Let C be a general element in the locus of curves in M_g lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukai's ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on C.
ARBARELLO, E., BRUNO, A., & SERNESI, E. (2014). Mukai's program for curves on a K3 surface. ALGEBRAIC GEOMETRY, 1, 532-557 [10.14231/AG-2014-023].