Can we please re-define the term FLOP

[Porcupine_I]

Kantor said:
According to Wikipedia, this seems to be the definition of a flop:

If f:X→Y is a morphism, and K is the canonical bundle of X, then the relative canonical ring of f is

oplus_m f_*(O_X(mK))

and is a sheaf of graded algebras over the sheaf OY of regular functions on Y. The blowup f+

f^+:X^+= Proj(oplus_m f_*(O_X(mK)))mapsto Y

of Y along the relative canonical ring is a morphism to Y. If the relative canonical ring is finitely generated (as an algebra over OY) then the morphism f+ is called the flip of f if −K is relatively ample, and the flop of f if K is relatively trivial. (Sometimes the induced birational morphism from X to X+ is called a flip or flop.)

In applications, f is often a small contraction of an extremal ray, which implies several extra properties:

* The exceptional sets of both maps f and f+ have codimension at least 2,
* X and X+ only have mild singularities, such as terminal singularities.
* f and f+ are birational morphisms onto Y, which is normal and projective.
* All curves in the fibers of f and f+ are numerically proportional.

yeah! what he said!