Computing certain Fano 3-folds. (English) Zbl 1166.14027

By definition, a \(\mathbb{Q}\)-Fano 3-fold \(X\) is a normal projective \(3\)-dimensional algebraic variety with terminal \(\mathbb{Q}\)-factorial singularities such that divisor \(-K_{X}\) is ample and \(\mathrm{rank}~\mathrm{Pic}(X)=1\). The Fano index \(f(X)\) of the \(\mathbb{Q}\)-Fano 3-fold \(X\) is \(\max\{m \in \mathbb{Z}_{>0}|-K_{X} = mA\), \(A \in \mathrm{Weil}(X)\}\), where equality means numerical equivalence of the \(\mathbb{Q}\)-Cartier divisors. A Weil divisor \(A\) such that \(-K_{X} = f(X)A\) is called a primitive divisor. By K. Suzuki [Manuscr. Math. 114, No. 2, 229–246 (2004; Zbl 1063.14049)], the bound \(f(X) \leq 19\) takes place.
In the paper under review, G. Brown and K. Suzuki study Hilbert series of the \(\mathbb{Q}\)-Fano 3-folds of Fano index \(\geq 3\). More precisely, for each \(3 \leq f \leq 19\) they investigate the number of power series that could be the Hilbert series of the form \(P_{X,A}(t) = \sum_{n=0}^{\infty}p_{n}t^{n}\), where \(p_{n}= \dim H^0(X, \mathcal{O}_{X}(nA))\), for some \(\mathbb{Q}\)-Fano 3-fold \(X\) of Fano index \(f=f(X)\) and primitive divisor \(A\) on \(X\). Among the power series in question the authors also distinguish which of them correspond to Kawamata–Bogomolov stable \(\mathbb{Q}\)-Fano 3-folds (i.e. for which the inequality \(f^2A^3 \leq 3Ac_{2}(X)\) holds).
From the obtained classification one can deduce that there are no \(\mathbb{Q}\)-Fano 3-folds of indices \(12\), \(14\), \(15\), \(16\) and \(18\). Moreover, the authors prove that \(H^0(X, \mathcal{O}_{X}(-K_{X})) \neq 0\). Some numerical examples are given in the last section of the paper.
The method which is used by the authors is based on the equivariant Riemann–Roch Theorem from [M. Reid, Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, 345–414 (1987; Zbl 0634.14003)]. This gives an expression of the above dimensions \(p_n\) in terms of \(n\), \(f(X)\), \(A^3\), \(A^2 \cdot c_{2}(X)\) and the basket of singularities \(\mathcal{B}\) which corresponds to \(X\). Here \(\mathcal{B}\) is a collection of quotient singularities of the form \(\frac{1}{r}(a,-a,f(X))\). Now, there are the following restrictions on the above numerical data:
According to [Y. Kawamata, Algebra, Proc. Int. Conf. Memory A. I. Mal’cev, Novosibirsk/USSR 1989, Contemp. Math. 131, Pt. 3, 439–445 (1992; Zbl 0785.14024)], one has \(-K_{X} \cdot c_{2}(X)>0\).
On the other hand, \(-K_{X} \cdot c_{2}(X) = 24 - \sum_{\frac{1}{r}(a,-a,f(X)) \in \mathcal{B}}\left(r-\frac{1}{r}\right)\);
The degree \(A^3\) is positive;
By Y. Kawamata [Zbl 0785.14024] and K. Suzuki [Zbl 1063.14049], the inequality \((4f(X)^2 - 3f(X))A^3 \leq 4f(X)A \cdot c_{2}(X)\) holds.
Using these restriction results and the computer algebra system Magma [see W. Bosma, J. Cannon and C. Playoust, J. Symb. Comput. 24, No. 3–4, 235–265 (1997; Zbl 0898.68039)] the authors compute all the possible power series. However, the main Theorem of the paper is only an “upper-bound” result, since it is not known which of the obtained power series is realized on the particular \(\mathbb{Q}\)-Fano 3-fold. Moreover, there is no example of the \(\mathbb{Q}\)-Fano 3-fold which is not Kawamata–Bogomolov stable (the authors construct only stable examples). There are many other interesting questions posed by the authors. All this gives a reach field to explore and makes the article interesting to read.


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


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