### 1. Special divisors and a conjecture of M. Green

Let C be a projective algebraic curve of genus g ≥ 2 over a field k and L be any
line bundle on C that is generated at stalks by its space of global sections. We
then consider

the homogeneous coordinate ring of C with respect to L. Then, we may write a
minimal free resolution of R as a graded module over S, the Symmetric algebra
on Γ(C,L),

where

H. H. Martens [15] has defined the Clifford Index γ_{C} of C as the
minimum of deg(D) - 2dim∣ D where D runs over all divisors on C such that
h^{0}(C,C(D)) > 1 and h^{1}(C,C(D)) > 1. We are concerned with the
following

Conjecture 1.1 (M. Green). With notation as above, M_{p,p+q} = 0, for all
p,q such that q > 1 and

Let E_{L} be the vector bundle on C defined by the following natural exact
sequence,

and V = Γ(C,L)^{*}. Then it is easily seen that,

Thus, the conjecture of Green is equivalent to the surjectivity of this morphism
for p,q in the given range. In Chapter 1 of the thesis we show

Theorem 1.2. Let C be an projective algebraic curve of genus g ≥ 2 over
a field k, and L a line bundle on it that is generated at stalks by its space
V = Γ(C,L)^{*} of global sections. Further, let E_{L} be the vector bundle on C
defined by the natural exact sequence

Then, the image of

contains all locally decomposable sections, whenever q > 1 and

As a result of this work the conjecture of M. Green now follows from

Conjecture 1.3. Let E_{L} be the ample vector bundle on a curve C as above
Then the locally decomposable sections of E_{L} span the vector space of all
sections, for all i.

In a later paper (see 5) we have proved this conjecture in some cases. Even for
a general ample bundle E which is generated by its space of global sections we
know of no situation contrary to this conjecture.