Atiyah flop, flip and related toric computation

Question

I am trying to understand the definitions of flips and flops by studying examples in this article of Hacon and McKernan. I would like to ask why the toric varieties constructed in Ex. 1.13 are indeed flips and flops. It is possible that my question can be solved simply answering how to compute relative canonical divisor of a toric morphism.

As far as I know (please correct me if I am wrong), a flip $X_{-}\dashrightarrow X_{+}$ between termnal, $\mathbb Q$-factorial varieties is a birational map which is an isomorphism in codimension one and factors as $\varphi_{+}^{-1}\circ \varphi_{-}$, where:

• $\varphi_{-}\colon X_{-}\to X$ is a small extremal contraction,
• $\varphi_{+}\colon X_{+}\to X$ is a small contraction such that $K_{X_{+}}$ is relatively ample.

A flop is defined in the same way, but the above two conditions are replaced by:

• $\varphi_{-}$, $\varphi_{+}$ are small contractions such that $K_{X_{+}}$ and $K_{X_{-}}$ are relatively trivial.

The variety $X$ above is only required to be normal.

The well known Atiyah flop is constructed as follows: let $X$ be a cone over a quadric surface, $\tilde{X}\to X$ be a blowup of its vertex, and $X_{+}$ and $X_{-}$ are given by contracting one of the two rulings of the exceptional divisor inside $\tilde{X}$. My first question is: is it immediate to see that $K_{X_{-}}$ and $K_{X_{+}}$ are relatively trivial in this case?

In the cited paper, Atiyah flop is given a following toric description: let $v_{1},v_{2},v_3,v_4\in \mathbb R^{3}$ be four vectors such that any three of them span the standard lattice $\mathbb Z^{3}$ and $$\tag{$*$} v_{1}+v_{3}=v_{2}+v_{4}.$$ These vectors generate a full-dimensional cone. Now the fan of $X_{-}$ is constructed from this cone by adding a wall joining $v_{1}$ with $v_{3}$, and the fan of $X_{+}$ is constructed by adding a wall between $v_{2}$ with $v_{4}$ (adding both these walls is the same as adding a ray $v_{1}+v_{3}$, which gives $\tilde{X}$). Now again, my question is, how to use this description to compute the relative canonical divisors and show that $X_{-}\dashrightarrow X_{+}$ is a flop?

Next, the authors say that one can obtain an example of a flip by replacing $(*)$ with, say, $$\tag{$**$} 2v_{1}+v_{3}=v_{2}+v_{4},$$ and follow the same construction. Now $X_{-}$ is singular, $X_{+}$ is smooth. My question is, again, how to see that this is a flip?

Answer 1

The following criterion is useful to determine whether a toric-invariant extremal ray is positive/negative/trivial without computing the exact intersection number (cf. Proposition 4.3, Decomposition of toric morphisms by Miles Reid):

(1) Let $F$ be a fan, and $X=X_{F}$; then $K_X$ is a $\mathbb{Q}$-Cartier divisor if and only if the roof of $F$ is flat over every $\sigma \in F$; that is, for all $\sigma=\langle e_{1},\dots,e_{s}\rangle\in F^{(k)}$, $\left[e_{1},\dots,e_{s} \right]$ is a polyhedron contained in a $(k-1)$-dimensional affine subspace of $N_{\mathbb{R}}$.

(2) Suppose that $K_{X}$ is $\mathbb{Q}$-Cartier, and let $w \in F^{(n-1)}$ be an interior wall. Then $K_{X}\cdot l_{w} < 0$ (resp. $=0$, $>0)$ if and only if $\mathrm{shed}\ F$ is strictly convex (resp. flat, strictly concave) in a small neighbourhood of a point $P \in (\mathrm{int}\ w) \cap \mathrm{roof}\ F$.

Roughly speaking, you can look at the polytope created by the origin and the primitive generators of the cones. In Atiyah's flop, the $4$ generators lie on the same plane, so the walls are flat and hence rays are trivial. In the second example, the wall of $v_1$ and $v_{3}$ is concave and the wall of $v_{2}$ and $v_{4}$ is convex. Hence if you remove the wall of $v_{2}$ and $v_{4}$ and replace it by the wall of $v_1$ and $v_{3}$, you get a flip.

If you do a little more work, you can compute the exact intersection number by the Cartier data of $K_{X}$, see section 6.3 and 6.4 of the book Toric Varieties by David Cox, John Little and Hal Schenck.