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

Abstract

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.