Simple fibrations in (1,2)-surfaces

TL;DR

This paper introduces simple fibrations in (1,2)-surfaces, classifies certain canonical threefolds on the Noether line, and explores their moduli space, revealing new threefolds and properties of known examples.

Contribution

It defines simple fibrations in (1,2)-surfaces, classifies Gorenstein cases on the Noether line, and analyzes their moduli space, including the discovery of new threefold components.

Findings

Almost all Gorenstein simple fibrations over P^1 are canonical threefolds on the Noether line.

The moduli space of these threefolds contains an open subset of Mori Dream Spaces.

A second component of the moduli space exists when the geometric genus is congruent to 6 mod 8.

Abstract

We introduce the notion of a simple fibration in $(1,2)$-surfaces. That is, a hypersurface inside a certain weighted projective space bundle over a curve such that the general fibre is a minimal surface of general type with $p_g=2$ and $K^2=1$. We prove that almost all Gorenstein simple fibrations over the projective line with at worst canonical singularities are canonical threefolds "on the Noether line" with $K^3=\frac43 p_g-\frac{10}3$, and we classify them. Among them, we find all the canonical threefolds on the Noether line that have previously appeared in the literature. The Gorenstein simple fibrations over $\mathbb{P}^1$ are Cartier divisors in a toric $4$-fold. This allows to us to show among other things, that the previously known canonical threefolds on the Noether line form an open subset of the moduli space of canonical threefolds, that the general element of this component is a Mori Dream Space, and that there is a second component when the geometric genus is congruent to $6$ modulo $8$; the threefolds in this component are new.