First, let me describe the premise. Let $X$ be a smooth projective curve of genus greater than 2 over $\mathbb{C}$, let $x,y\in X$ two closed points, and let $M_{y-x}$ denote the moduli space of S-equivalence classes of rank 2 semistable vector bundles on $X$ of fixed determinant $\mathcal{O}(y-x).$ $M_{y-x}$ is constructed as the GIT quotient $\mathcal{R}//SL(p),$ where $p$ is a large even number, and $\mathcal{R}$ is a smooth open subvariety of the Grassmannian of 2-dimensional quotients of $\mathbb{C}^p$. There exists a surjective map $\mathcal{R}^{ss}\rightarrow M_{y-x}$, where $\mathcal{R}^{ss}$ denotes the semistable points under the $SL(p)$-action. Let $l$ be a strictly semistable (i.e. semistable but not stable) bundle. Using Jordan-Holder filtration, it is easy to see that the S-equivalence class of $l$ is of the form $[l] = [L\oplus L^{-1}(y-x)]$. We know that There is a unique closed orbit lying over $[l]$ in $\mathcal{R}^{ss}$.

In a paper that I'm reading, the following has been claimed:

for each point $h$ on this closed orbit, the normal space of this orbit at $h$ is isomorphic to $\mathcal{N}=H^1(X,End_0(L\oplus L^{-1}(y-x)).$ Moreover, the stabilizer of the point $h$ is isomorphic to $\mathbb{C}^{*}$.

My question is: Does the above follow from general theory of GIT? Can someone give some idea on why it should be true? any reference would be highly appreciated.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.