## Nagoya Mathematical Journal

### On classification of $\mathbb{Q}$-Fano $3$-folds of Gorenstein index 2. II.

Hiromichi Takagi

#### Abstract

In the previous paper, we obtained a list of numerical possibilities of $\mathbb{Q}$-Fano $3$-folds $X$ with Pic $X = \mathbb{Z} (-2K_{X})$ and $h^{0} (-K_{X}) \geq 4$ containing index $2$ points $P$ such that $(X, P) \simeq (\{ xy+z^{2}+u^{a} = 0 \} / \mathbb{Z}_{2}(1, 1, 1, 0), o)$ for some $a \in \mathbb{N}$. Moreover we showed that such an $X$ is birational to a simpler Mori fiber space. In this paper, we prove their existence except for a few cases by constructing a Mori fiber space with desired properties and reconstructing $X$ from it.

#### Article information

Source
Nagoya Math. J., Volume 167 (2002), 157-216.

Dates
First available in Project Euclid: 27 April 2005

https://projecteuclid.org/euclid.nmj/1114649296

Mathematical Reviews number (MathSciNet)
MR1924722

Zentralblatt MATH identifier
1048.14023

#### Citation

Takagi, Hiromichi. On classification of $\mathbb{Q}$-Fano $3$-folds of Gorenstein index 2. II. Nagoya Math. J. 167 (2002), 157--216. https://projecteuclid.org/euclid.nmj/1114649296

#### References

• F. Ambro, Ladders on Fano varieties , J. Math. Sci., 94 (1999), 1126–1135.
• A. Grothendieck, Cohomologie Local des Faisceaux Cohérent et Théorème de Lefschetz Locaux et Globaux - SGA2, North Holland (1968).
• J. Kollár and S. Mori, Classification of three dimensional flips , J. of Amer. Math. Soc., 5 (1992), 533–703.
• Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimal model problem , Adv. St. Pure Math., vol. 10 (1987), 287–360.
• K. Kodaira, On stability of compact submanifolds of complex manifolds , Amer. J. Math., 85 (1963), 79–94.
• M. Mella, Existence of good divisors on Mukai varieties , J. Alg. Geom., 8 (1999), 197–206.
• T. Minagawa, Global smoothing of singular weak Fano $3$-folds , preprint (1999).
• T. Minagawa, Deformations of weak Fano $3$-folds with only terminal singularities , Osaka. J. Math., 38 , no. 3 (2001), 533–540.
• S. Mori and S. Mukai, Classification of Fano $3$-folds with $b_2 \geq 2$ , Manuscripta Math., 36 (1981), 147–162.
• S. Mori and S. Mukai, On Fano $3$-folds with $b_{2} \geq 2$ , Algebraic and Analytic Varieties, Adv. Stud. in Pure Math., vol. 1 (1983), 101–129.
• S. Mori and S. Mukai, Classification of Fano $3$-folds with $b_2 \geq 2$, I , Algebraic and Topological Theories, 1985, to the memory of Dr. Takehiko MIYATA, 496–545.
• S. Mori, Threefolds whose canonical bundles are not numerically effective , Ann. of Math., 116 (1982), 133–176.
• S. Mukai, Biregular classification of Fano threefolds and Fano manifolds of coindex $3$ , Proc. Nat'l. Acad. Sci. USA, 86 (1989), 3000–3002.
• S. Mukai, Fano $3$-folds , London Math. Soc. Lecture Notes, vol. 179, Cambridge Univ. Press (1992), 255–263.
• S. Mukai, Curves and Grassmannians , Algebraic Geometry and Related Topics, the Proceedings of the International Symposium, Inchoen, Republic of Korea, International Press (1993), 19–40.
• S. Mukai, New development of the theory of Fano threefolds : Vector bundle method and moduli problem , in Japanese, Sugaku, 47 (1995), 125–144.
• Y. Namikawa, Smoothing Fano $3$-folds , J. Alg. Geom., 6 (1997), 307–324.
• I. Reider, Vector bundles of rank $2$ and linear systems on algebraic surfaces , Ann. of Math., 127 (1988), 309–316.
• M. Reid, Projective morphisms according to Kawamata , preprint (1983); available at http://www.maths.warwick.ac.uk/$^{\sim}$miles/3folds/.
• M. Reid, The moduli space of $3$-folds with $K \equiv 0$ may nevertheless be irreducible , Math. Ann., 278 (1987), 329–334.
• M. Reid, Young person's guide to canonical singularities , Algebraic Geometry, Bowdoin, 1985, Proc. Symp. Pure Math., vol. 46 (1987), 345–414.
• M. Reid, Infinitesimal view of extending a hyperplane section –- deformation theory and computer algebra , Lecture Notes in Math., vol. 1417, Springer-Verlag, Berlin-New York (1990), 214–286.
• M. Reid, Nonnormal del Pezzo surface , Publ. RIMS Kyoto Univ., 30 (1994), 695–728.
• T. Sano, On classification of non-Gorenstein $\mQ$-Fano $3$-folds of Fano index $1$ , J. Math. Soc. Japan, 47 (1995, no. 2), 369–380.
• T. Sano, Classification of non-Gorenstein $\mQ$-Fano $d$-folds of Fano index greater than $d-2$ , Nagoya Math. J., 142 (1996), 133–143.
• V. V. Shokurov, The existence of a straight line on Fano $3$-folds , Izv. Akad. Nauk SSSR Ser. Mat, 43 (1979), 921–963 ; English transl. in Math. USSR Izv. 15 (1980), 173–209.
• V. V. Shokurov, Smoothness of the anticanonical divisor on a Fano $3$-folds , Math. USSR. Izvestija, 43 (1979), 430–441 ; English transl. in Math. USSR Izv. 14 (1980) 395–405.
• H. Takagi, On classification of, $\mQ$-Fano $3$-folds of Gorenstein index $2$. I , Nagoya Math. Journal, 167 (2002), 117–155.
• K. Takeuchi, Some birational maps of Fano $3$-folds , Compositio Math., 71 (1989), 265–283.
• K. Takeuchi, Del Pezzo fiber spaces whose total spaces are weak Fano $3$-folds , in Japanese, Proceedings, Hodge Theory and Algebraic Geometry, 1995 in Kanazawa Univ. (1996), 84–95.
• K. Takeuchi, Weak Fano $3$-folds with del Pezzo fibration , preprint (1999).