On the existence of certain weak Fano threefolds of Picard number two


  • Maxim Arap
  • Joseph Cutrone
  • Nicholas Marshburn




This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number $1$ with the exception of $12$ numerical cases.


Arap, M. and Marshburn, N., Brill-Noether general curves on Knutsen K3 surfaces, C. R. Math. Acad. Sci. Paris 352 (2014), no. 2, 133–135. http://dx.doi.org/10.1016/j.crma.2013.11.020

Beauville, A., Complex algebraic surfaces, second ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. http://dx.doi.org/10.1017/CBO9780511623936

Blanc, J. and Lamy, S., Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links, Proc. Lond. Math. Soc. (3) 105 (2012), no. 5, 1047–1075. http://dx.doi.org/10.1112/plms/pds023

Corti, A., Haskins, M., Nordström, J., and Pacini, T., Asymptotically cylindrical Calabi-Yau $3$-folds from weak Fano $3$-folds, Geom. Topol. 17 (2013), no. 4, 1955–2059. http://dx.doi.org/10.2140/gt.2013.17.1955

Corti, A., Haskins, M., Nordström, J., and Pacini, T., $mathrm G_2$-manifolds and associative submanifolds via semi-Fano 3-folds, Duke Math. J. 164 (2015), no. 10, 1971–2092. http://dx.doi.org/10.1215/00127094-3120743

Cutrone, J. W. and Marshburn, N. A., Towards the classification of weak Fano threefolds with $rho =2$, Cent. Eur. J. Math. 11 (2013), no. 9, 1552–1576. http://dx.doi.org/10.2478/s11533-013-0261-5

Green, M. and Lazarsfeld, R., Special divisors on curves on a $K3$ surface, Invent. Math. 89 (1987), no. 2, 357–370. http://dx.doi.org/10.1007/BF01389083

Gushel, N. P., Fano varieties of genus $6$, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1159–1174.

Gushel, N. P., Fano $3$-folds of genus $8$, Algebra i Analiz 4 (1992), no. 1, 120–134, English translation: St. Petersberg Math. J. 4 (1993) no. 1, 115–129.

Iskovskih, V. A., Fano threefolds. I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 3, 516–562.

Iskovskih, V. A., Fano threefolds. II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 3, 506–549.

Iskovskikh, V. A. and Prokhorov, Y. G., Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1--247.

Jahnke, P., Peternell, T., and Radloff, I., Threefolds with big and nef anticanonical bundles. I, Math. Ann. 333 (2005), no. 3, 569–631. http://dx.doi.org/10.1007/s00208-005-0682-y

Jahnke, P., Peternell, T., and Radloff, I., Threefolds with big and nef anticanonical bundles II, Cent. Eur. J. Math. 9 (2011), no. 3, 449–488. http://dx.doi.org/10.2478/s11533-011-0023-1

Kleiman, S. L. and Laksov, D., Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082.

Knutsen, A. L., Smooth curves on projective $K3$ surfaces, Math. Scand. 90 (2002), no. 2, 215–231. http://dx.doi.org/10.7146/math.scand.a-14371

Knutsen, A. L., Smooth, isolated curves in families of Calabi-Yau threefolds in homogeneous spaces, J. Korean Math. Soc. 50 (2013), no. 5, 1033–1050. http://dx.doi.org/10.4134/JKMS.2013.50.5.1033

Kollár, J., Flops, Nagoya Math. J. 113 (1989), 15–36.

Lazarsfeld, R., Brill-Noether-Petri without degenerations, J. Differential Geom. 23 (1986), no. 3, 299–307.

Lazarsfeld, R., Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 48, Springer-Verlag, Berlin, 2004. http://dx.doi.org/10.1007/978-3-642-18808-4

Le Barz, P., Formules multisécantes pour les courbes gauches quelconques, Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., vol. 24, Birkhäuser, Boston, Mass., 1982, pp. 165--197.

Maruyama, M., On a family of algebraic vector bundles, Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 95--146.

Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133–176. http://dx.doi.org/10.2307/2007050

Mori, S. and Mukai, S., Classification of Fano $3$-folds with $B_2geq 2$. I, Algebraic and topological theories (Kinosaki, 1984), Kinokuniya, Tokyo, 1986, pp. 496--545.

Mukai, S., Curves and Grassmannians, Algebraic geometry and related topics (Inchon, 1992), Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 19--40.

Mukai, S., New developments in Fano manifold theory related to the vector bundle method and moduli problems, Sūgaku 47 (1995), no. 2, 125–144, English translation: Sugaku Expositions 15 (2002), no. 2, 125–150.

Saint-Donat, B., Projective models of $K-3$ surfaces, Amer. J. Math. 96 (1974), 602–639.

Šokurov, V. V., The existence of a line on Fano varieties, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 922–964.

Takeuchi, K., Weak Fano threefolds with del Pezzo fibration, preprint arXiv:0910.2188 [math.AG], October 2009.




How to Cite

Arap, M., Cutrone, J., & Marshburn, N. (2017). On the existence of certain weak Fano threefolds of Picard number two. MATHEMATICA SCANDINAVICA, 120(1), 68–86. https://doi.org/10.7146/math.scand.a-25505