On the moduli space of rank $3$ vector bundles on a genus $2$ curve and the Coble cubic

We prove a conjecture about the moduli space $\mathcal {SU}_C(3)$ of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve $C$, due to I. Dolgachev. Given $C$ a smooth projective curve of genus 2, and the embedding of the Jacobian $JC$ into $|3\Theta |$, A. Coble proved, at the beginning of the 20th century, that there exists a unique cubic hypersurface $\mathcal {C}$ in $|3\Theta |^* \simeq \mathbb {P}^8$, $JC[3]$-invariant and singular along $JC$. On the other hand, we have a map of degree 2 from $\mathcal {SU}_C(3)$ over $|3\Theta | \simeq \mathbb {P}^{8*}$, ramified along a sextic hypersurface $\mathcal {B}$. Dolgachev’s conjecture affirms that the sextic $\mathcal {B}$ is the dual variety of Coble’s cubic $\mathcal {C}$.