# Automorphism of moduli space of stable vector bundles over a curve

Let $$C$$ be a smooth genus two hyperelliptic curve and $$\mathcal{M}_C$$ be a moduli space of stable rank two vector bundles of fixed degree(or fixed determinant). Then is $$\mathrm{Aut}(\mathcal{M}_C)\subset\mathrm{Aut}(C)?$$. The motivation is following. Let $$Y$$ be an index two prime Fano threefold of degree $$4$$, one can show that such threefold is isomorphic to a moduli space of stable vector bundle of rank two fixed determinant over a genus two curve $$C$$ such that its degree is odd(say degree=1). Then, I am wondering if $$\mathrm{Aut}(Y)\subset\mathrm{Aut}(C)?$$. In the paper Hilbert scheme of lines and conics and automorphism groups of Fano threefolds, the authors show that $$\mathrm{Aut}(Y)\subset\mathrm{Aut}(J(C))$$. But I think $$\mathrm{Aut}(J(C))$$ is not contained in $$\mathrm{Aut}(C)$$ since $$J(C)$$ has some translation which is not coming from $$\mathrm{Aut}(C)$$. On the other hand, for index one degree 12 prime Fano threefold $$X$$. The authors show that $$\mathrm{Aut}(X)\subset\mathrm{Aut}(C)$$, in this case $$X$$ is also purely determined by $$C$$.

The moduli space of rank 2 vector bundles on $$C$$ with fixed determinant of odd degree is a smooth complete intersection of two quadrics in $$\ \mathbb{P}^5\qquad$$ (P. Newstead, Topology 7 (1968), 205-215); in an appropriate system of coordinates, it is given by $$\sum X_i^2=\sum \lambda _iX_i^2=0$$, for $$\lambda_0,\ldots ,\lambda_5\in\mathbb{C}$$ distinct. It always admits a group $$(\mathbb{Z}/2)^5$$ of automorphisms (acting by changing the sign of the coordinates). On the other hand, $$\operatorname{Aut}(C)=\mathbb{Z}/2$$ for $$C$$ general.