The Sp3-grassmannian and duality for prime Fano threefolds of genus 9

By a result of Mukai, the non-abelian Brill-Noether locus X=MC(2, K : 3F) of type II, defined by a stable rank 2 vector bundle F of invariant 3 over a plane quartic curve C, is a prime Fano 3-fold X=X16 of degree 16. The associate ruled surface SX=P(F) is uniquely defined by X, and we see that for the general X=X16, SX is isomorphic to the Fano surface of conics on X. The argument uses the geometry of the Sp3-grassmannian and the double projection from a line on X16.