##
**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.

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.

Reviewer: Ilya Karzhemanov (Vladimir)

### Software:

Magma
PDFBibTeX
XMLCite

\textit{G. Brown} and \textit{K. Suzuki}, Japan J. Ind. Appl. Math. 24, No. 3, 241--250 (2007; Zbl 1166.14027)

### References:

[1] | Altinok, S.; Brown, G.; Reid, M., Fano 3-folds,K3 surfaces and graded rings. Topology and geometry: commemorating SISTAG, Contemp. Math., 314, 25-53 (2002) · Zbl 1047.14026 |

[2] | G. Brown, A database of polarised K3 surfaces. Exp. Math., 2006, to appear. |

[3] | G. Brown and K. Suzuki, Lists of examples and Magma code available for download at www.kent.ac.uk/ims/grdb. |

[4] | A.R. Iano-Fletcher, Working with weighted complete intersections. Explicit birational geometry of 3-folds (A. Corti and M. Reid eds.), LMS Lecture Note Ser.,281, CUP, 2000, 101-173. · Zbl 0960.14027 |

[5] | Iskovskikh, V. A.; Prokhorov, Yu. G., Fano Varieties, Algebraic geometry V, Encyclopaedia of Mathematical Sciences,47 (1999), Berlin: Springer-Verlag, Berlin · Zbl 0912.14013 |

[6] | Kawamata, Y., Boundedness of ℚ-Fano Threefolds, Contemp. Math., 131, 439-445 (1992) · Zbl 0785.14024 |

[7] | Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system I: The user language, J. Symb. Comp., 24, 235-265 (1997) · Zbl 0898.68039 |

[8] | M. Reid, Young person’s guide to canonical singularities. Algebraic Geometry (Bowdoin 1985), vol. 1, S. Bloch ed., Proc. of Symposia in Pure Math.,46, AMS, 1987, 345-414. |

[9] | M. Reid, Graded rings and birational geometry, Proc. of algebraic geometry symposium (Kinosaki, Oct 2000), K. Ohno (ed.), 1-72. |

[10] | K. Suzuki, On ℚ-Fano 3-folds with Fano index ≥ 9. Manuscripta Mathematica,114, Springer, 229-246. · Zbl 1063.14049 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.