A property of five lines in ℙ3 and four generated 4-instantons

Abstract

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 existence of 4-instantons with the above property using the method of Hartshorne and Hirschowitz [Ann. Scient. Éc. Norm. Sup. (4) 15 (1982), 365–390] and a geometric result, of classical flavor, stating that the union of five general lines in the projective 3-space is the zero scheme of a global section of T(2), where T is the tangent bundle of the space. This approach sheds, also, some light on the set of jumping lines of a 4-instanton.