Let $L$ be a line bundle of degree $d$ on a curve $X$ and let $x$ be a point of $X$. I want to describe the intermediate extension of the constant sheaf from the stack of line bundles of degree $d$ to the point given by the coherent sheaf $L(-x) \oplus O_x$ of rank $1$ and degree $d$ with determinant $L$. In order to do this, can I simply calculate the intermediate extension of the constant sheaf from $Ext^1(O_x, L(-x)) \setminus 0$ to the vector space $Ext^1(O_x, L(-x))$? I.e. does $Ext^1(O_x, L(-x))$ form an atlas for the neighborhood of the point $L(-x) \oplus O_x$ in the stack of coherent sheaves of rank $1$ and degree $d$ with determinant $L$?

Let $L$ be a line bundle of degree $d$ on a curve $X$ and let $x$ be a point of $X$. I want to describe the intermediate extension of the constant sheaf from the stack of line bundles of degree $d$ to the point given by the coherent sheaf $L(-x) \oplus O_x$ of rank $1$ and degree $d$ with determinant $L$. In order to do this, can I simply calculate the intermediate extension of the constant sheaf from $Ext^1(O_x, L(-x)) \setminus \{0\}$ to the vector space $Ext^1(O_x, L(-x))$? I.e. does $Ext^1(O_x, L(-x))$ form an atlas for the neighborhood of the point $L(-x) \oplus O_x$ in the stack of coherent sheaves of rank $1$ and degree $d$ with determinant $L$?