Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps of degree $d_1$ and $d_2$ respectively. Suppose they agree at some point (say $[1,0]$), i.e. $$ v_1([1,0]) = v_2([1,0]).$$ Then we can think of this whole collection as a stable map $v$ from a nodal riemann surface (the two spheres identified at $[1,0]$). Suppose $v_1$ and $v_2$ are explicitly given as $$v_1([X,Y]):= [p_0(X,Y), p_1(X,Y), p_2(X,Y)],$$ $$ v_2([X,Y]):= [q_0(X,Y), q_1(X,Y), q_2(X,Y)],$$ where $p_i$ and $q_i$ are homogeneous polynomials of degrees $d_1$ and $d_2$ respectively.
$\textbf{Question:}$ Can one explicitly construct a sequence of holomorphic maps $u_t:\mathbb{CP}^1\rightarrow \mathbb{CP}^2$ of degree $d_1+d_2$ that converges to the stable map $v$? I am looking for an explicit formula for the family $u_t$ in terms of the polynomials $p_i$ and $q_i$ and $t$.
The convergence is in the sense of Gromov Gonvergence for $J$-holomorphic maps.
$\textbf{Remark:}$ I am aware that abstractly the $u_t$ exists; it follows from general gluing theorem for $J$-holomorphic maps. I am just wondering in the special case of degree $d$ rational maps in $\mathbb{CP}^2$ if one can explicitly write down that family of maps $u_t$ in terms of polynomials.
