Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and $\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of points of $M$, where $d\mu|_p$ fails to be injective. Assume that $D$ is a smooth complex submanifold of $M$ of the expected dimension $m-1$ (more precisely, $d\mu|_p$ is not injective means a certain determinant is zero, assume that determinant vanishes transversally). Furthermore assume that on all points of $D$, the Kernel of $d\mu|_p$ is $\textit{exactly}$ one dimensional.

$\textbf{Question:}$ Define the line bundle over $D$, given by $L:= Ker(d\mu) \rightarrow D$. How does one compute $c_1(L)$?

The specific example where I need to compute $c_1(L)$ is as follows: $M:= \mathbb{P}^1 \times \mathbb{P}^1$, $N:= \mathbb{P}^2$ and $\mu:M \rightarrow N$ is a map of type $(d,k)$, i.e. $\mu^*\mathcal{O}(1) = \mathcal{O}(d,k)$.

$\textbf{Added Later:}$ My main interest is in the specific example I asked. Its being pointed out that in general there may not be any explicit/reasonable formula for $c_1(L)$.