Simple fibrations in (1,2)-surfaces

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.