  In Butler, J.Differential Geom. 39 (1):1--34,1994, the author gives a
sufficient condition for a line bundle associated with a divisor D to be
normally generated on $X=P(E)$ where E is a vector bundle over a smooth curve
C. A line bundle which is ample and normally generated is automatically very
ample. Therefore the condition found in Butler's work, together with Miyaoka's
well known ampleness criterion, give a sufficient condition for the very
ampleness of D on X. This work is devoted to the study of numerical criteria
for very ampleness of divisors D which do not satisfy the above criterion, in
the case of C elliptic. Numerical conditions for the very ampleness of D are
proved,improving existing results. In some cases a complete numerical
characterization is found.
