Let $X, X^{\prime}\colon$ smooth projective curve on $\mathbb{C}$ (genus $\geq 3$),

$M(r,d)\colon$ coarse moduli of stable vector bundles with rank $r\geq2$, and degree $d$ , and

$M(r,\xi)\colon$ coarse moduli of stable vector bundles which have fixed determinant $\xi$ with rank $r$.

In other words, define
\begin{eqnarray}
\operatorname{det}\colon &M(r,d)&\rightarrow &\operatorname{Jac}^{d}(X)\\
&E&\mapsto&\wedge^{r} E
\end{eqnarray}
then,
\begin{equation}
M(r,\xi)=\operatorname{det}^{-1}(\xi)
\end{equation}

There is a famous fact,
$M(r,\xi)_{X}\cong M(r,\xi^{\prime})_{X^{\prime}}$ then $X\cong X^{\prime}$.

I want to know whether $M(r,d)_{X}\cong M(r,d)_{X^{\prime}}$ implies $X\cong X^{\prime}$.

I think it is possible to prove that by almost the same proof.

But if not, why I must consider fixed determinant ?

Thanks in advance.