Degeneracy locus

Question

In the paper Non-Abelian Brill-Noether theory and Fano 3-folds, Proposition 1.2, part 3, the author uses a homomorphism of vector bundles $f:E\rightarrow F$ to produce a map $\ker f_x\otimes (\operatorname{coker} f_x)^\ast\rightarrow m_x/m_x^2$ of (I imagine) vector spaces over the ground field. I guess $m_x/m_x^2$ is fibre of the conormal sheaf. How is this map defined (he doesn't say that), and why is that part of the proposition true (he gives a reference, but that part of the proposition cannot be found in the reference)?

Answer 1

Choose bases in $E_x$ and $F_x$ such that the first vectors span $\mathrm{ker} f_x$ and $\mathrm{im} f_x$ respectively. Then the matrix of $f_x$ can be written in the block form as $$ \begin{pmatrix} A & B \\ C & D \end{pmatrix}, $$ where $A$, $B$, $C$, $D$ are matrices of functions on $X$, $B(x) = C(x) = D(x) = 0$, while $A(x)$ in invertible. Then the map you need is given by partial derivatives of the matrix $D$ at $x$. One can check that the result is independent of all choices.

EDIT. To get a coordinate free description of the map, let us consider a tangent vector at $x$, i.e., a morphism from $\mathrm{Spec}(k[\epsilon]/\epsilon^2)$ to the variety in question that takes the closed point to $x$, pullback $f$ to $\mathrm{Spec}(k[\epsilon]/\epsilon^2)$, and consider the composition $$ (\mathrm{ker} f\vert_{\mathrm{Spec}(k[\epsilon]/\epsilon^2)}) \to E\vert_{\mathrm{Spec}(k[\epsilon]/\epsilon^2)} \stackrel{f\vert_{\mathrm{Spec}(k[\epsilon]/\epsilon^2)}}\to F\vert_{\mathrm{Spec}(k[\epsilon]/\epsilon^2)} \to (\mathrm{coker} f\vert_{\mathrm{Spec}(k[\epsilon]/\epsilon^2)}). $$ It vanishes at the closed point, hence is divisible by $\epsilon$. After division you get a map $\mathrm{ker} f_x \to \mathrm{coker} f_x$. Since it is defined for each tangent vector, altogether you get a morphism $$ (\mathfrak{m}_x/\mathfrak{m}_x^2)^* \otimes \mathrm{ker} f_x \to \mathrm{coker} f_x, $$ as desired.