Virtual Complete Intersections in ${\mathbb{P}}^1\times {\mathbb{P}}^1$

The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces ${\mathbb{P}}^1\times {\mathbb{P}}^1$, we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.