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I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the literature. Unfortunately, I could not find them. Here is the construction.

Let $\mathbb P^4$ over $\mathbb C$ with homogenous coordinates $[x_1:\cdots:x_5]$, let $Z=\{x_1=x_2=0\} \subset \mathbb P^4$. Set $$\pi: P={\rm Bl}_Z \mathbb P^4 \to \mathbb P^4$$ be the blowup of $Z$. The linear system $|-K_{P}|$ is base point free and let $X \in |-K_P|$ be a general element. By adjunction $K_X=0$.

Let $Y = \pi(X) \subset \mathbb P^4$. Then the natural contraction $\pi^-: X \to Y$ is a flop contraction ($X \cap {\rm Exc}(\pi)$ maps generally finite to $Z$, hence $\pi^-$ is a small contraction).

Now I want to get an explicit description of its flop $\pi^+: X^+ \to Y$. (For example, can $X^+$ be construct as some hypersurface? )

The above construction can be describe in equations:

As $Z \subset Y$ and $Y \in |-K_{\mathbb P^4}|$, we have $$ Y = \{f=\sum_{1 \leq a+b \leq 5} x_1^a x_2^b f_{ab}=0\} \subset \mathbb P^4.$$ Here $a, b \in \mathbb{Z}_{\geq 0}$ and $f_{ab}$ are general polynomials of degree $(5-a-b)$. For $z \in Z$ such that $X_z$ is a curve, we have $$z \in \{x_1=x_2=f_{10}=f_{01}\} \subset \mathbb P^4.$$ In fact, at such point, the relative tangent space $T_{Y/Z}$ has dimension $1$. Thus, as $\deg f_{10}=\deg f_{01}=4$, we have $16$ flopped curves. As a result, the flop $\pi^+: X^+ \to Z$ should also contract $16$ curves.

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You have a threefold hypersurface $Y\subset \mathbb{P}^4$ of the form $V(ax_1 - bx_2)$, for some quartic polynomials $a,b\in\mathbb{C}[\mathbb{P}^4]$.

You blow up the plane $Z=V(x_1,x_2)$ in the ambient space, to obtain $X\subset \mathbb{P}^1_{\lambda:\mu}\times\mathbb{P}^4$. As you say, this makes a small resolution of 16 nodes and it amounts to introducing blowup parameters $\tfrac{\lambda}{\mu}=\tfrac{x_1}{x_2}=\tfrac{b}{a}$.

To obtain the flop $X^+$ you can blow up the quartic surface $Z^+=V(x_1,b)$ in the ambient space instead. To describe $X^+$ explicitly, consider the standard affine patches $U_i=Y\cap\{x_i=1\}$. Then, over $U_i$, the blowup is given by $Bl_{Z^+}U_i\subset\mathbb{P}^1_{\lambda':\mu'}\times\mathbb{A}^4 \to U_i$ defined by the equations $$ ax_1 - bx_2 = \lambda'x_1 - \mu'b = \lambda'x_2 - \mu'a = 0 $$ (after setting $x_i=1$).

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  • $\begingroup$ Thank you very much!! But I don't familiar with the terminology you are using (blowup parameters...). Could you provide some related references? $\endgroup$
    – Li Yutong
    Jan 12, 2019 at 3:25
  • $\begingroup$ Sorry, I think making a weighted blowup was entirely unnecessary and probably caused some confusion, so I have rewritten my answer. $\endgroup$
    – Tom Ducat
    Jan 12, 2019 at 18:47
  • $\begingroup$ Thank you again! But why do you think the variety you constructed is the flop $X^+$? $\endgroup$
    – Li Yutong
    Jan 13, 2019 at 14:06
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    $\begingroup$ Your example is basically just the Atiyah flop in disguise. To construct the Atiyah flop we start with the singularity $W=V(xy-zt)\subset \mathbb{A}^4$. We can blowup the ideal $(x,z)$ in the ambient space to make one small resolution of singularities, and blowup the ideal $(x,t)$ to make the other. Note that whether we choose to blow up $(x,z)$ or $(y,t)$ we get isomorphic resolutions of $X$. Similarly with $(x,t)$ and $(y,z)$. In your example this is happening locally at every point of $V(x_1,x_2,a,b)$. (As long as this defines 16 distinct points. Something more complicated if not.) $\endgroup$
    – Tom Ducat
    Jan 13, 2019 at 15:29

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