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arXiv · 2110.10705

Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces

Juliette Bruce·Lauren Cranton Heller·Mahrud Sayrafi
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Abstract

We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces $X$. After proving a uniqueness theorem for certain virtual resolutions, we show that the multigraded regularity region of a module $M$ is determined by the minimal graded free resolutions of the truncations $M_{\geq\mathbf d}$ for $\mathbf d\in\operatorname{Pic} X$. Further, by relating the minimal graded free resolutions of $M$ and $M_{\geq\mathbf d}$ we provide a new bound on multigraded regularity of $M$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo--Mumford regularity for a wide class of complete intersections in products of projective spaces.

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Juliette Bruce, Lauren Cranton Heller, Mahrud Sayrafi. 2021-10-20. Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces. https://arxiv.org/abs/2110.10705

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