The virtual resolutions package for Macaulay2

Almousa, Ayah; Bruce, Juliette; Loper, Michael C.; Sayrafi, Mahrud

doi:10.2140/jsag.2020.10.51

Cited by 9 publications

(10 citation statements)

References 4 publications

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“…Proof. For statement (1), [11,Lemma 3.5] proves the case deg(L) = (1, 0) and the proof can be easily adapted to our case. Statement ( 2) is [11,Lemma 4.2].…”

Section: Introductionmentioning

confidence: 90%

“…In this paper we focus on virtual resolutions of points in P 1 × P 1 . By focusing on this family, we can take advantage of many known algebraic and geometric properties about them, e.g., as developed in [5,7,8,10,11].…”

Section: Introductionmentioning

confidence: 99%

“…The authors thank Ayah Almousa, Daniel Erman, and Michael Loper for useful discussions. The authors used Macaulay2 [9], and in particular, the package VirtualResolutions [1] in their computer experiments. An early version of the results in this paper appeared in the second author's MSc thesis [16].…”

Section: Introductionmentioning

confidence: 99%

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

Harada¹,

Nowroozi²,

Tuyl³

2021

Preprint

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We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of sets of points in P 1 × P 1 . Specifically, we describe a virtual resolution for a sufficiently general set of points X in P 1 × 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 P 1 × 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 P 1 × P 1 .

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“…Proof. For statement (1), [11,Lemma 3.5] proves the case deg(L) = (1, 0) and the proof can be easily adapted to our case. Statement ( 2) is [11,Lemma 4.2].…”

Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Harada¹,

Nowroozi²,

Tuyl³

2021

Preprint

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“…Michael Kemeny helped me understand some of the arguments in [GLP83], and Rob Lazarsfeld alerted me to a different treatment of the n = 2 case by his student in [Loz08]. The computer algebra system Macaulay2 [M2] was used extensively, in particular the VirtualResolutions package [ABLS20]. 0.…”

Section: Introductionmentioning

confidence: 99%

Syzygies of Curves in Products of Projective Spaces

Cobb¹

2023

Preprint

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Motivated by toric geometry, we lift machinery for understanding syzygies of curves in projective space to the setting of products of projective spaces. Using this machinery, we show an analogue of an influential result of Gruson, Peskine, and Lazarsfeld that gives a bound on the regularity of a possibly singular curve given its degree and the dimension of the ambient projective space. To do so, we show new results linking the shape of multigraded resolutions of a sheaf to its regularity region.

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“…In this paper we focus on virtual resolutions of finite sets of points in P 1 × P 1 . By focusing on this family, we can take advantage of many known algebraic and geometric properties about them, e.g., as developed in [6,8,9,11,12].…”

Section: Introductionmentioning

confidence: 99%