### 5. Further work on syzygies

In collaboration with K. Hulek and S. Ramanan we have explored a conjecture
made along with S. Ramanan on the syzygies of canonical curves. Specifically, we
have conjectured that (with notation as in 1)

Conjecture 5.1. The vector bundle E_{K} is such that the sections of E_{K}
are spanned by the decomposable sections.

This conjecture implies the conjecture of M. Green mentioned in 1.

In this work we have proved this conjecture for trigonal curves and also for
plane quintics. This covers the case of Clifford Index 1. The case of hyperelliptic
curves (Clifford Index 0) is elementary.