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3-dimensional Fano varieties with canonical singularities. (English) Zbl 0708.14025

Let \(X\) be a Fano 3-fold with canonical singularities; let \(r=r(X)\) be the index of \(X\) (see definition (0.2)), and let \(-K_X\sim r.H\). Starting from Miles Reid’s description of the singularities of the general member \(S\in | -K_X|\) and using the methods of V. V. Shokurov and T. Fujita, the author proves:
(1) If \(r>1\), then \(Bs| -K_X| =\emptyset\).
(2) If \(\dim (Bs| -K_X|)=1\), then \(Bs| -K_X| \cap \mathrm{Sing}(S)=\emptyset\).
(3) If \(\dim (Bs| -K_X|)=0\), then \(Bs| -K_X| \subset \mathrm{Sing}(S)\) consists of exactly one point (see theorem (0.5)). It follows from the Fujita’s \(\Delta\)-genera theory that \(r=r(X)\) is not greater than 4 (see corollary (3.6)); moreover, \(r(X)=4\) iff \(X\simeq P^3\), and \(r(X)=3\) iff \(X\simeq\) [a quadric in \(P^4]\) (see theorem (3.9)). The class of the Fano 3-folds of index \(r=2\) corresponds in some sense to the del Pezzo surfaces (see for example definitions (0.3)).
The main result in this paper is the classifying theorem for the Fano 3-folds of index 2, which possess canonical singularities (see theorem (0.6) and corollary (0.7)). More concretely:
Let \(S\) be a general member of \(| H|\). Then \(S\) has at worst rational double points (theorem (0.6)). Let \(d=H^3\); then:
(1) \(d=1\) iff \(Bs| H| =\) (one point) iff \(W=\phi_H(X)\) has dimension 2 iff \(X\simeq\) [a hypersurface of degree 6 in \(P(1,1,1,2,3)]\).
(2) \(d>1\) iff \(Bs| H| =\emptyset\) iff \(\dim (W=\Phi_ H(X))=3\); moreover:
(2.a) If \(d\geq 3\) then \(H\) is very ample and \(X\simeq\) [a hypersurface of degree \(d\) in \(P^{d-1}]\).
(2.b) If \(d=2\), then \(\Phi_H\) is a double cover over \(P^3\) and \(X\simeq\) [a hypersurface of degree 4 in \(P(1,1,1,1,2)]\).

MSC:

14J30 \(3\)-folds
14J10 Families, moduli, classification: algebraic theory
14B05 Singularities in algebraic geometry
14J45 Fano varieties
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