Biregular classification of Fano 3-folds and Fano manifolds of coindex 3.
Biregular classification of Fano 3-folds and Fano manifolds of coindex 3.
复制标题
DOI：
10.1073/pnas.86.9.3000
复制
发表时间：
1989-05
影响因子：
11.1
通讯作者：
S. Mukai
中科院分区：
文献类型：
--
作者：
S. Mukai
通过应用向量束理论（在B(2) = 1的情况下）和极值射线理论（在B(2) >/= 2的情况下），在任意场k [unk] C上对Fano 3-褶皱及其高维类似物进行了分类。如果一个n维光滑射影变量X / k的第一类c(1)(X) H(2)（X， Z）在Kodaira [Kodaira， k . (1954) Ann]意义上是正的，那么它就是一个Fano流形。[数学。60,28-48]（或样本）。如果n = 3 c (1) (X)产生H (2) (X, Z),然后要么(我)X是一个完整的十字路口Grassmann品种G对一个齐次向量包E G: E = codim的秩(G) X, X是同构的零位点全球部分E, (ii) X是一个线性部分的十维旋量不同X (12) (10) [unk] P (k)(15),或(3)X是同构的双封面P (k)(3),一个三维二次曲面Q (k)(3),或五次del Pezzo三倍V (5) [unk] P (k)(6)。如果n = 4且c(1)(X)可被2整除，则X [unk] c同构于(a)齐次空间或其双覆盖上的完全交，(b) P(1)与Fano 3-fold的乘积，(c) Q(4) [unk] P(5)沿直线或沿二次曲线的膨胀，或(d) P(1)-束紧化P(3)或Q(3) [unk] P(4)上的直线束。
The Fano 3-folds and their higher dimensional analogues are classified over an arbitrary field k [unk] C by applying the theory of vector bundles (in the case B(2) = 1) and the theory of extremal rays (in the case B(2) >/= 2). An n-dimensional smooth projective variety X over k is a Fano manifold if its first Chern class c(1)(X) epsilon H(2)(X, Z) is positive in the sense of Kodaira [Kodaira, K. (1954) Ann. Math. 60, 28-48] (or ample). If n = 3 and c(1)(X) generates H(2)(X, Z), then either (i) X is a complete intersection in a Grassmann variety G with respect to a homogeneous vector bundle E on G: the rank of E is equal to codim(G)X and X is isomorphic to the zero locus of a global section of E, (ii) X is a linear section of a 10-dimensional spinor variety X(12) (10) [unk] P(k) (15), or (iii) X is isomorphic to a double cover of P(k) (3), a 3-dimensional quadric Q(k) (3), or a quintic del Pezzo 3-fold V(5) [unk] P(k) (6). If n = 4 and c(1)(X) is divisible by 2, then X [unk] C is isomorphic to (a) a complete intersection in a homogeneous space or its double cover, (b) a product of P(1) and a Fano 3-fold, (c) the blow-up of Q(4) [unk] P(5) along a line or along a conic, or (d) a P(1)-bundle compactifying a line bundle on P(3) or on Q(3) [unk] P(4).