  We show that there exist mathematical 4-instanton bundles F on the projective
3-space such that F(2) is globally generated (by four global sections). This is
equivalent to the existence of elliptic space curves of degree 8 defined by
quartic equations. There is a (possibly incomplete) intersection theoretic
argument for the existence of such curves in D'Almeida [Bull. Soc. Math. France
128 (2000), 577-584] and another argument, using results of Mori [Nagoya Math.
J. 96 (1984), 127-132], in Chiodera and Ellia [Rend. Istit. Univ. Trieste 44
(2012), 413-422]. Our argument is quite different. We prove directly the former
fact, using the method of Hartshorne and Hirschowitz [Ann. Scient. Ec. Norm.
Sup. (4) 15 (1982), 365-390] and the geometry of five lines in the projective
3-space.
