Hilbert–Burch Virtual Resolutions for Points in P1×P1

Abstract

Based on the work of Harada, Nowroozi, and Van Tuyl, who provided particular length two virtual resolutions for finite sets of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$, we prove that the vast majority of virtual resolutions of a pair for minimal elements of the multigraded regularity in this setting are of Hilbert–Burch type. We give explicit descriptions of these short virtual resolutions that depend only on the number of points. Moreover, despite initial evidence, we show that these virtual resolutions are not always short and give sufficient conditions for their length to be three.