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On the classification of degree 1 elliptic threefolds with constant \(j\)-invariant. (English) Zbl 1283.14014

This paper deals with the classification of a class of threefolds over \(\mathbb{C}\) admitting elliptic fibrations to \(\mathbb{P}^2\). More precisely, the threefolds considered here are the ones admitting a degree 1 elliptic fibration to \(\mathbb{P}^2\), that is, that are birational to a degree 6 hypersurface in the weighted projective space \(\mathbb{P}(2,3,1,1,1)\). Further it is also assumed that they are neither birational to a cone over an elliptic surface in \(\mathbb{P}(2,3,1,1)\) nor to a product \(\mathbb{P}^2 \times E\). For such threefolds all possible Mordell-Weil groups are determined. Further the ones with fibers having \(j\)-invariant 1728 the equation is determined for two possible cases of the Mordell-Weil group, and similar results are obtained for \(j\)-invariant \(0\) as well.
The main method is the following. Treating a given threefold as an elliptic surface \(S\) over the function field of \(\mathbb{P}^1\), the Mordell-Weil group is the subgroup of the Mordell-Weil group of \(S_{\overline{K(\mathbb{P}^1)}}\) fixed by the Galois action of \(\mathrm{Gal} \left(\overline{K(\mathbb{P}^1)}, K(\mathbb{P}^1) \right)\). By the classification of elliptic surfaces according to K. Oguiso and T. Shioda [Comment. Math. Univ. St. Pauli 40, No. 1, 83–99 (1991; Zbl 0757.14011)], there is a small list fo the possible Mordell-Weil groups of such surfaces, and then geometric considerations lead to the right subgroups. The analysis of the special cases of the \(j\)-invariants uses a Hodge theoretic description of the Mordell-Weil group.

MSC:

14J30 \(3\)-folds
11G05 Elliptic curves over global fields
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations

Citations:

Zbl 0757.14011
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Full Text: arXiv Euclid

References:

[1] E. Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen , Manuscripta Math. 2 (1970), 103-161. · Zbl 0186.26101
[2] A. Dimca, Betti numbers of hypersurfaces and defects of linear systems , Duke Math. J. 60 (1990), 285-298. · Zbl 0729.14017
[3] A. Dimca, Singularities and topology of hypersurfaces , Universitext, Springer-Verlag, New York, 1992. · Zbl 0753.57001
[4] M. Grooten and J. H. M. Steenbrink, Quartic double solids with ordinary singularities , preprint, 2008; available at · Zbl 1209.14012
[5] K. Hulek and R. Kloosterman, Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces , preprint, 2008; available at · Zbl 1246.14057
[6] M. Ishikawa, T. C. Nguyen and M. Oka, On topological types of reduced sextics , Kodai Math. J. 27 (2004), 237-260. · Zbl 1067.14024
[7] R. Kloosterman, A different method to calculate the rank of an elliptic threefold , Rocky Mountain J. Math. 42 (2012), 643-655. · Zbl 1257.14027
[8] R. Miranda, Smooth models for elliptic threefolds , The birational geometry of degenerations (Cambridge, Mass., 1981), Progr. Math., vol. 29, Birkhäuser Boston, Boston, MA, 1983, pp. 85-133. · Zbl 0583.14014
[9] K. Oguiso and T. Shioda, The Mordell-Weil lattice of a rational elliptic surface , Comment. Math. Univ. St. Pauli. 40 (1991), 83-99. · Zbl 0757.14011
[10] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 52, Springer-Verlag, Berlin, 2008. · Zbl 1138.14002
[11] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves , Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. · Zbl 0911.14015
[12] J. H. M. Steenbrink, Adjunction conditions for one-forms on surfaces in projective three-space , Singularities and computer algebra, London Math. Soc. Lecture Note Ser., vol. 324, Cambridge Univ. Press, Cambridge, 2006, pp. 301-314. · Zbl 1105.14008
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