Birational Geometry of 3-fold Mori Fibre Spaces

In this paper we study standard 3-fold conic bundles X/P and stable cubic del Pezzo fibrations X/P. These are the key examples of 3-fold Mori fibre spaces (Mfs). We begin systematically to chart the geography of these varieties, namely the classification of their deformation families and discrete invariants. Given a 3-fold Mfs X/S, we aim to understand the set of all Mfs Y/T with X birational to Y . For example, we say that X/S is birationally rigid if this is a 1-element set. We state conjectures on the rigidity of Mfs, with the goal of approaching optimal criteria. Our main results are summarised in Tables 1 and 2 and Figure 1. We want this to become your standard guidebook to Mori fibre spaces and their birational geometry. [email protected] [email protected] [email protected]