Virtual resolutions of points in \(\mathbb{P}^1 \times \mathbb{P}^1\). (English) Zbl 1523.13022
Summary: We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of finite 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 \(\mathbb{P}^1 \times \mathbb{P}^1\).
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
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