TITLE: How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?
QUESTION [5 upvotes]: 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 $c_1(L)$.

REPLY [3 votes]: On $M$ there is an exact sequence
$$
0 \to \mu^*\Omega_N \to \Omega_M \to i_*L^\vee \to 0,
$$
where $i:D \to M$ is the embedding. This allows to understand the class of $D$ since $D = c_1(i_*L) = c_1(\Omega_M) - \mu^*c_1(N)$. In your case it is equal to
$$
(-2,-2) - (-3d,-3k) = (3d-2,3k-2).
$$
Thus $D$ is a curve of bidegree $(3d-2,3k-2)$. In particular, its genus is equal to $g = 9(d-1)(k-1)$. To understand the degree of $L$ you can use Riemann--Roch:
$$
\deg(L^\vee) + 1 - g = \chi(i_*L^\vee) = \chi(\Omega_M) - \chi(\mu^*\Omega_N). 
$$
In your case $\chi(\Omega_M) = -2$, while the pullback of the Euler sequence
$$
0 \to \mu^*\Omega_N \to O(-d,-k)^{\oplus 3} \to O \to 0
$$
allows to compute $\chi(\Omega_N) = 3(d-1)(k-1) - 1$. In the end
$$
\deg(L^\vee) = -2 + 1 - 3(d-1)(k-1) + 9(d-1)(k-1) - 1 = 6(d-1)(k-1)-2.
$$