Threefolds Fibred by Mirror Sextic Double Planes

TL;DR

This paper systematically studies threefolds fibred by mirror sextic double plane K3 surfaces, introducing invariants, explicit models, and analyzing their geometric properties, akin to elliptic surface theory.

Contribution

It introduces generalized invariants, constructs explicit Weierstrass models, and describes the minimal forms and geometric properties of these threefolds.

Findings

Explicit birational Weierstrass models derived

Minimal forms with mild singularities obtained

Descriptions of singular fibers, canonical divisor, Betti numbers

Abstract

We present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces. We show that the geometry of such threefolds is controlled by a pair of invariants, called the generalized functional and generalized homological invariants, and we derive an explicit birational model for them, which we call the Weierstrass form. We then describe how to resolve the singularities of the Weierstrass form to obtain the "minimal form", which has mild singularities and is unique up to birational maps in codimension 2. Finally we describe some of the geometric properties of threefolds in minimal form, including their singular fibres, canonical divisor, and Betti numbers.