Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

Abstract

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Revised paper; final section is shortened; correction of typos

References in corpus (5)

• Defect and Hodge numbers of hypersurfaces
• Elliptic K3 surfaces with geometric Mordell-Weil rank 15
• On the classification of degree 1 elliptic threefolds with constant $j$-invariant
• The elliptic threefold y^2=x^3+16s^6+16t^6-32(t^3s^3+t^3+s^3)+16
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Cited by in corpus (13)

• Anomaly Equations and Intersection Theory
• TASI Lectures on Supergravity and String Vacua in Various Dimensions
• Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves
• 6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces
• Vanishing homology of projective hypersurfaces with 1-dimensional singularities
• Mordell-Weil groups and Zariski triples
• Cuspidal plane curves, syzygies and a bound on the MW-rank
• On the classification of degree 1 elliptic threefolds with constant $j$-invariant
• Elliptic Calabi-Yau threefolds over a del Pezzo surface
• Defect formula for nodal complete intersection threefolds
• Reduction of symbolic first integrals of planar vector fields
• Flops and Mordell-Weil group of Elliptic Threefolds with (4,6,12)-singular fibers
• The average rank of elliptic $n$-folds