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Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces

Juliette Bruce, Lauren Cranton Heller, Mahrud Sayrafi

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 minimal 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.

Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces

Abstract

We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces . After proving a uniqueness theorem for certain minimal virtual resolutions, we show that the multigraded regularity region of a module is determined by the minimal graded free resolutions of the truncations for . Further, by relating the minimal graded free resolutions of and we provide a new bound on multigraded regularity of 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.

Paper Structure

This paper contains 21 sections, 31 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Truncations and Multigraded Regularity
  3. Betti Numbers and Multigraded Regularity
  4. Notation and Background
  5. Multigraded Regularity
  6. Truncations and Local Cohomology
  7. Koszul Complexes and Cotangent Sheaves
  8. Virtual Resolutions
  9. Subcomplexes of Minimal Free Resolutions
  10. The Fourier--Mukai Transform
  11. The Beilinson Spectral Sequence
  12. Beilinson's Collection and its Endomorphism Algebra
  13. Construction and Uniqueness of Virtual Resolutions
  14. A Criterion for Multigraded Regularity
  15. Regularity Implies Quasilinearity
  16. ...and 6 more sections

Key Result

Theorem A

Let $M$ be a finitely generated $\mathds{Z}^r$-graded $S$-module with $H_B^0(M) = 0$. Then $M$ is $\mathbf d$-regular if and only if $M_{\geq\mathbf d}$ has a quasilinear resolution $F_\bullet$ with $F_0$ generated in degree $\mathbf d$.

Figures (2)

  • Figure 1: The top row shows the regions $L_i(1,2)$ in green, and the bottom row $Q_i(1,2)$ in pink for $i=0,1,2,3$, from left to right, as defined in Section \ref{['sec:regularity']}.
  • Figure 2: The four regions for Example \ref{['ex:hyperelliptic-curve']} inside $\operatorname{Pic}\mathopen{}\left({\mathds{P}^{1}\!}\times{\mathds{P}^{2}\!}\right)\mathclose{}$.

Theorems & Definitions (71)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Example 2.1
  • Definition 2.3
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.8
  • proof
  • ...and 61 more