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.



Explicit descriptions of a flop
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 $x\_i=1$). 


