For points $p_1=[1,0,0,0], p_2=[0,1,0,0], p_3=[0,0,1,0]$ and $p_4=[0,0,0,1]$
in the projective space $\mathbb P^3$, Let $l_{ij}$ be the line through $p_i, p_j$. Let

$$C=l_{12} \cup l_{13} \cup l_{24} \cup l_{34}.$$

Can one construct an explicit family of curve $\pi : \mathcal C \rightarrow B$ such that $\pi^{-1}(0)=C$ and $\pi^{-1}(t)$ is a smooth elliptic curve in $\mathbb P^3$ for $0<|t|<\epsilon$, where $\epsilon$ is a sufficiently small positive number?

More concretely, I would like to know the defining equations of $\pi^{-1}(t)$ in $\mathbb P^3$ for each $t$.

For points $p_1=[1,0,0,0], p_2=[0,1,0,0], p_3=[0,0,1,0]$, $p_4=[0,0,0,1]$ and $p_5=[1,1,1,1]$
in the projective space $\mathbb P^3$, Let $l_{ij}$ be the line through $p_i, p_j$. Let

$$C=l_{12} \cup l_{23} \cup l_{34} \cup l_{45} \cup l_{51}.$$

Can one construct an explicit family of curve $\pi : \mathcal C \rightarrow B$ such that $\pi^{-1}(0)=C$ and $\pi^{-1}(t)$ is a smooth elliptic curve in $\mathbb P^3$ for $0<|t|<\epsilon$, where $\epsilon$ is a sufficiently small positive number?

More concretely, I would like to know the defining equations of $\pi^{-1}(t)$ in $\mathbb P^3$ for each $t$.