 The major technical problem is that, at some stage, the variety formula_5 may become 'too singular', in the sense that the canonical divisor formula_3 is no longer a Cartier divisor, so the intersection number formula_7 with a curve formula_8 is not even defined.The (conjectural) solution to this problem is the "flip".Given a problematic formula_5 as above, the flip of formula_5 is a birational map (in fact an isomorphism in codimension 1) formula_11 to a variety whose singularities are 'better' than those of formula_5.