## Mathematics, Department of

#### Date of this Version

9-2008

#### Abstract

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).

## Comments

Article in submission to

Journal of Algebraic Geometry.Originally published in arXiv math.AG/0611014, November, 2006; revised September 2, 2008. Copyright 2006 Carina Curto and David R. Morrison.