Virtual resolutions of points in P1×P1

A short proof of the Hanlon-Hicks-Lazarev Theorem

“…Hilbert’s Syzygy Theorem gives a bound of ; Theorem 1.4 implies that the added flexibility of virtual resolutions allows for significantly shorter resolutions, especially when is large. See [BES20, HNVT22, BKLY21] and elsewhere for many examples of this phenomenon. Prior to [HHL], Theorem 1.4 had been proven in several special cases: when , it essentially follows from Hilbert’s Syzygy Theorem; for products of projective spaces, it was shown in [BES20, Theorem 1.2] (see also [EES15, Corollary 1.14]); Yang proved it for any monomial ideal in the Cox ring of a smooth toric variety [Yan21]; and Brown-Sayrafi proved it for smooth projective toric varieties of Picard rank 2 [BS22].…”