It has recently been proven that the volume of a threefold of general type is bounded from below from 4/3p_g-10/3 when p_g is big enough. This is known as the Noether inequality, the 3-dimensional analogue of the famous Noether inequality for the surfaces of general type.

Classical works of Horikawa and others show that surfaces on the Noether line are exactly fibrations over the projective line such that every fibre is algebraically like a smooth curve of genus 2. Inspired by this we introduced the concept of "simple" fibrations in (1,2)-surfaces and conjectured that all the threefolds "near" the Noether line are simple fibrations in (1,2)-surfaces.  I will discuss our first results on this conjecture and give a detailed description of the corresponding moduli spaces of threefolds which have striking similarities with the moduli spaces of surfaces on the Noether line.

This is joint work in progress with Yong Hu, Roberto Pignatelli and Tong Zhang.