Advances in Theoretical and Mathematical Physics

Volume 17 (2013)

Number 4

Weierstrass models of elliptic toric $K3$ hypersurfaces and symplectic cuts

Pages: 741 – 770



Antonella Grassi (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

Vittorio Perduca (MAP5, Laboratory of Applied Mathematics, Paris Descartes University & CNRS, Paris, France)


We study elliptically fibered $K3$ surfaces, with sections, in toric Fano 3-folds which satisfy certain combinatorial properties relevant to F-theory/heterotic duality. We show that some of these conditions are equivalent to the existence of an appropriate notion of a Weierstrass model adapted to the toric context. Moreover, we show that if in addition other conditions are satisfied, there exists a toric semistable degeneration of the elliptic $K3$ surface which is compatible with the elliptic fibration and F-theory/Heterotic duality.

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