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