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.