Threefold flops via matrix factorization

The structure of birational maps between algebraic varieties becomes increasingly complicated as the dimension of the varieties increases. There is no birational geometry to speak of in dimension one: if two complete algebraic curves are birationally isomorphic then they are biregularly isomorphic. In dimension two we encounter the phenomenon of the blowup of a point, and every birational isomorphism can be factored into a sequence of blowups and blowdowns. In dimension three, however, we first encounter birational maps which are biregular outside of a subvariety of codimension two (called the center of the birational map). When the center has a neighborhood with trivial canonical bundle, the birational map is called a flop. The focus of this paper will be the case of a three-dimensional simple flop, in which the center is an irreducible curve (necessarily a smooth rational curve). One of the motivations for studying this case is a theorem of Kawamata [17], which says that all birational maps between Calabi–Yau threefolds can be expressed as the composition of simple flops (in fact, of simple flops between nonsingular varieties). Important examples of simple flops were provided by Laufer [22], and three-dimensional simple flops were studied in general by Reid [25] and by Pinkham [24]. One fundamental property is that the center of the flop can be contracted, leaving a (singular) variety X which is dominated by both of the varieties involved in the original flop. X has a hypersurface singularity, and thus can be locally described as {f = 0} for some polynomial f in which