Abstract
Global F-theory compactifications whose fibers are realized as complete inter-sections form a richer set of models than just hypersurfaces. The detailed study of the physics associated with such geometries depends crucially on being able to put the elliptic fiber into Weierstrass form. While such a transformation is always guaranteed to exist, its explicit form is only known in a few special cases. We present a general algorithm for computing the Weierstrass form of elliptic curves defined as complete intersections of different codimensions and use it to solve all cases of complete intersections of two equations in an ambient toric variety. Using this result, we determine the toric Mordell-Weil groups of all 3134 nef partitions obtained from the 4319 three-dimensional reflexive polytopes and find new groups that do not exist for toric hypersurfaces. As an application, we construct several models that cannot be realized as toric hypersurfaces, such as the first toric SU(5) GUT model in the literature with distinctly charged 10 representations and an F-theory model with discrete gauge group ℤ4 whose dual fiber has a Mordell-Weil group with ℤ4 torsion.
Highlights
Of the main quantities of interest. τ and especially the locus of its singularities can be obtained if the defining equation of the T 2 is given in Weierstrass form y2 = x3 + f x + g
We present a general algorithm for computing the Weierstrass form of elliptic curves defined as complete intersections of different codimensions and use it to solve all cases of complete intersections of two equations in an ambient toric variety
If a nef partition has one component ∇i that is spanned only by a single vertex v, the complete intersection can be reduced to a complete intersection in a toric variety of one dimension less whose reflexive polytope is obtained by projecting ∆◦ along v
Summary
The one indispensable tool for studying complete intersections is the Koszul complex and the associated hypercohomology spectral sequence. The polynomial pj defines a divisor Dj = V (pj) = {pj = 0}, and the cohomology groups of the line bundle O(Dj) precisely involve differential forms of the same degree of homogeneity as pj. This is not the end of it and even a d1-cohomology class need not survive to a non-zero element of Hp+q(Y, L|Y ) This is the case when two different k-forms α1, α2 on X are related via a double residue of a (k + 1)-form, α1 − α2 = d Res Res2(ω). One can reconstruct the dimension of the line bundle cohomology groups on the complete intersection from the knowledge of the dimensions of the E∞ tableau entries
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
