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.


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


Zbl 0757.14011
Full Text: arXiv Euclid


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.