  We describe the spaces of minimal rank last syzygies for the Mukai Varieties
of sectional genus 6,7 and 8. Based on this we show:
  1. The first geometric syzygies of a general canonical curve of genus 6 form
a non degenerate configuration of 5 lines in P^4.
  2. The first geometric syzygies of a general canonical curve of genus 7 form
a non degenerate, linearly normal, ruled surface of degree 84 on a spinor
variety S in P^15.
  3. The second geometric syzygies of a general canonical curve of genus 8 form
a non degenerate configuration of 14 conics on a 2-uple embedded P^5 in P^20.
  This proves a natural generalization of Green's conjecture [1984], namely
that the geometric syzygies should span the space of all syzygies, in these
cases. We have generalized results 1 and 3 to general curves of even genus in
math.AG/0108078. Result 2 is the main new result of this paper.
