762:
of algebraic vector bundles on a smooth quadric; it is the free abelian group
3312:
2546:
2264:
305:
56:
2538:
514:
has characteristic not 2, smoothness of a quadric is also equivalent to the
1280:{\displaystyle x_{0}x_{1}+x_{2}x_{3}+\cdots +x_{2m-2}x_{2m-1}+x_{2m}^{2}=0}
1148:
is split, then it can be written (after a linear change of coordinates) as
790:-rational point then it has infinitely many. This equivalence is proved by
31:
4879:{\displaystyle K_{0}(X)=\mathbb {Z} \{S_{+},S_{-},O,O(1),\ldots ,O(n-1)\}}
4634:{\displaystyle D^{b}(X)=\langle S_{+},S_{-},O,O(1),\ldots ,O(n-1)\rangle }
4134:), and there is a standard way to extend the spin representations of Spin(
462:
3190: + 1)/2. (Another description of the pure spinor variety is as
2092:
1529:
is even or odd. (The dimension of the empty set is taken to be −1 here.)
2119:
in the orthogonal group maps one component isomorphically to the other.
5561:
5454:
5272:(reprint ed.). Columbia University Press (1954); Springer (1996).
2877:{\displaystyle CH^{*}(X)\cong \mathbb {Z} /(h^{m+1}-2hl,l^{2}-ah^{m}l)}
1432:
of all linear subspaces of maximal dimension in a given smooth quadric
25:
2189:
of a smooth projective cellular variety is trivial, in the sense that
2537:
Using the Bruhat decomposition, it is straightforward to compute the
809:-rational point. An example of an anisotropic quadric is the quadric
5553:
151:, and so the study of quadrics can be considered as a descendant of
4284:-equivariant vector bundles associated to these representations of
1666:, but their closures in the projective plane are all isomorphic to
1588:
461:
30:
455:
defined by homogeneous polynomial equations with coefficients in
4990:{\displaystyle K_{0}(X)=\mathbb {Z} \{S,O,O(1),\ldots ,O(n-1)\}}
4740:{\displaystyle D^{b}(X)=\langle S,O,O(1),\ldots ,O(n-1)\rangle }
3654:
For example, the complex projectivized pure spinor variety OGr(
1553:
and the linear spaces it contains can be described as follows.
4468:{\displaystyle X\cong \mathbf {P} ^{1}\times \mathbf {P} ^{1}}
3057:
1456:
has odd dimension, whereas it has two connected components if
1405:{\displaystyle x_{0}x_{1}+x_{2}x_{3}+\cdots +x_{2m}x_{2m+1}=0}
593:
It is a fundamental result that a smooth quadric over a field
466:
A singular quadric surface, the cone over a smooth conic curve
1428:
For many applications, it is important to describe the space
35:
The two families of lines on a smooth (split) quadric surface
2932:, the class of the other type of maximal linear subspace is
2719:{\displaystyle CH^{*}(X)\cong \mathbb {Z} /(h^{m}-2l,l^{2})}
71:
rather than affine space. An example is the quadric surface
5133:
Elman, Karpenko, & Merkurjev (2008), Proposition 7.28.
3049:) together with the class of a maximal linear subspace of
5507:
Ottaviani, Giorgio (1988), "Spinor bundles on quadrics",
5208:
Elman, Karpenko & Merkurjev (2008), Proposition 68.1.
5115:
Elman, Karpenko & Merkurjev (2008), Proposition 22.9.
4425:
For example, the two spinor bundles on a quadric surface
4086:, and therefore so does its double cover, the spin group
3361:
3168: − 1)-planes in a smooth quadric of dimension 2
1914:
can be identified with the space LGr(2,4) of 2-planes in
891:{\displaystyle x_{0}^{2}+x_{1}^{2}+\cdots +x_{n+1}^{2}=0}
5217:
Elman, Karpenko, & Merkurjev (2008), Exercise 68.3.
4510:
involving the spinor bundles, along with the "obvious"
3426:{\displaystyle G=\operatorname {SO} (n+2,\mathbf {C} )}
3381:
is a homogeneous space for the complex algebraic group
1745:{\displaystyle \mathbf {P} ^{1}\times \mathbf {P} ^{1}}
586:
is an iterated cone over a smooth quadric of dimension
448:{\displaystyle {\mathbf {P} }^{N}(k)=(k^{N+1}-0)/k^{*}}
1839:). (This is related to the exceptional isomorphism of
5270:
The
Algebraic Theory of Spinors and Clifford Algebras
5061:
5051:(of continuous complex vector bundles on the quadric
5021:
4902:
4771:
4657:
4531:
4431:
4379:
4334:
4294:
4241:
4196:
4148:
4017:
3987:
3956:
3687:
3572:
3454:
3387:
3337:
3268:
3196:
3095:
3018:
2979:
2938:
2757:
2627:
2585:
2555:
2492:
2440:
2402:
2356:
2316:
2243:
2195:
2062:
2003:
1970:
1928:
1853:
1795:
1758:
1714:
1672:
1635:
1602:
1563:
1509:
1311:
1159:
1108:
1059:
1021:
990:
959:
907:
818:
749:
685:
627:
375:
314:
238:
187:
115:
80:
4486:
To indicate the significance of the spinor bundles:
3172: − 1 may also be viewed as the variety of
162:. Another generalization of quadrics is provided by
158:
Many properties of quadrics hold more generally for
4754:is odd. Concretely, this implies the split case of
5083:
5043:
4989:
4878:
4739:
4633:
4467:
4398:
4353:
4320:
4260:
4215:
4174:
4030:
4003:
3969:
3936:
3643:
3540:
3425:
3350:
3287:
3243:{\displaystyle \operatorname {OGr} _{+}(m+1,2m+2)}
3242:
3142:{\displaystyle \operatorname {OGr} _{+}(m+1,2m+2)}
3141:
3041:
3004:
2957:
2876:
2718:
2601:
2571:
2513:
2466:
2417:
2371:
2342:
2255:
2229:
2151:has an algebraic cell decomposition, known as the
2077:
2045:{\displaystyle \operatorname {SL} (4,k)/\{\pm 1\}}
2044:
1985:
1943:
1895:{\displaystyle \operatorname {Sp} (4,k)/\{\pm 1\}}
1894:
1810:
1773:
1744:
1687:
1650:
1617:
1578:
1521:
1500:are distinguished by whether the dimension of the
1404:
1279:
1128:
1079:
1033:
1007:
976:
930:
890:
778:minus a lower-dimensional subset. For example, if
766:
723:
671:
617:. That is, if there is a solution of the equation
447:
326:
276:
208:
132:
98:
5509:Transactions of the American Mathematical Society
5344:Algebraic and geometric theory of quadratic forms
2056:has two connected components, each isomorphic to
1789:has two connected components, each isomorphic to
2143:. Like any projective homogeneous variety for a
2052:.) The space of 2-planes in the quadric 4-fold
2396:/2, both Schubert varieties are linear spaces
1902:.) Namely, given a 4-dimensional vector space
5540:(1985), "K-theory of quadric hypersurfaces",
2920:is the class of a maximal linear subspace of
2111:− 1)-planes in a split quadric of dimension 2
284:. (A homogeneous polynomial is also called a
169:Property of quadric By definition, a quadric
8:
4984:
4930:
4873:
4799:
4734:
4680:
4628:
4554:
2039:
2030:
1889:
1880:
1785:. The space of lines in the quadric surface
1123:
1109:
1074:
1060:
5304:Mimura & Toda (1991), Theorem III.6.11.
2609:. (The cohomology in odd degrees is zero.)
1102:if it contains a linear space of dimension
4268:.) Then the spinor bundles on the quadric
3307:, but rather from a representation of its
2139:), viewed as linear algebraic groups over
5520:
5291:
5289:
5066:
5060:
5026:
5020:
4926:
4925:
4907:
4901:
4819:
4806:
4795:
4794:
4776:
4770:
4662:
4656:
4574:
4561:
4536:
4530:
4459:
4454:
4444:
4439:
4430:
4384:
4378:
4339:
4333:
4312:
4299:
4293:
4246:
4240:
4201:
4195:
4166:
4153:
4147:
4022:
4016:
3995:
3986:
3961:
3955:
3920:
3905:
3895:
3861:
3845:
3823:
3807:
3791:
3786:
3774:
3765:
3746:
3735:
3734:
3695:
3686:
3594:
3571:
3476:
3453:
3415:
3386:
3342:
3336:
3273:
3267:
3250:.) To explain the name: the smallest SO(2
3201:
3195:
3100:
3094:
3027:
3021:
3020:
3017:
2984:
2978:
2943:
2937:
2916:is the class of a hyperplane section and
2862:
2846:
2815:
2803:
2784:
2783:
2765:
2756:
2707:
2685:
2673:
2654:
2653:
2635:
2626:
2590:
2584:
2563:
2554:
2499:
2494:
2491:
2444:
2439:
2409:
2404:
2401:
2363:
2358:
2355:
2332:
2315:
2242:
2200:
2194:
2103:-planes in a split quadric of dimension 2
2069:
2064:
2061:
2025:
2002:
1977:
1972:
1969:
1935:
1930:
1927:
1875:
1852:
1802:
1797:
1794:
1765:
1760:
1757:
1736:
1731:
1721:
1716:
1713:
1679:
1674:
1671:
1642:
1637:
1634:
1609:
1604:
1601:
1570:
1565:
1562:
1508:
1381:
1368:
1349:
1339:
1326:
1316:
1310:
1265:
1257:
1235:
1216:
1197:
1187:
1174:
1164:
1158:
1115:
1107:
1066:
1058:
1020:
999:
993:
992:
989:
968:
962:
961:
958:
916:
910:
909:
906:
876:
865:
846:
841:
828:
823:
817:
758:
752:
751:
748:
709:
690:
684:
654:
635:
626:
582:of the Hessian matrix. A quadric of rank
439:
430:
409:
384:
378:
377:
374:
313:
300:is the product of two linear forms, then
262:
243:
237:
194:
189:
186:
124:
118:
117:
114:
79:
67:. The theory is simplified by working in
5186:
5184:
2099:As these examples suggest, the space of
1053:contains a linear subspace of dimension
147:. A quadric has a natural action of the
5099:
3262:lands in projective space of dimension
672:{\displaystyle (a_{0},\ldots ,a_{n+1})}
63:. Quadrics are fundamental examples in
4142:. (There are two spin representations
2969:is generated by the "obvious" element
2135:for the orthogonal group (and for the
3662: + 1) can be viewed as SO(2
1596:is isomorphic to the projective line
774:minus a lower-dimensional subset and
724:{\displaystyle a_{0},\ldots ,a_{n+1}}
526:, or to the associated bilinear form
353:are considered as a special class of
277:{\displaystyle x_{0},\ldots ,x_{n+1}}
7:
5055:) is given by the same formula, and
3042:{\displaystyle {\mathbf {P} }^{n+1}}
2474:, the Schubert variety of dimension
2302:has only one cell of each dimension
931:{\displaystyle {\mathbf {P} }^{n+1}}
342:, which excludes that special case.
4479:is the natural rank-2 subbundle on
1129:{\displaystyle \lfloor n/2\rfloor }
1080:{\displaystyle \lfloor n/2\rfloor }
5406:Algebraic geometry: a first course
5160:Harris (1995), Lecture 22, p. 285.
5151:Harris (1995), Lecture 22, p. 284.
4522:restricted from projective space:
4288:. So there are two spinor bundles
4074:. The special orthogonal group SO(
3602:
3484:
2514:{\displaystyle \mathbf {P} ^{n+1}}
1008:{\displaystyle {\mathbf {P} }^{a}}
977:{\displaystyle {\mathbf {P} }^{N}}
767:{\displaystyle {\mathbf {P} }^{n}}
209:{\displaystyle \mathbf {P} ^{n+1}}
133:{\displaystyle {\mathbf {P} }^{3}}
16:This article is about quadrics in
14:
5522:10.1090/S0002-9947-1988-0936818-5
5485:, American Mathematical Society,
5346:, American Mathematical Society,
4038:is understood to mean 0 for
2482:with a linear space of dimension
1444:.) A striking phenomenon is that
4455:
4440:
4066:be a split quadric of dimension
3416:
3022:
2541:of a split quadric of dimension
2495:
2418:{\displaystyle \mathbf {P} ^{r}}
2405:
2372:{\displaystyle \mathbf {P} ^{r}}
2359:
2167:, and their closures are called
2078:{\displaystyle \mathbf {P} ^{3}}
2065:
1986:{\displaystyle \mathbf {P} ^{3}}
1973:
1944:{\displaystyle \mathbf {P} ^{3}}
1931:
1811:{\displaystyle \mathbf {P} ^{1}}
1798:
1774:{\displaystyle \mathbf {P} ^{3}}
1761:
1732:
1717:
1688:{\displaystyle \mathbf {P} ^{1}}
1675:
1651:{\displaystyle \mathbf {P} ^{2}}
1638:
1618:{\displaystyle \mathbf {P} ^{1}}
1605:
1579:{\displaystyle \mathbf {P} ^{2}}
1566:
1541:be a split quadric over a field
1094:, a smooth quadric of dimension
994:
963:
911:
782:is infinite, it follows that if
753:
379:
190:
160:projective homogeneous varieties
119:
55:-dimensional space defined by a
5199:Fulton (1998), Example 19.1.11.
2549:ring of a smooth quadric, with
2275:on the set of cells, as is the
5313:Kapranov (1988), Theorem 4.10.
5078:
5072:
5038:
5032:
4981:
4969:
4954:
4948:
4919:
4913:
4870:
4858:
4843:
4837:
4788:
4782:
4731:
4719:
4704:
4698:
4674:
4668:
4625:
4613:
4598:
4592:
4548:
4542:
4223:, and one spin representation
4058:play a special role among all
3928:
3892:
3882:
3779:
3771:
3739:
3728:
3707:
3635:
3632:
3626:
3614:
3608:
3599:
3591:
3579:
3532:
3529:
3514:
3502:
3490:
3481:
3473:
3461:
3420:
3400:
3331: + 1), of dimension
3237:
3210:
3136:
3109:
2999:
2993:
2871:
2808:
2800:
2788:
2777:
2771:
2713:
2678:
2670:
2658:
2647:
2641:
2467:{\displaystyle n/2<r\leq n}
2343:{\displaystyle 0\leq r<n/2}
2263:. For a cellular variety, the
2218:
2212:
2133:projective homogeneous variety
2127:A smooth quadric over a field
2022:
2010:
1872:
1860:
666:
628:
427:
402:
396:
390:
308:. It is common to assume that
1:
5190:Harris (1995), Theorem 22.14.
5169:Harris (1995), Exercise 22.6.
5142:Harris (1995), Theorem 22.14.
5124:Harris (1995), Theorem 22.13.
574:of characteristic not 2, the
498:is not a cone if and only if
5601:Algebraic homogeneous spaces
5295:Ottaviani (1988), section 1.
5178:Harris (1995), Example 22.7.
2973:(pulled back from the class
2230:{\displaystyle h^{p,q}(X)=0}
1436:. (For clarity, assume that
949:Linear subspaces of quadrics
59:equation of degree 2 over a
5106:Harris (1995), Example 3.3.
4321:{\displaystyle S_{+},S_{-}}
4175:{\displaystyle V_{+},V_{-}}
3981:vector bundle are equal to
3152:The isotropic Grassmannian
5617:
4050:Spinor bundles on quadrics
2579:mapping isomorphically to
15:
3005:{\displaystyle c_{1}O(1)}
5435:Inventiones Mathematicae
5084:{\displaystyle K^{1}(X)}
5044:{\displaystyle K^{0}(X)}
4490:showed that the bounded
4369:, and one spinor bundle
4138:) to representations of
3435:maximal compact subgroup
3258:projective embedding of
2137:special orthogonal group
2123:The Bruhat decomposition
1704:A split quadric surface
1533:Low-dimensional quadrics
1488:. Then the two types of
792:stereographic projection
20:. For quadrics over the
5322:Swan (1985), Theorem 1.
4399:{\displaystyle 2^{m-1}}
4354:{\displaystyle 2^{m-1}}
4261:{\displaystyle 2^{m-1}}
4216:{\displaystyle 2^{m-1}}
4130:is the spin group Spin(
4102:is a homogeneous space
3288:{\displaystyle 2^{m}-1}
3175:Projective pure spinors
3075:orthogonal Grassmannian
2958:{\displaystyle h^{m}-l}
2478:is the intersection of
2267:of algebraic cycles on
2256:{\displaystyle p\neq q}
2095:for the group Spin(8).)
1956:A split quadric 4-fold
1841:linear algebraic groups
1823:A split quadric 3-fold
1522:{\displaystyle P\cap Q}
1034:{\displaystyle a\leq N}
797:A quadric over a field
578:of a quadric means the
327:{\displaystyle n\geq 1}
5483:Topology of Lie groups
5227:Cartan, Élie (1981) ,
5085:
5045:
4991:
4880:
4758:'s calculation of the
4741:
4635:
4508:exceptional collection
4469:
4400:
4355:
4322:
4262:
4217:
4176:
4032:
4005:
4004:{\displaystyle 2e_{j}}
3971:
3938:
3645:
3542:
3427:
3352:
3289:
3244:
3143:
3071:isotropic Grassmannian
3043:
3006:
2959:
2878:
2720:
2603:
2602:{\displaystyle H^{2j}}
2573:
2572:{\displaystyle CH^{j}}
2515:
2468:
2419:
2373:
2344:
2257:
2231:
2079:
2046:
1987:
1945:
1896:
1829:isotropic Grassmannian
1812:
1775:
1746:
1689:
1652:
1619:
1580:
1523:
1406:
1281:
1130:
1081:
1035:
1009:
978:
932:
892:
768:
725:
673:
467:
449:
328:
278:
226:homogeneous polynomial
210:
134:
100:
36:
5542:Annals of Mathematics
5230:The theory of spinors
5086:
5046:
4992:
4881:
4742:
4636:
4470:
4401:
4356:
4323:
4263:
4218:
4177:
4033:
4031:{\displaystyle e_{j}}
4006:
3972:
3970:{\displaystyle c_{j}}
3939:
3646:
3543:
3428:
3377:-dimensional quadric
3365: + 1,
3353:
3351:{\displaystyle 2^{m}}
3323:. This is called the
3319: + 1) over
3290:
3245:
3180:simple spinor variety
3164: + 1) of (
3144:
3081: + 1,
3044:
3007:
2960:
2879:
2721:
2604:
2574:
2516:
2469:
2420:
2374:
2345:
2258:
2232:
2145:split reductive group
2080:
2047:
1988:
1960:can be viewed as the
1946:
1910:, the quadric 3-fold
1897:
1813:
1776:
1747:
1690:
1653:
1620:
1592:. A split conic over
1581:
1524:
1407:
1282:
1131:
1082:
1036:
1010:
979:
933:
893:
769:
726:
674:
465:
450:
329:
279:
211:
135:
101:
99:{\displaystyle xy=zw}
34:
5375:, Berlin, New York:
5340:Merkurjev, Alexander
5338:; Karpenko, Nikita;
5059:
5019:
4900:
4769:
4655:
4529:
4429:
4377:
4332:
4292:
4239:
4194:
4190:, each of dimension
4146:
4015:
3985:
3977:of the natural rank-
3954:
3685:
3570:
3452:
3385:
3335:
3295:. The action of SO(2
3266:
3194:
3093:
3065:-planes in a smooth
3016:
2977:
2936:
2755:
2625:
2583:
2553:
2490:
2438:
2400:
2354:
2314:
2241:
2193:
2153:Bruhat decomposition
2060:
2001:
1968:
1926:
1851:
1827:can be viewed as an
1793:
1756:
1712:
1670:
1633:
1600:
1561:
1507:
1309:
1157:
1106:
1057:
1019:
988:
957:
905:
901:in projective space
816:
747:
683:
625:
373:
367:algebraically closed
312:
304:is the union of two
236:
185:
113:
109:in projective space
78:
49:quadric hypersurface
5596:Projective geometry
5447:1988InMat..92..479K
5408:, Springer-Verlag,
5373:Intersection Theory
5014:topological K-group
4498:on a split quadric
4280:are defined as the
3922: for all
3796:
3670:), and also as SO(2
3433:, and also for its
3369: + 2) of
3325:spin representation
3012:of a hyperplane in
1557:A quadric curve in
1270:
881:
851:
833:
486:is not a cone. For
347:algebraic varieties
181:is the subspace of
51:is the subspace of
5591:Algebraic geometry
5455:10.1007/BF01393744
5235:Dover Publications
5081:
5041:
4987:
4876:
4760:Grothendieck group
4737:
4631:
4465:
4396:
4351:
4318:
4258:
4235:− 1, of dimension
4213:
4172:
4116:parabolic subgroup
4098:. In these terms,
4028:
4001:
3967:
3934:
3782:
3666: + 1)/U(
3641:
3538:
3423:
3348:
3311:double cover, the
3285:
3240:
3139:
3039:
3002:
2955:
2874:
2716:
2599:
2569:
2511:
2464:
2415:
2369:
2340:
2273:free abelian group
2253:
2227:
2170:Schubert varieties
2147:, a split quadric
2075:
2042:
1983:
1941:
1892:
1808:
1771:
1742:
1685:
1660:Veronese embedding
1648:
1615:
1576:
1545:. (In particular,
1519:
1402:
1277:
1253:
1126:
1077:
1031:
1005:
974:
928:
888:
861:
837:
819:
764:
737:rational functions
721:
669:
570:. In general, for
468:
445:
324:
274:
206:
153:Euclidean geometry
130:
96:
65:algebraic geometry
37:
18:algebraic geometry
5431:Kapranov, Mikhail
5386:978-0-387-98549-7
5353:978-0-8218-4329-1
5279:978-3-540-57063-9
5266:Chevalley, Claude
5244:978-0-486-64070-9
3923:
3439:compact Lie group
2350:, a linear space
2277:integral homology
1922:is isomorphic to
1708:is isomorphic to
984:is isomorphic to
228:of degree 2 over
5608:
5572:
5533:
5524:
5503:
5477:Mimura, Mamoru;
5473:
5426:
5397:
5364:
5323:
5320:
5314:
5311:
5305:
5302:
5296:
5293:
5284:
5283:
5262:
5256:
5255:
5224:
5218:
5215:
5209:
5206:
5200:
5197:
5191:
5188:
5179:
5176:
5170:
5167:
5161:
5158:
5152:
5149:
5143:
5140:
5134:
5131:
5125:
5122:
5116:
5113:
5107:
5104:
5090:
5088:
5087:
5082:
5071:
5070:
5050:
5048:
5047:
5042:
5031:
5030:
4996:
4994:
4993:
4988:
4929:
4912:
4911:
4885:
4883:
4882:
4877:
4824:
4823:
4811:
4810:
4798:
4781:
4780:
4746:
4744:
4743:
4738:
4667:
4666:
4640:
4638:
4637:
4632:
4579:
4578:
4566:
4565:
4541:
4540:
4496:coherent sheaves
4492:derived category
4488:Mikhail Kapranov
4474:
4472:
4471:
4466:
4464:
4463:
4458:
4449:
4448:
4443:
4405:
4403:
4402:
4397:
4395:
4394:
4360:
4358:
4357:
4352:
4350:
4349:
4327:
4325:
4324:
4319:
4317:
4316:
4304:
4303:
4267:
4265:
4264:
4259:
4257:
4256:
4222:
4220:
4219:
4214:
4212:
4211:
4181:
4179:
4178:
4173:
4171:
4170:
4158:
4157:
4042: >
4037:
4035:
4034:
4029:
4027:
4026:
4010:
4008:
4007:
4002:
4000:
3999:
3976:
3974:
3973:
3968:
3966:
3965:
3943:
3941:
3940:
3935:
3924:
3921:
3913:
3912:
3900:
3899:
3872:
3871:
3856:
3855:
3834:
3833:
3818:
3817:
3795:
3790:
3778:
3770:
3769:
3751:
3750:
3738:
3700:
3699:
3650:
3648:
3647:
3642:
3598:
3547:
3545:
3544:
3539:
3480:
3432:
3430:
3429:
3424:
3419:
3357:
3355:
3354:
3349:
3347:
3346:
3309:simply connected
3294:
3292:
3291:
3286:
3278:
3277:
3254: + 1)-
3249:
3247:
3246:
3241:
3206:
3205:
3148:
3146:
3145:
3140:
3105:
3104:
3048:
3046:
3045:
3040:
3038:
3037:
3026:
3025:
3011:
3009:
3008:
3003:
2989:
2988:
2964:
2962:
2961:
2956:
2948:
2947:
2883:
2881:
2880:
2875:
2867:
2866:
2851:
2850:
2826:
2825:
2807:
2787:
2770:
2769:
2725:
2723:
2722:
2717:
2712:
2711:
2690:
2689:
2677:
2657:
2640:
2639:
2608:
2606:
2605:
2600:
2598:
2597:
2578:
2576:
2575:
2570:
2568:
2567:
2520:
2518:
2517:
2512:
2510:
2509:
2498:
2473:
2471:
2470:
2465:
2448:
2424:
2422:
2421:
2416:
2414:
2413:
2408:
2378:
2376:
2375:
2370:
2368:
2367:
2362:
2349:
2347:
2346:
2341:
2336:
2294:A split quadric
2262:
2260:
2259:
2254:
2236:
2234:
2233:
2228:
2211:
2210:
2084:
2082:
2081:
2076:
2074:
2073:
2068:
2051:
2049:
2048:
2043:
2029:
1992:
1990:
1989:
1984:
1982:
1981:
1976:
1950:
1948:
1947:
1942:
1940:
1939:
1934:
1901:
1899:
1898:
1893:
1879:
1833:symplectic group
1817:
1815:
1814:
1809:
1807:
1806:
1801:
1780:
1778:
1777:
1772:
1770:
1769:
1764:
1751:
1749:
1748:
1743:
1741:
1740:
1735:
1726:
1725:
1720:
1694:
1692:
1691:
1686:
1684:
1683:
1678:
1657:
1655:
1654:
1649:
1647:
1646:
1641:
1624:
1622:
1621:
1616:
1614:
1613:
1608:
1585:
1583:
1582:
1577:
1575:
1574:
1569:
1528:
1526:
1525:
1520:
1411:
1409:
1408:
1403:
1395:
1394:
1376:
1375:
1354:
1353:
1344:
1343:
1331:
1330:
1321:
1320:
1286:
1284:
1283:
1278:
1269:
1264:
1249:
1248:
1230:
1229:
1202:
1201:
1192:
1191:
1179:
1178:
1169:
1168:
1135:
1133:
1132:
1127:
1119:
1086:
1084:
1083:
1078:
1070:
1040:
1038:
1037:
1032:
1014:
1012:
1011:
1006:
1004:
1003:
998:
997:
983:
981:
980:
975:
973:
972:
967:
966:
937:
935:
934:
929:
927:
926:
915:
914:
897:
895:
894:
889:
880:
875:
850:
845:
832:
827:
773:
771:
770:
765:
763:
762:
757:
756:
730:
728:
727:
722:
720:
719:
695:
694:
678:
676:
675:
670:
665:
664:
640:
639:
621:= 0 of the form
454:
452:
451:
446:
444:
443:
434:
420:
419:
389:
388:
383:
382:
333:
331:
330:
325:
292:may be called a
283:
281:
280:
275:
273:
272:
248:
247:
215:
213:
212:
207:
205:
204:
193:
149:orthogonal group
139:
137:
136:
131:
129:
128:
123:
122:
105:
103:
102:
97:
69:projective space
5616:
5615:
5611:
5610:
5609:
5607:
5606:
5605:
5576:
5575:
5554:10.2307/1971371
5536:
5506:
5493:
5476:
5429:
5416:
5400:
5387:
5377:Springer-Verlag
5369:Fulton, William
5367:
5354:
5334:
5331:
5326:
5321:
5317:
5312:
5308:
5303:
5299:
5294:
5287:
5280:
5264:
5263:
5259:
5245:
5226:
5225:
5221:
5216:
5212:
5207:
5203:
5198:
5194:
5189:
5182:
5177:
5173:
5168:
5164:
5159:
5155:
5150:
5146:
5141:
5137:
5132:
5128:
5123:
5119:
5114:
5110:
5105:
5101:
5097:
5062:
5057:
5056:
5022:
5017:
5016:
4903:
4898:
4897:
4815:
4802:
4772:
4767:
4766:
4658:
4653:
4652:
4570:
4557:
4532:
4527:
4526:
4453:
4438:
4427:
4426:
4380:
4375:
4374:
4335:
4330:
4329:
4308:
4295:
4290:
4289:
4242:
4237:
4236:
4197:
4192:
4191:
4162:
4149:
4144:
4143:
4052:
4018:
4013:
4012:
3991:
3983:
3982:
3957:
3952:
3951:
3901:
3891:
3857:
3841:
3819:
3803:
3761:
3742:
3691:
3683:
3682:
3568:
3567:
3450:
3449:
3383:
3382:
3338:
3333:
3332:
3269:
3264:
3263:
3197:
3192:
3191:
3182:, of dimension
3096:
3091:
3090:
3059:
3019:
3014:
3013:
2980:
2975:
2974:
2939:
2934:
2933:
2858:
2842:
2811:
2761:
2753:
2752:
2703:
2681:
2631:
2623:
2622:
2586:
2581:
2580:
2559:
2551:
2550:
2493:
2488:
2487:
2436:
2435:
2403:
2398:
2397:
2357:
2352:
2351:
2312:
2311:
2239:
2238:
2196:
2191:
2190:
2125:
2063:
2058:
2057:
1999:
1998:
1971:
1966:
1965:
1929:
1924:
1923:
1908:symplectic form
1849:
1848:
1796:
1791:
1790:
1783:Segre embedding
1759:
1754:
1753:
1730:
1715:
1710:
1709:
1673:
1668:
1667:
1636:
1631:
1630:
1603:
1598:
1597:
1564:
1559:
1558:
1535:
1505:
1504:
1421:has dimension 2
1377:
1364:
1345:
1335:
1322:
1312:
1307:
1306:
1296:has dimension 2
1231:
1212:
1193:
1183:
1170:
1160:
1155:
1154:
1104:
1103:
1090:Over any field
1055:
1054:
1017:
1016:
991:
986:
985:
960:
955:
954:
951:
908:
903:
902:
814:
813:
750:
745:
744:
705:
686:
681:
680:
650:
631:
623:
622:
605:if and only if
522:having nonzero
480:projective cone
435:
405:
376:
371:
370:
310:
309:
258:
239:
234:
233:
188:
183:
182:
142:complex numbers
116:
111:
110:
76:
75:
29:
12:
11:
5:
5614:
5612:
5604:
5603:
5598:
5593:
5588:
5578:
5577:
5574:
5573:
5548:(1): 113–153,
5534:
5504:
5492:978-0821813423
5491:
5474:
5441:(3): 479–508,
5427:
5414:
5398:
5385:
5365:
5352:
5336:Elman, Richard
5330:
5327:
5325:
5324:
5315:
5306:
5297:
5285:
5278:
5257:
5243:
5219:
5210:
5201:
5192:
5180:
5171:
5162:
5153:
5144:
5135:
5126:
5117:
5108:
5098:
5096:
5093:
5080:
5077:
5074:
5069:
5065:
5040:
5037:
5034:
5029:
5025:
4998:
4997:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4928:
4924:
4921:
4918:
4915:
4910:
4906:
4887:
4886:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4822:
4818:
4814:
4809:
4805:
4801:
4797:
4793:
4790:
4787:
4784:
4779:
4775:
4748:
4747:
4736:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4670:
4665:
4661:
4642:
4641:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4577:
4573:
4569:
4564:
4560:
4556:
4553:
4550:
4547:
4544:
4539:
4535:
4462:
4457:
4452:
4447:
4442:
4437:
4434:
4393:
4390:
4387:
4383:
4348:
4345:
4342:
4338:
4315:
4311:
4307:
4302:
4298:
4255:
4252:
4249:
4245:
4210:
4207:
4204:
4200:
4169:
4165:
4161:
4156:
4152:
4060:vector bundles
4056:spinor bundles
4051:
4048:
4025:
4021:
3998:
3994:
3990:
3964:
3960:
3945:
3944:
3933:
3930:
3927:
3919:
3916:
3911:
3908:
3904:
3898:
3894:
3890:
3887:
3884:
3881:
3878:
3875:
3870:
3867:
3864:
3860:
3854:
3851:
3848:
3844:
3840:
3837:
3832:
3829:
3826:
3822:
3816:
3813:
3810:
3806:
3802:
3799:
3794:
3789:
3785:
3781:
3777:
3773:
3768:
3764:
3760:
3757:
3754:
3749:
3745:
3741:
3737:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3698:
3694:
3690:
3652:
3651:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3597:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3549:
3548:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3479:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3422:
3418:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3373:-planes in an
3345:
3341:
3284:
3281:
3276:
3272:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3204:
3200:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3103:
3099:
3058:
3055:
3036:
3033:
3030:
3024:
3001:
2998:
2995:
2992:
2987:
2983:
2954:
2951:
2946:
2942:
2928: = 2
2910:
2909:
2904:odd and 1 for
2892:| =
2873:
2870:
2865:
2861:
2857:
2854:
2849:
2845:
2841:
2838:
2835:
2832:
2829:
2824:
2821:
2818:
2814:
2810:
2806:
2802:
2799:
2796:
2793:
2790:
2786:
2782:
2779:
2776:
2773:
2768:
2764:
2760:
2740:
2739:
2734:| =
2715:
2710:
2706:
2702:
2699:
2696:
2693:
2688:
2684:
2680:
2676:
2672:
2669:
2666:
2663:
2660:
2656:
2652:
2649:
2646:
2643:
2638:
2634:
2630:
2596:
2593:
2589:
2566:
2562:
2558:
2535:
2534:
2521:; so it is an
2508:
2505:
2502:
2497:
2463:
2460:
2457:
2454:
2451:
2447:
2443:
2431:
2430:
2412:
2407:
2385:
2384:
2366:
2361:
2339:
2335:
2331:
2328:
2325:
2322:
2319:
2252:
2249:
2246:
2226:
2223:
2220:
2217:
2214:
2209:
2206:
2203:
2199:
2165:Schubert cells
2124:
2121:
2097:
2096:
2087:
2086:
2072:
2067:
2041:
2038:
2035:
2032:
2028:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
1980:
1975:
1953:
1952:
1938:
1933:
1891:
1888:
1885:
1882:
1878:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1820:
1819:
1805:
1800:
1768:
1763:
1752:, embedded in
1739:
1734:
1729:
1724:
1719:
1701:
1700:
1682:
1677:
1645:
1640:
1629:, embedded in
1612:
1607:
1573:
1568:
1534:
1531:
1518:
1515:
1512:
1472:of dimension 2
1440:is split over
1415:
1414:
1413:
1412:
1401:
1398:
1393:
1390:
1387:
1384:
1380:
1374:
1371:
1367:
1363:
1360:
1357:
1352:
1348:
1342:
1338:
1334:
1329:
1325:
1319:
1315:
1290:
1289:
1288:
1287:
1276:
1273:
1268:
1263:
1260:
1256:
1252:
1247:
1244:
1241:
1238:
1234:
1228:
1225:
1222:
1219:
1215:
1211:
1208:
1205:
1200:
1196:
1190:
1186:
1182:
1177:
1173:
1167:
1163:
1125:
1122:
1118:
1114:
1111:
1076:
1073:
1069:
1065:
1062:
1030:
1027:
1024:
1002:
996:
971:
965:
950:
947:
925:
922:
919:
913:
899:
898:
887:
884:
879:
874:
871:
868:
864:
860:
857:
854:
849:
844:
840:
836:
831:
826:
822:
761:
755:
718:
715:
712:
708:
704:
701:
698:
693:
689:
668:
663:
660:
657:
653:
649:
646:
643:
638:
634:
630:
615:rational point
516:Hessian matrix
492:characteristic
442:
438:
433:
429:
426:
423:
418:
415:
412:
408:
404:
401:
398:
395:
392:
387:
381:
323:
320:
317:
294:quadratic form
271:
268:
265:
261:
257:
254:
251:
246:
242:
203:
200:
197:
192:
164:Fano varieties
127:
121:
107:
106:
95:
92:
89:
86:
83:
13:
10:
9:
6:
4:
3:
2:
5613:
5602:
5599:
5597:
5594:
5592:
5589:
5587:
5584:
5583:
5581:
5571:
5567:
5563:
5559:
5555:
5551:
5547:
5543:
5539:
5538:Swan, Richard
5535:
5532:
5528:
5523:
5518:
5514:
5510:
5505:
5502:
5498:
5494:
5488:
5484:
5480:
5475:
5472:
5468:
5464:
5460:
5456:
5452:
5448:
5444:
5440:
5436:
5432:
5428:
5425:
5421:
5417:
5415:0-387-97716-3
5411:
5407:
5403:
5399:
5396:
5392:
5388:
5382:
5378:
5374:
5370:
5366:
5363:
5359:
5355:
5349:
5345:
5341:
5337:
5333:
5332:
5328:
5319:
5316:
5310:
5307:
5301:
5298:
5292:
5290:
5286:
5281:
5275:
5271:
5267:
5261:
5258:
5254:
5250:
5246:
5240:
5236:
5232:
5231:
5223:
5220:
5214:
5211:
5205:
5202:
5196:
5193:
5187:
5185:
5181:
5175:
5172:
5166:
5163:
5157:
5154:
5148:
5145:
5139:
5136:
5130:
5127:
5121:
5118:
5112:
5109:
5103:
5100:
5094:
5092:
5075:
5067:
5063:
5054:
5035:
5027:
5023:
5015:
5011:
5007:
5003:
4978:
4975:
4972:
4966:
4963:
4960:
4957:
4951:
4945:
4942:
4939:
4936:
4933:
4922:
4916:
4908:
4904:
4896:
4895:
4894:
4892:
4867:
4864:
4861:
4855:
4852:
4849:
4846:
4840:
4834:
4831:
4828:
4825:
4820:
4816:
4812:
4807:
4803:
4791:
4785:
4777:
4773:
4765:
4764:
4763:
4761:
4757:
4753:
4728:
4725:
4722:
4716:
4713:
4710:
4707:
4701:
4695:
4692:
4689:
4686:
4683:
4677:
4671:
4663:
4659:
4651:
4650:
4649:
4648:is even, and
4647:
4622:
4619:
4616:
4610:
4607:
4604:
4601:
4595:
4589:
4586:
4583:
4580:
4575:
4571:
4567:
4562:
4558:
4551:
4545:
4537:
4533:
4525:
4524:
4523:
4521:
4519:
4515:
4512:line bundles
4509:
4505:
4502:over a field
4501:
4497:
4493:
4489:
4484:
4482:
4478:
4460:
4450:
4445:
4435:
4432:
4423:
4421:
4417:
4413:
4409:
4391:
4388:
4385:
4381:
4372:
4368:
4364:
4346:
4343:
4340:
4336:
4313:
4309:
4305:
4300:
4296:
4287:
4283:
4279:
4275:
4271:
4253:
4250:
4247:
4243:
4234:
4230:
4226:
4208:
4205:
4202:
4198:
4189:
4185:
4167:
4163:
4159:
4154:
4150:
4141:
4137:
4133:
4129:
4125:
4121:
4117:
4114:is a maximal
4113:
4109:
4105:
4101:
4097:
4093:
4089:
4085:
4081:
4077:
4073:
4070:over a field
4069:
4065:
4061:
4057:
4049:
4047:
4045:
4041:
4023:
4019:
3996:
3992:
3988:
3980:
3962:
3958:
3950:
3949:Chern classes
3931:
3925:
3917:
3914:
3909:
3906:
3902:
3896:
3888:
3885:
3879:
3876:
3873:
3868:
3865:
3862:
3858:
3852:
3849:
3846:
3842:
3838:
3835:
3830:
3827:
3824:
3820:
3814:
3811:
3808:
3804:
3800:
3797:
3792:
3787:
3783:
3775:
3766:
3762:
3758:
3755:
3752:
3747:
3743:
3731:
3725:
3722:
3719:
3716:
3713:
3710:
3704:
3701:
3696:
3692:
3688:
3681:
3680:
3679:
3677:
3673:
3669:
3665:
3661:
3657:
3638:
3629:
3623:
3620:
3617:
3611:
3605:
3595:
3588:
3585:
3582:
3576:
3573:
3566:
3565:
3564:
3562:
3558:
3557:unitary group
3554:
3535:
3526:
3523:
3520:
3517:
3511:
3508:
3505:
3499:
3496:
3493:
3487:
3477:
3470:
3467:
3464:
3458:
3455:
3448:
3447:
3446:
3444:
3440:
3436:
3412:
3409:
3406:
3403:
3397:
3394:
3391:
3388:
3380:
3376:
3372:
3368:
3364:
3359:
3343:
3339:
3330:
3326:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3282:
3279:
3274:
3270:
3261:
3257:
3253:
3234:
3231:
3228:
3225:
3222:
3219:
3216:
3213:
3207:
3202:
3198:
3189:
3185:
3181:
3177:
3176:
3171:
3167:
3163:
3159:
3155:
3150:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3106:
3101:
3097:
3089:, one writes
3088:
3084:
3080:
3076:
3072:
3068:
3064:
3061:The space of
3056:
3054:
3052:
3034:
3031:
3028:
2996:
2990:
2985:
2981:
2972:
2968:
2952:
2949:
2944:
2940:
2931:
2927:
2923:
2919:
2915:
2907:
2903:
2899:
2895:
2891:
2887:
2868:
2863:
2859:
2855:
2852:
2847:
2843:
2839:
2836:
2833:
2830:
2827:
2822:
2819:
2816:
2812:
2804:
2797:
2794:
2791:
2780:
2774:
2766:
2762:
2758:
2750:
2746:
2742:
2741:
2737:
2733:
2729:
2708:
2704:
2700:
2697:
2694:
2691:
2686:
2682:
2674:
2667:
2664:
2661:
2650:
2644:
2636:
2632:
2628:
2620:
2616:
2612:
2611:
2610:
2594:
2591:
2587:
2564:
2560:
2556:
2548:
2544:
2540:
2532:
2528:
2524:
2506:
2503:
2500:
2485:
2481:
2477:
2461:
2458:
2455:
2452:
2449:
2445:
2441:
2433:
2432:
2428:
2425:contained in
2410:
2395:
2391:
2387:
2386:
2382:
2379:contained in
2364:
2337:
2333:
2329:
2326:
2323:
2320:
2317:
2309:
2308:
2307:
2305:
2301:
2298:of dimension
2297:
2292:
2290:
2286:
2282:
2278:
2274:
2270:
2266:
2250:
2247:
2244:
2224:
2221:
2215:
2207:
2204:
2201:
2197:
2188:
2184:
2180:
2176:
2172:
2171:
2166:
2162:
2158:
2154:
2150:
2146:
2142:
2138:
2134:
2130:
2122:
2120:
2118:
2114:
2110:
2106:
2102:
2094:
2089:
2088:
2070:
2055:
2036:
2033:
2026:
2019:
2016:
2013:
2007:
2004:
1996:
1978:
1963:
1959:
1955:
1954:
1936:
1921:
1917:
1913:
1909:
1905:
1886:
1883:
1876:
1869:
1866:
1863:
1857:
1854:
1846:
1843:between SO(5,
1842:
1838:
1834:
1830:
1826:
1822:
1821:
1803:
1788:
1784:
1766:
1737:
1727:
1722:
1707:
1703:
1702:
1698:
1680:
1665:
1661:
1643:
1628:
1610:
1595:
1591:
1590:
1571:
1556:
1555:
1554:
1552:
1548:
1544:
1540:
1532:
1530:
1516:
1513:
1510:
1503:
1499:
1496:contained in
1495:
1491:
1487:
1484:contained in
1483:
1479:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1447:
1443:
1439:
1435:
1431:
1426:
1424:
1420:
1399:
1396:
1391:
1388:
1385:
1382:
1378:
1372:
1369:
1365:
1361:
1358:
1355:
1350:
1346:
1340:
1336:
1332:
1327:
1323:
1317:
1313:
1305:
1304:
1303:
1302:
1301:
1299:
1295:
1274:
1271:
1266:
1261:
1258:
1254:
1250:
1245:
1242:
1239:
1236:
1232:
1226:
1223:
1220:
1217:
1213:
1209:
1206:
1203:
1198:
1194:
1188:
1184:
1180:
1175:
1171:
1165:
1161:
1153:
1152:
1151:
1150:
1149:
1147:
1144:over a field
1143:
1139:
1120:
1116:
1112:
1101:
1097:
1093:
1088:
1071:
1067:
1063:
1052:
1048:
1044:
1028:
1025:
1022:
1000:
969:
948:
946:
944:
941:
923:
920:
917:
885:
882:
877:
872:
869:
866:
862:
858:
855:
852:
847:
842:
838:
834:
829:
824:
820:
812:
811:
810:
808:
804:
800:
795:
793:
789:
785:
781:
777:
759:
742:
738:
734:
716:
713:
710:
706:
702:
699:
696:
691:
687:
661:
658:
655:
651:
647:
644:
641:
636:
632:
620:
616:
612:
608:
604:
600:
596:
591:
589:
585:
581:
577:
573:
569:
568:nondegenerate
565:
561:
557:
553:
549:
545:
541:
537:
533:
529:
525:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
464:
460:
458:
440:
436:
431:
424:
421:
416:
413:
410:
406:
399:
393:
385:
368:
364:
360:
356:
352:
349:over a field
348:
343:
341:
337:
321:
318:
315:
307:
303:
299:
295:
291:
287:
269:
266:
263:
259:
255:
252:
249:
244:
240:
232:in variables
231:
227:
224:is a nonzero
223:
219:
201:
198:
195:
180:
177:over a field
176:
173:of dimension
172:
167:
165:
161:
156:
154:
150:
146:
143:
125:
93:
90:
87:
84:
81:
74:
73:
72:
70:
66:
62:
58:
54:
50:
46:
42:
33:
27:
23:
19:
5545:
5541:
5512:
5508:
5482:
5479:Toda, Hirosi
5438:
5434:
5405:
5372:
5343:
5318:
5309:
5300:
5269:
5260:
5233:, New York:
5229:
5222:
5213:
5204:
5195:
5174:
5165:
5156:
5147:
5138:
5129:
5120:
5111:
5102:
5052:
5009:
5005:
5001:
4999:
4890:
4888:
4756:Richard Swan
4751:
4749:
4645:
4643:
4517:
4513:
4503:
4499:
4485:
4480:
4476:
4424:
4419:
4415:
4411:
4407:
4370:
4366:
4362:
4285:
4281:
4277:
4273:
4269:
4232:
4228:
4224:
4187:
4183:
4139:
4135:
4131:
4127:
4119:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4083:
4079:
4075:
4071:
4067:
4063:
4055:
4053:
4043:
4039:
3978:
3946:
3675:
3671:
3667:
3663:
3659:
3655:
3653:
3560:
3552:
3550:
3442:
3378:
3374:
3370:
3366:
3362:
3360:
3328:
3320:
3316:
3304:
3300:
3296:
3259:
3251:
3187:
3183:
3179:
3173:
3169:
3165:
3161:
3157:
3153:
3151:
3086:
3082:
3078:
3074:
3070:
3066:
3062:
3060:
3050:
2970:
2966:
2929:
2925:
2921:
2917:
2913:
2911:
2905:
2901:
2897:
2893:
2889:
2885:
2748:
2744:
2735:
2731:
2727:
2618:
2614:
2542:
2536:
2530:
2526:
2522:
2483:
2479:
2475:
2426:
2393:
2389:
2380:
2303:
2299:
2295:
2293:
2288:
2284:
2280:
2268:
2187:Hodge theory
2182:
2178:
2168:
2164:
2160:
2156:
2148:
2140:
2128:
2126:
2112:
2108:
2104:
2100:
2098:
2053:
1994:
1962:Grassmannian
1957:
1919:
1915:
1911:
1903:
1844:
1836:
1824:
1786:
1705:
1696:
1663:
1626:
1593:
1587:
1586:is called a
1550:
1546:
1542:
1538:
1536:
1502:intersection
1497:
1493:
1489:
1485:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1445:
1441:
1437:
1433:
1429:
1427:
1422:
1418:
1416:
1297:
1293:
1291:
1145:
1141:
1137:
1099:
1095:
1091:
1089:
1050:
1046:
1042:
952:
942:
940:real numbers
900:
806:
805:if it has a
802:
798:
796:
787:
783:
779:
775:
740:
732:
618:
610:
606:
602:
594:
592:
587:
583:
575:
571:
563:
559:
555:
551:
547:
543:
539:
535:
531:
527:
519:
511:
507:
499:
495:
487:
483:
475:
471:
469:
456:
362:
358:
350:
344:
335:
301:
297:
289:
285:
229:
221:
217:
178:
174:
170:
168:
157:
144:
108:
52:
48:
44:
38:
22:real numbers
5515:: 301–316,
5402:Harris, Joe
4506:has a full
3555:+1) is the
3256:equivariant
3156:= OGr(
2888:| = 1 and |
2730:| = 1 and |
2177:, and (for
1658:by the 2nd
524:determinant
340:irreducible
306:hyperplanes
220:= 0, where
216:defined by
41:mathematics
5580:Categories
5329:References
5004:odd. When
4893:even, and
4124:semisimple
3947:where the
3313:spin group
2547:cohomology
2265:Chow group
2117:reflection
1476:, fix one
1098:is called
801:is called
57:polynomial
5471:119584668
5268:(1996) .
5091:is zero.
4976:−
4961:…
4865:−
4850:…
4821:−
4735:⟩
4726:−
4711:…
4681:⟨
4629:⟩
4620:−
4605:…
4576:−
4555:⟨
4451:×
4436:≅
4414:− 1. For
4389:−
4344:−
4314:−
4251:−
4206:−
4168:−
4094:+2) over
4078:+2) over
3886:−
3877:⋯
3874:−
3850:−
3812:−
3798:−
3756:…
3732:≅
3705:
3697:∗
3624:
3618:×
3606:
3577:
3521:−
3512:
3506:×
3488:
3459:
3398:
3327:of Spin(2
3303:+1) over
3280:−
3208:
3107:
2950:−
2900:is 0 for
2884:, where |
2853:−
2828:−
2781:≅
2767:∗
2726:, where |
2692:−
2651:≅
2637:∗
2539:Chow ring
2459:≤
2321:≤
2248:≠
2115:− 1. Any
2034:±
2008:
1884:±
1858:
1728:×
1514:∩
1450:connected
1359:⋯
1243:−
1224:−
1207:⋯
1124:⌋
1110:⌊
1075:⌋
1061:⌊
1026:≤
1015:for some
938:over the
856:⋯
803:isotropic
700:…
645:…
441:∗
422:−
319:≥
288:, and so
253:…
140:over the
5586:Quadrics
5481:(1992),
5404:(1995),
5371:(1998),
5342:(2008),
4373:of rank
4328:of rank
4126:part of
4110:, where
4082:acts on
3658:, 2
3551:where U(
2175:rational
2093:triality
1831:for the
1492:-planes
1300:− 1, or
786:has one
743:between
599:rational
566:) being
5570:0799254
5562:1971371
5531:0936818
5501:1122592
5463:0939472
5443:Bibcode
5424:1416564
5395:1644323
5362:2427530
5253:0631850
4090:= Spin(
4011:. Here
2924:. (For
2486:+ 1 in
2271:is the
1906:with a
1781:by the
1480:-plane
510:. When
494:not 2,
478:is the
361:. When
355:schemes
45:quadric
26:quadric
5568:
5560:
5529:
5499:
5489:
5469:
5461:
5422:
5412:
5393:
5383:
5360:
5350:
5276:
5251:
5241:
5012:, the
4122:. The
3674:+2)/U(
3559:. For
3437:, the
3315:Spin(2
2896:, and
2185:) the
1997:) and
1847:) and
609:has a
504:smooth
296:.) If
24:, see
5558:JSTOR
5467:S2CID
5095:Notes
3178:, or
2912:Here
2908:even.
2621:− 1,
2131:is a
1835:Sp(4,
1695:over
1625:over
1589:conic
1464:when
1136:over
1100:split
1049:over
739:over
679:with
601:over
590:− 2.
506:over
357:over
345:Here
61:field
5487:ISBN
5410:ISBN
5381:ISBN
5348:ISBN
5274:ISBN
5239:ISBN
5000:for
4889:for
4406:for
4361:for
4227:for
4182:for
4054:The
3077:OGr(
2743:For
2613:For
2453:<
2434:For
2388:For
2327:<
2310:For
2283:(if
2237:for
1537:Let
580:rank
576:rank
558:) –
550:) –
538:) =
334:and
286:form
43:, a
5550:doi
5546:122
5517:doi
5513:307
5451:doi
4750:if
4644:if
4494:of
4410:= 2
4365:= 2
4231:= 2
4186:= 2
4118:of
3702:OGr
3441:SO(
3199:OGr
3098:OGr
3073:or
2747:= 2
2617:= 2
2291:).
2279:of
1452:if
1448:is
1417:if
1292:if
731:in
597:is
518:of
502:is
490:of
470:If
365:is
338:is
47:or
39:In
5582::
5566:MR
5564:,
5556:,
5544:,
5527:MR
5525:,
5511:,
5497:MR
5495:,
5465:,
5459:MR
5457:,
5449:,
5439:92
5437:,
5420:MR
5418:,
5391:MR
5389:,
5379:,
5358:MR
5356:,
5288:^
5249:MR
5247:,
5237:,
5183:^
5008:=
4422:.
4272:=
4046:.
3621:SO
3574:SO
3509:SO
3456:SO
3395:SO
3358:.
3160:,2
3053:.
2751:,
2529:−
2392:=
2287:=
2181:=
2005:SL
1855:Sp
1699:.)
1087:.
945:.
459:.
166:.
155:.
5552::
5519::
5453::
5445::
5282:.
5079:)
5076:X
5073:(
5068:1
5064:K
5053:X
5039:)
5036:X
5033:(
5028:0
5024:K
5010:C
5006:k
5002:n
4985:}
4982:)
4979:1
4973:n
4970:(
4967:O
4964:,
4958:,
4955:)
4952:1
4949:(
4946:O
4943:,
4940:O
4937:,
4934:S
4931:{
4927:Z
4923:=
4920:)
4917:X
4914:(
4909:0
4905:K
4891:n
4874:}
4871:)
4868:1
4862:n
4859:(
4856:O
4853:,
4847:,
4844:)
4841:1
4838:(
4835:O
4832:,
4829:O
4826:,
4817:S
4813:,
4808:+
4804:S
4800:{
4796:Z
4792:=
4789:)
4786:X
4783:(
4778:0
4774:K
4752:n
4732:)
4729:1
4723:n
4720:(
4717:O
4714:,
4708:,
4705:)
4702:1
4699:(
4696:O
4693:,
4690:O
4687:,
4684:S
4678:=
4675:)
4672:X
4669:(
4664:b
4660:D
4646:n
4626:)
4623:1
4617:n
4614:(
4611:O
4608:,
4602:,
4599:)
4596:1
4593:(
4590:O
4587:,
4584:O
4581:,
4572:S
4568:,
4563:+
4559:S
4552:=
4549:)
4546:X
4543:(
4538:b
4534:D
4520:)
4518:j
4516:(
4514:O
4504:k
4500:X
4481:X
4477:X
4461:1
4456:P
4446:1
4441:P
4433:X
4420:X
4416:n
4412:m
4408:n
4392:1
4386:m
4382:2
4371:S
4367:m
4363:n
4347:1
4341:m
4337:2
4310:S
4306:,
4301:+
4297:S
4286:P
4282:G
4278:P
4276:/
4274:G
4270:X
4254:1
4248:m
4244:2
4233:m
4229:n
4225:V
4209:1
4203:m
4199:2
4188:m
4184:n
4164:V
4160:,
4155:+
4151:V
4140:P
4136:n
4132:n
4128:P
4120:G
4112:P
4108:P
4106:/
4104:G
4100:X
4096:k
4092:n
4088:G
4084:X
4080:k
4076:n
4072:k
4068:n
4064:X
4044:m
4040:j
4024:j
4020:e
3997:j
3993:e
3989:2
3979:m
3963:j
3959:c
3932:,
3929:)
3926:j
3918:0
3915:=
3910:j
3907:2
3903:e
3897:j
3893:)
3889:1
3883:(
3880:+
3869:2
3866:+
3863:j
3859:e
3853:2
3847:j
3843:e
3839:2
3836:+
3831:1
3828:+
3825:j
3821:e
3815:1
3809:j
3805:e
3801:2
3793:2
3788:j
3784:e
3780:(
3776:/
3772:]
3767:m
3763:e
3759:,
3753:,
3748:1
3744:e
3740:[
3736:Z
3729:)
3726:1
3723:+
3720:m
3717:2
3714:,
3711:m
3708:(
3693:H
3689:C
3676:m
3672:m
3668:m
3664:m
3660:m
3656:m
3639:.
3636:)
3633:)
3630:n
3627:(
3615:)
3612:1
3609:(
3603:U
3600:(
3596:/
3592:)
3589:2
3586:+
3583:n
3580:(
3561:r
3553:r
3536:,
3533:)
3530:)
3527:r
3524:2
3518:n
3515:(
3503:)
3500:1
3497:+
3494:r
3491:(
3485:U
3482:(
3478:/
3474:)
3471:2
3468:+
3465:n
3462:(
3443:n
3421:)
3417:C
3413:,
3410:2
3407:+
3404:n
3401:(
3392:=
3389:G
3379:X
3375:n
3371:r
3367:n
3363:r
3344:m
3340:2
3329:m
3321:k
3317:m
3305:k
3301:m
3297:m
3283:1
3275:m
3271:2
3260:W
3252:m
3238:)
3235:2
3232:+
3229:m
3226:2
3223:,
3220:1
3217:+
3214:m
3211:(
3203:+
3188:m
3186:(
3184:m
3170:m
3166:m
3162:m
3158:m
3154:W
3137:)
3134:2
3131:+
3128:m
3125:2
3122:,
3119:1
3116:+
3113:m
3110:(
3102:+
3087:m
3083:n
3079:r
3067:n
3063:r
3051:X
3035:1
3032:+
3029:n
3023:P
3000:)
2997:1
2994:(
2991:O
2986:1
2982:c
2971:h
2967:X
2953:l
2945:m
2941:h
2930:m
2926:n
2922:X
2918:l
2914:h
2906:m
2902:m
2898:a
2894:m
2890:l
2886:h
2872:)
2869:l
2864:m
2860:h
2856:a
2848:2
2844:l
2840:,
2837:l
2834:h
2831:2
2823:1
2820:+
2817:m
2813:h
2809:(
2805:/
2801:]
2798:l
2795:,
2792:h
2789:[
2785:Z
2778:)
2775:X
2772:(
2763:H
2759:C
2749:m
2745:n
2738:.
2736:m
2732:l
2728:h
2714:)
2709:2
2705:l
2701:,
2698:l
2695:2
2687:m
2683:h
2679:(
2675:/
2671:]
2668:l
2665:,
2662:h
2659:[
2655:Z
2648:)
2645:X
2642:(
2633:H
2629:C
2619:m
2615:n
2595:j
2592:2
2588:H
2565:j
2561:H
2557:C
2543:n
2533:.
2531:n
2527:r
2523:r
2507:1
2504:+
2501:n
2496:P
2484:r
2480:X
2476:r
2462:n
2456:r
2450:2
2446:/
2442:n
2427:X
2411:r
2406:P
2394:n
2390:r
2383:.
2381:X
2365:r
2360:P
2338:2
2334:/
2330:n
2324:r
2318:0
2304:r
2300:n
2296:X
2289:C
2285:k
2281:X
2269:X
2251:q
2245:p
2225:0
2222:=
2219:)
2216:X
2213:(
2208:q
2205:,
2202:p
2198:h
2183:C
2179:k
2161:k
2157:X
2149:X
2141:k
2129:k
2113:m
2109:m
2105:m
2101:m
2085:.
2071:3
2066:P
2054:X
2040:}
2037:1
2031:{
2027:/
2023:)
2020:k
2017:,
2014:4
2011:(
1995:k
1979:3
1974:P
1958:X
1951:.
1937:3
1932:P
1920:X
1916:V
1912:X
1904:V
1890:}
1887:1
1881:{
1877:/
1873:)
1870:k
1867:,
1864:4
1861:(
1845:k
1837:k
1825:X
1818:.
1804:1
1799:P
1787:X
1767:3
1762:P
1738:1
1733:P
1723:1
1718:P
1706:X
1697:R
1681:1
1676:P
1664:R
1644:2
1639:P
1627:k
1611:1
1606:P
1594:k
1572:2
1567:P
1551:X
1547:X
1543:k
1539:X
1517:Q
1511:P
1498:X
1494:P
1490:m
1486:X
1482:Q
1478:m
1474:m
1470:X
1466:X
1462:X
1458:X
1454:X
1446:Y
1442:k
1438:X
1434:X
1430:Y
1423:m
1419:X
1400:0
1397:=
1392:1
1389:+
1386:m
1383:2
1379:x
1373:m
1370:2
1366:x
1362:+
1356:+
1351:3
1347:x
1341:2
1337:x
1333:+
1328:1
1324:x
1318:0
1314:x
1298:m
1294:X
1275:0
1272:=
1267:2
1262:m
1259:2
1255:x
1251:+
1246:1
1240:m
1237:2
1233:x
1227:2
1221:m
1218:2
1214:x
1210:+
1204:+
1199:3
1195:x
1189:2
1185:x
1181:+
1176:1
1172:x
1166:0
1162:x
1146:k
1142:X
1138:k
1121:2
1117:/
1113:n
1096:n
1092:k
1072:2
1068:/
1064:n
1051:k
1047:n
1043:k
1029:N
1023:a
1001:a
995:P
970:N
964:P
943:R
924:1
921:+
918:n
912:P
886:0
883:=
878:2
873:1
870:+
867:n
863:x
859:+
853:+
848:2
843:1
839:x
835:+
830:2
825:0
821:x
807:k
799:k
788:k
784:X
780:k
776:X
760:n
754:P
741:k
733:k
717:1
714:+
711:n
707:a
703:,
697:,
692:0
688:a
667:)
662:1
659:+
656:n
652:a
648:,
642:,
637:0
633:a
629:(
619:q
613:-
611:k
607:X
603:k
595:k
588:r
584:r
572:k
564:y
562:(
560:q
556:x
554:(
552:q
548:y
546:+
544:x
542:(
540:q
536:y
534:,
532:x
530:(
528:b
520:q
512:k
508:k
500:X
496:X
488:k
484:X
476:X
472:q
457:k
437:k
432:/
428:)
425:0
417:1
414:+
411:N
407:k
403:(
400:=
397:)
394:k
391:(
386:N
380:P
363:k
359:k
351:k
336:q
322:1
316:n
302:X
298:q
290:q
270:1
267:+
264:n
260:x
256:,
250:,
245:0
241:x
230:k
222:q
218:q
202:1
199:+
196:n
191:P
179:k
175:n
171:X
145:C
126:3
120:P
94:w
91:z
88:=
85:y
82:x
53:N
28:.
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