Elliptic three-folds II: Multiple fibres

Gross, Mark

doi:10.1090/S0002-9947-97-01845-X

Elliptic three-folds II: Multiple fibres
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Trans. Amer. Math. Soc. 349 (1997), 3409-3468 Request permission

Abstract:

Let $f:X\rightarrow S$ be an elliptic fibration with a section, where $S$ is a projective surface and $X$ is a projective threefold. We determine when it is possible to perform a logarithmic transformation along a closed subset $Z\subseteq S$ to obtain a new elliptic fibration $f’:X’\rightarrow S$ which now has multiple fibres along $Z$. This is done in the setting of Ogg-Shafarevich theory. We find a number of obstructions to performing such a logarithmic transformation, the very last of which takes values in the torsion part of the codimension 2 Chow group of $X$.

References

W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574, DOI 10.1007/978-3-642-96754-2
S. Bloch, Torsion algebraic cycles and a theorem of Roitman, Compositio Math. 39 (1979), no. 1, 107–127. MR 539002
Spencer Bloch and Arthur Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181–201 (1975). MR 412191
François R. Cossec and Igor V. Dolgachev, Enriques surfaces. I, Progress in Mathematics, vol. 76, Birkhäuser Boston, Inc., Boston, MA, 1989. MR 986969, DOI 10.1007/978-1-4612-3696-2
Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, and Christophe Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J. 50 (1983), no. 3, 763–801 (French). MR 714830, DOI 10.1215/S0012-7094-83-05038-X
David A. Cox and Steven Zucker, Intersection numbers of sections of elliptic surfaces, Invent. Math. 53 (1979), no. 1, 1–44. MR 538682, DOI 10.1007/BF01403189
P. Deligne, Courbes elliptiques: formulaire d’après J. Tate, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 53–73 (French). MR 0387292
Igor Dolgachev and Mark Gross, Elliptic threefolds. I. Ogg-Shafarevich theory, J. Algebraic Geom. 3 (1994), no. 1, 39–80. MR 1242006
Yoshio Fujimoto, Logarithmic transformations on elliptic fiber spaces, J. Math. Kyoto Univ. 28 (1988), no. 1, 91–110. MR 929211, DOI 10.1215/kjm/1250520560
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Grassi, A., Minimal Models of Elliptic Threefolds, Ph.D. Thesis, Duke University, 1990.
Mark Gross, A finiteness theorem for elliptic Calabi-Yau threefolds, Duke Math. J. 74 (1994), no. 2, 271–299. MR 1272978, DOI 10.1215/S0012-7094-94-07414-0
L. C. Young, On an inequality of Marcel Riesz, Ann. of Math. (2) 40 (1939), 567–574. MR 39, DOI 10.2307/1968941
K. Kodaira, On compact complex analytic surfaces. I, Ann. of Math. (2) 71 (1960), 111–152. MR 132556, DOI 10.2307/1969881
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
Rick Miranda, Smooth models for elliptic threefolds, The birational geometry of degenerations (Cambridge, Mass., 1981) Progr. Math., vol. 29, Birkhäuser, Boston, Mass., 1983, pp. 85–133. MR 690264
David Mumford and Kalevi Suominen, Introduction to the theory of moduli, Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer-School in Math.), Wolters-Noordhoff, Groningen, 1972, pp. 171–222. MR 0437531
Noboru Nakayama, On Weierstrass models, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 405–431. MR 977771
Nakayama, N., Local Structure of an Elliptic Fibration, Preprint, Univ. of Tokyo, 1991.
A. P. Ogg, Cohomology of abelian varieties over function fields, Ann. of Math. (2) 76 (1962), 185–212. MR 155824, DOI 10.2307/1970272
I. R. Šafarevič, Principal homogeneous spaces defined over a function field, Trudy Mat. Inst. Steklov. 64 (1961), 316–346 (Russian). MR 0162806