Virtual resolutions of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$

Abstract

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of sets of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Specifically, we describe a virtual resolution for a sufficiently general set of points $X$ in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ that only depends on $|X|$. We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$; more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

19 pages