Year: 2020
Type: article
Abstract: We introduce the VirtualResolutions package for the computer algebra system Macaulay2.This package has tools to construct, display, and study virtual resolutions for products of projective spaces.The package also has tools for generating curves in ސ 1 × ސ 2 , providing sources of interesting virtual resolutions.1. INTRODUCTION.Recently, Berkesch, Erman, and Smith [Berkesch et al. 2017] introduced the notion of virtual resolutions for subvarieties of smooth projective toric varieties as an analogue to minimal graded free resolutions for subvarieties of projective space.A virtual resolution is a complex of finitely generated graded free modules over the Cox ring of a smooth projective toric variety that becomes exact upon passing to the corresponding complex of coherent sheaves.While graded minimal free resolutions are useful for studying quasicoherent sheaves on projective spaces, when working over a product of projective spaces or, more generally, over smooth projective toric varieties, they are often long and cumbersome to compute.By allowing a limited amount of homology, virtual resolutions offer a more flexible alternative for studying toric subvarieties when compared to graded minimal free resolutions.This article introduces the VirtualResolutions package for [Macaulay2], which provides tools for constructing and studying virtual resolutions over products of projective spaces.The VirtualResolutions project began in 2018 at the Macaulay2 Workshop at the University of Wisconsin -Madison, building on previous work of Christine Berkesch, David Eisenbud, Daniel Erman, and Gregory G. Smith.Along with them, the authors of this paper developed the VirtualResolutions package to generate examples of virtual resolutions in products of projective spaces.In particular, the package provides tools to generate examples of virtual resolutions and to check whether chain complexes are virtual resolutions.These methods are introduced and demonstrated in Section 2.In Section 3, we concentrate on virtual resolutions arising from curves in ސ 1 × ސ 2 .This is the simplest case after points in a product of projective spaces.We demonstrate several methods for finding the
Authors: Ayah Almousa, Juliette Bruce, Michael Loper, Mahrud Sayrafi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funder: National Science Foundation
Awards: Macaulay2 Programming Workshop, RTG: Combinatorics and Algebra, Graduate Reserach Fellowship Program (GRFP), RTG: Algebraic Geometry, Applied Algebra, and Number Theory at the University of Wisconsin, Graduate Workshop in Commutative Algebra for Underrepresented Minorities, Graduate Research Fellowship Program (GRFP), G8229174675
OpenAlex ID: W2945369674