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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
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