Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts

TL;DR

This paper explores the construction of Weierstrass models for elliptic K3 surfaces within toric Fano threefolds, establishing conditions for their existence and compatibility with duality theories, and introduces toric semistable degenerations.

Contribution

It identifies combinatorial conditions for elliptic K3 surfaces in toric Fano threefolds that ensure the existence of adapted Weierstrass models and compatible degenerations.

Findings

Conditions equivalent to Weierstrass model existence.

Existence of toric semistable degenerations under certain conditions.

Compatibility with F-theory/Heterotic duality.

Abstract

We study elliptically fibered K3 surfaces, with sections, in toric Fano threefolds 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.