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Let $L\in Pic^{2d}C$ be a line bundle on a curve $C$ such that $d\geq 2\,g(C)$, $Mat_{N}(H^{0}(L))$ the set of $N\times N$ matrices with entries in $H^{0}(L)$, $Mat_{N,1}(H^{0}(L))$ the subset of matrices of rank one over $k(C)$. The groupe $GL(N,N)=GL(N)\times GL(N)/G_{m}$ is acting on $Mat_{N}(H^{0}(L))$ by left and right multiplication, and we denote by $Mat^{s}_{N,1}(H^{0}(L))$ the subset of stable points.

In ordre to prove that the projective variety $Mat^{s}_{N,1}(H^{0}(L))/GL(N,N)=Proj\,k[Mat_{N,1}(H^{0}(L))]^{SL(N,N)}$ is nonsingular, Mukai (An introduction to invariants and moduli, 323-327) defined the following map $$ \begin{array}{llll} \pi_{\psi}:&T_{\psi}\,Mat^{s}_{N,1}(H^{0}(L))&\longrightarrow & H^{1}(O_{C})\\ &A&\longmapsto &(h_{p})_{p\in C}, \end{array} $$ by some complicated methode. Indeed, let $\xi\in Pic^{d}\,C$ and $\widehat{\xi}=L\otimes \xi^{-1}\in Pic^{d}(C)$ be two line bundles. By Riemann Roch one can show that dim $H^{0}(\xi)=$ dim $H^{0}(\widehat{\xi})$, so take $S=\{s_{1},\dots,s_{N}\}$ and $T=\{t_{1},\dots,t_{N}\}$ to be the bases of $H^{0}(\xi)$ and $H^{0}(\widehat{\xi})$, respectively. So, the matrix $$ \psi(\xi, S,T)=\left( \begin{array}{ccc} s_{1}t_{1} & \dots & s_{1}t_{N}\\ \vdots & \cdots& \vdots\\ s_{N}t_{1} & \cdots & s_{N}t_{N}\\ \end{array} \right). $$ is in $Mat_{N,1}(H^{0}(L))$. Let $T_{\psi}\,Mat^{s}_{N,1}(H^{0}(L))$ be the tangent space to $Mat^{s}_{N,1}(H^{0}(L))$ at the point $\psi(\xi, S,T)$. Mukai shows that there is a correspondance between tangent vectors to the affine space $Mat_{N}(H^{0}(L))$ at $\psi(\xi,S,T)$ and $N\times N$ matrices $A$ with entries in $H^{0}(L)$. Then, by identification, he takes $A$ in $T_{\psi}\,Mat^{s}_{N,1}(H^{0}(L))$, he proves that on can write $A$ as $$ A=\begin{pmatrix}s_{1}^{'}\\\vdots\\s_{N}^{'}\end{pmatrix}(t_{1},\dots,t_N)+\begin{pmatrix}s_{1}\\\vdots\\s_{N}\end{pmatrix}(t^{'}_{1},\dots,t^{'}_N), $$ where $s_{i}^{'}$ and $t_{i}^{'}$ are in $k(C)$. Finally, he shows that $\forall p\in C$, there is $h_p\in k(C)$ such taht $$ \begin{pmatrix}s_{1}^{'}\\\vdots\\s_{N}^{'}\end{pmatrix}\equiv h_{p}\begin{pmatrix}s_{1}\\\vdots\\s_{N}\end{pmatrix}\,\text{mod }\xi_{p}, $$ and $(t_{1}^{'},\dots,t_{N}^{'})\equiv -h_{p}(t_{1},\dots,t_{N})$ mod $\widehat{\xi}_{p}$. He defines $(h_{p})_{p\in C}\in H^{1}(O_{C})$ to be the image $\pi_{\psi}(A)=(h_{p})_{p\in C}$ of $A$ by this map.

My question for experts, I feel that there are no complete details about this map on Mukai's book. So, is this map construction original? or it come from an other reference? I would be grateful if someone share such reference with us.

Thank you.

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    $\begingroup$ It is better to explain your notation and to give a precise reference to the statement in the book of Mukai. $\endgroup$
    – Sasha
    Nov 28, 2016 at 20:51
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    $\begingroup$ ok, I will do that. $\endgroup$
    – M.Souf
    Nov 28, 2016 at 21:01

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