The talk is about QQ-Fano 3-folds, esp. codimension 4, where there is no tractable structure theory. I discuss the algebraic idea of Gorenstein unprojection, and some
particular cases called Tom and Jerry. The database lists 145 numerical candidates for codimension 1 QQ-Fanos, of
which around 130 have a Type 1 centre. Every one of these candidates gives rise to at least 2 families of QQ-Fano
3-folds that have the same numerical specification (that is, Hilbert series), but are topologically different (have different Betti numbers).
See http://www.grdb.co.uk/ + Fano 3-folds and Gavin Brown, Michael Kerber and Miles Reid, Fano 3-folds in codimension 4, Tom and Jerry, Part I, Compositio Math 148 (2012), 1171-1194, preprint arXiv:1009.4313, 33 pp.