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Hyperelliptic and trigonal Fano threefolds. (English. Russian original) Zbl 1081.14059

Izv. Math. 69, No. 2, 365-421 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 2, 145-204 (2005).
The normal complex projective variety \(X\) is Fano if its anticanonical divisor \(-K_X\) is an ample \({\mathbb Q}\)-Cartier divisor. In addition \(X\) is Gorenstein (and the singularities of \(X\) are called Gorenstein) if the local ring of any point \(x \in X\) is Cohen-Macaulay, and the dualizing sheaf \(\omega_X\) is invertible. In this case the canonical class \(-K_X \in \text{Pic}(X)\), i.e. the divisor \(-K_X\) is Cartier. Such \(X\) has canonical (terminal) singularities if for any resolution \(f: X' \rightarrow X\), the divisor \(K_{X'} - f^*(K_X)\) is a sum of exceptional divisors with non-negative (positive) rational coefficients, [see e.g. V. Iskovskikh, Yu. Prokhorov, “Fano varieties”, Encycl. Math. Sci., 47 (1999; Zbl 0912.14013)]. Consider Gorenstein Fano threefolds \(X\) with canonical singularities, or in brief \(GC\)-Fano 3-folds. Let \(| -K_X| \) be the anticanonical system of \(X\), and denote by \(\text{Bas}(-K_X)\) its base locus. The \(GC\)-Fano 3-folds \(X\) with non-empty \(\text{Bas}(-K_X)\) are classified by P. Jahnke and I. Radloff [Gorenstein Fano threefolds with base points in the anticanonical system, preprint, http://arxiv.org/abs/math.AG/0404156]. The \(CG\)-Fano 3-folds with empty \(\text{Bas}(-K_X)\), i.e. with base-point-free \(| -K_X| \), are divided into three cases: (1) Hyperelliptic, i.e. these \(X\) for which the morphism \(\varphi_X\), defined by \(| -K_X| \), is not an embedding; (2) Trigonal, i.e. when \(\varphi_X\) is an embedding but the image \(\varphi_X(X)\) is not an intersction of quadrics; (3) these \(X\) for which \(\varphi_X\) is an embedding and \(\varphi_X(X)\) is an intersection of quadrics. In this paper are classified the \(GC\)-Fano 3-folds in cases (1) and (2), i.e. the hyperelliptic and the trigonal \(CG\)-Fano 3-folds. By theorem 1.5, the list of hyperelliptic \(CG\)-Fano 3-folds contains \(47\) families \((H_1)-(H_{47})\). The threefolds \(X\) from \((H_1)\) and \((H_2)\) are hypersurfaces of degree \(6\) in the weighted projective spaces \({\mathbb P}(1^3,2,3)\) and \({\mathbb P}(1^4,3)\), and any \(X \in (H_3)\) is a complete intersection of a quadric cone and a quartic in \({\mathbb P}(1^5,2)\). The other hyperelliptic \(CG\)-Fano 3-folds all happen to be anticanonical models of special double coverings \(V\) of rational scrolls \({\mathbb F}(d_1,d_2,d_3)\) for \(44\) choices of the triples of integers \((d_1,d_2,d_3)\), corresponding to the cases \((H_4)-(H_{47})\). Theorem 1.6 gives the list of all trigonal \(CG\)-Fano 3-folds. It contains \(69\) families \((T_1)-(T_{69})\). The threefolds \(X\) from \((T_1)\) and \((T_2)\) are hypersurfaces of degree \(4\) in \({\mathbb P}^4\) and complete intersections of a quadric and a cubic in \({\mathbb P}^5\), while any \(X \in (T_3)\) is represented as the anticanonical image of a certain divisor \(Y \subset {\mathbb F}(2,0,0)\). The other trigonal \(X\) are all anticanonical images of special divisors \(V \subset {\mathbb F}(d_1,d_2,d_3,d_4)\) for \(65\) choices of the 4-ples of integers \((d_1,d_2,d_3,d_4)\), thus giving the families \((T_4)-(T_{69})\). As known by [Y. Namikawa, J. Algebr. Geom., 6, 307–324 (1997; Zbl 0906.14019)], any terminal Gorenstein Fano 3-fold can be deformed to a smooth Fano 3-fold. In contrast, among the \(CG\)-Fano 3-folds from the lists of theorems 1.5 and 1.6 only the families \((H_1)-(H_4),(H_6),(H_9)\), \((T_1), (T_2),(T_5),(T_8)\) and \((T_{14})\) contain smooth threefolds, see remark 1.9.

MSC:

14J45 Fano varieties
14J30 \(3\)-folds
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