Mordell-Weil torsion in the mirror of multi-sections
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We give further evidence that genus-one fibers with multi-sections are mirror dual to fibers with Mordell-Weil torsion. In the physics of F-theory compactifications this implies a relation between models with a non-simply connected gauge group and those with discrete symmetries. We provide a combinatorial explanation of this phenomenon for toric hypersurfaces. In particular this leads to a criterion to deduce Mordell-Weil torsion directly from the polytope. For all 3134 complete intersection genus-one curves in three-dimensional toric ambient spaces we confirm the conjecture by explicit calculation. We comment on several new features of these models: the Weierstrass forms of many models can be identified by relabeling the coefficient sections. This reduces the number of models to 1024 inequivalent ones. We give an example of a fiber which contains only non-toric sections one of which becomes toric when the fiber is realized in a different ambient space. Similarly a singularity in codimension one can have a toric resolution in one representation while it is non-toric in another. Finally we give a list of 24 inequivalent genus-one fibers that simultaneously exhibit multi-sections and Mordell-Weil torsion in the Jacobian. We discuss a self-mirror example from this list in detail.
KeywordsDifferential and Algebraic Geometry F-Theory Global Symmetries String Duality
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- W. Stein et al., Sage Mathematics Software (Version 7.1), The Sage Development Team (2016) http://www.sagemath.org.
- A. Novoseltsev, Lattice polytope module for Sage, The Sage Development Team (2010) http://www.sagemath.org/doc/reference/geometry/sage/geometry/lattice polytope.html.
- V. Braun and A. Novoseltsev, Toric geometry module for Sage, The Sage Development Team (2013) http://www.sagemath.org/doc/reference/schemes/sage/schemes/toric/variety.html.
- W. Fulton, Introduction to toric varieties, volume 131, Annals of Mathematics Studies, Princeton University Press (1993).Google Scholar
- D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, volume 68, Mathematical surveys and monographs, American Mathematical Society (1999) [INSPIRE].
- D. Cox, J. Little and H. Schenck, Toric Varieties, Graduate studies in mathematics, American Mathematical Society (2011).Google Scholar
- L. Borisov, Towards the Mirror Symmetry for Calabi-Yau Complete intersections in Gorenstein Toric Fano Varieties, alg-geom/9310001.