Open Access Open Access  Restricted Access Subscription Access

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.

Full Text:



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.

Beauville, A., Complex algebraic surfaces, second ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996.

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.

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.

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.

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.

Green, M. and Lazarsfeld, R., Special divisors on curves on a $K3$ surface, Invent. Math. 89 (1987), no. 2, 357–370.

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.

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.

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.

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.

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.

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.

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.



  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.

ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library