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Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. (English) Zbl 0679.14020
This paper is simply an announcement of results. The details will be published elsewhere. From the abstract:
“The Fano 3-folds and their higher dimensional analogues are classified over an arbitrary field $$k\subset {\mathbb{C}}$$ by applying the theory of vector bundles (in the case $$B_ 2=1)$$ and the theory of extremal rays (in the case $$B_ 2\geq 2)$$. An n-dimensional smooth projective variety X over k is a Fano manifold if its first Chern class $$c_ 1(X)\in H^ 2(X,{\mathbb{Z}})$$ is ample.
If $$n=3$$ and $$c_ 1(X)$$ generates $$H^ 2(X,{\mathbb{Z}})$$, 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 $$co\dim_ GX$$ and X is isomorphic to the zero locus of a global section of E, $$(ii)\quad X\quad is$$ a linear section of a 10-dimensional spinor variety $$X^{10}_{12}\subset {\mathbb{P}}_ k^{15}$$, or (iii) X is isomorphic to a double cover of $${\mathbb{P}}^ 3_ k$$, a 3-dimensional quadric $$Q^ 3_ k$$, or a quintic del Pezzo 3-fold $$V_ 5\subset {\mathbb{P}}^ 6_ k.$$
If $$n=4$$ and $$c_ 1(X)$$ is divisible by 2, then $$X\otimes {\mathbb{C}}$$ is isomorphic to (a) a complete intersection in a homogeneous space or its double cover, (b) a product of $${\mathbb{P}}^ 1$$ and a Fano 3-fold; (c) the blow-up of $$Q^ 4\subset {\mathbb{P}}^ 5$$ along a line or along a conic, or (d) a $${\mathbb{P}}^ 1$$-bundle compactifying a line bundle on $${\mathbb{P}}^ 3$$ or on $$Q^ 3\subset {\mathbb{P}}^ 4.''$$
Reviewer: M.Beltrametti

##### MSC:
 14J30 $$3$$-folds 14E05 Rational and birational maps 14J10 Families, moduli, classification: algebraic theory
##### Keywords:
Fano 3-folds; vector bundles; extremal rays; Grassmann variety
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