Abstract
We prove that a terminal three-dimensional del Pezzo fibration has no fibers of multiplicity >6. We also obtain a rough classification of possible configurations of singular points on multiple fibers and give some examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Alexeev, “General Elephants of Q-Fano 3-folds,” Compos. Math. 91(1), 91–116 (1994).
P. Griffiths and J. Harris, Principles of Algebraic Geometry (J. Wiley & Sons, New York., 1978), Pure Appl. Math.
P. Hacking and Yu. Prokhorov, “Degenerations of del Pezzo Surfaces. I,” arXiv:math.AG/0509529.
F. Hidaka and K. Watanabe, “Normal Gorenstein Surfaces with Ample Anti-canonical Divisor,” Tokyo J. Math. 4(2), 319–330 (1981).
Y. Kawamata, “Crepant Blowing-up of 3-Dimensional Canonical Singularities and Its Application to Degenerations of Surfaces,” Ann. Math., Ser. 2, 127(1), 93–163 (1988).
Y. Kawamata, “On Fujita’s Freeness Conjecture for 3-folds and 4-folds,” Math. Ann. 308(3), 491–505 (1997).
Y. Kawamata, “Subadjunction of Log Canonical Divisors for a Subvariety of Codimension 2,” in Birational Algebraic Geometry, Baltimore, MD, 1996 (Am. Math. Soc., Providence, RI, 1997), Contemp. Math. 207, pp. 79–88.
Y. Kawamata, K. Matsuda, and K. Matsuki, “Introduction to the Minimal Model Problem,” in Algebraic Geometry, Sendai, 1985 (North-Holland, Amsterdam, 1987), Adv. Stud. Pure Math. 10, pp. 283–360.
K. Kodaira, “On Compact Analytic Surfaces. II,” Ann. Math., Ser. 2, 77, 563–626 (1963).
K. Kodaira, “On the Structure of Compact Complex Analytic Surfaces. I,” Am. J. Math. 86, 751–798 (1964).
Flips and Abundance for Algebraic Threefolds, Ed. by J. Kollár (Soc. Math. France, Paris, 1992), Astérisque 211.
J. Kollár and N. I. Shepherd-Barron, “Threefolds and Deformations of Surface Singularities,” Invent. Math. 91(2), 299–338 (1988).
S. Mori, “On 3-Dimensional Terminal Singularities,” Nagoya Math. J. 98, 43–66 (1985).
S. Mori, “Flip Theorem and the Existence of Minimal Models for 3-folds,” J. Am. Math. Soc. 1(1), 117–253 (1988).
S. Mori and Yu. Prokhorov, “On ℚ-Conic Bundles,” Publ. Res. Inst. Math. Sci. 44(2), 315–369 (2008).
M. Miyanishi and D. Q. Zhang, “Gorenstein Log del Pezzo Surfaces of Rank One,” J. Algebra 118(1), 63–84 (1988).
M. Miyanishi and D. Q. Zhang, “Gorenstein Log del Pezzo Surfaces. II,” J. Algebra 156(1), 183–193 (1993).
N. Nakayama, “Hodge Filtrations and the Higher Direct Images of Canonical Sheaves,” Invent. Math. 85(1), 217–221 (1986).
M. Reid, “Young Person’s Guide to Canonical Singularities,” in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Am. Math. Soc., Providence, RI, 1987), Part 1, Proc. Symp. Pure Math. 46, pp. 345–414.
V. V. Shokurov, “3-fold Log Flips,” Izv. Ross. Akad. Nauk, Ser. Mat. 56(1), 105–203 (1992) [Russ. Acad. Sci., Izv. Math. 40, 95–202 (1993)].
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Professor Vasily Alekseevich Iskovskikh
Rights and permissions
About this article
Cite this article
Mori, S., Prokhorov, Y. Multiple fibers of del Pezzo fibrations. Proc. Steklov Inst. Math. 264, 131–145 (2009). https://doi.org/10.1134/S0081543809010167
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543809010167