Four generated 4-instantons

TL;DR

This paper proves the existence of certain 4-instanton bundles on projective 3-space with globally generated twists, linking algebraic geometry and elliptic space curves of degree 8 defined by quartic equations.

Contribution

It provides a new direct proof of the existence of 4-instanton bundles with specific properties, using geometric methods involving lines in projective space.

Findings

Existence of 4-instanton bundles with globally generated twist F(2)

Correspondence with elliptic space curves of degree 8

New geometric proof using lines in projective space

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 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.