The curve of lines on a prime Fano threefold of genus 8

We show that a general prime Fano threefold X of genus 8 can be reconstructed from the pair $(\Gamma,L)$, where $\Gamma$ is its Fano curve of lines and $L=O_{\Gamma}(1)$ is the theta-characteristic which gives a natural embedding $\Gamma \subset \matbb{P}^5$.