Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves
-
Jose-Ignacio Cogolludo-Agustín
and
Anatoly Libgober
Abstract
We establish a correspondence between the rank of Mordell–Weil group of the complex elliptic threefold associated with a plane curve 𝒞 ⊂ ℙ2(𝔻) with equation F = 0, certain roots of the Alexander polynomial associated with the fundamental group π1(ℙ2(𝔻)∖𝒞) and the polynomial solutions for the functional equation of type h1pF1 + h2qF2 + h3rF3 = 0 where F = F1F2F3. This correspondence is obtained for curves in a certain class which includes the curves having introduced here δ-essential singularities and in particular for all curves with ADE singularities.
As a consequence we find a linear bound for the degree of the Alexander polynomial in terms of the degree of 𝒞 for curves with δ-essential singularities and in particular arbitrary ADE singularities.
Funding source: Spanish Ministry of Education
Award Identifier / Grant number: MTM2010-21740-C02-02
Funding source: NSF
© 2014 by De Gruyter
Articles in the same Issue
- Frontmatter
- Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces
- Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves
- Reflexive differential forms on singular spaces. Geometry and cohomology
- Special cycles on unitary Shimura varieties II: Global theory
- Batalin–Vilkovisky structures on Ext and Tor
- Tangent cones and regularity of real hypersurfaces
Articles in the same Issue
- Frontmatter
- Holomorphic one-forms, integral and rational points on complex hyperbolic surfaces
- Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves
- Reflexive differential forms on singular spaces. Geometry and cohomology
- Special cycles on unitary Shimura varieties II: Global theory
- Batalin–Vilkovisky structures on Ext and Tor
- Tangent cones and regularity of real hypersurfaces
