Virtual resolutions for a product of projective spaces

Christine Berkesch, Daniel Erman, Gregory G. Smith

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically super uous summands. In this paper, we construct some much shorter free complexes that better encode the geometry.

Original languageEnglish (US)
Pages (from-to)460-481
Number of pages22
JournalAlgebraic Geometry
Issue number4
StatePublished - 2020

Bibliographical note

Funding Information:
Some of this research was completed during visits to the Banff International Research Station (BIRS) and the Mathematical Sciences Research Institute (MSRI), and we are very grateful for their hospitality. We thank Lawrence Ein, David Eisenbud, Craig Huneke, Nathan Ilten, Rob Lazarsfeld, Mike Loper, Diane Maclagan, Frank-Olaf Schreyer, and Ian Shipman for helpful conversations. We also thank an anonymous referee for their valuable suggestions.

Funding Information:
The first author was partially supported by the NSF Grant DMS-1440537, the second author was partially supported by the NSF Grants DMS-1302057 and DMS-1601619, and the third author was partially supported by the NSERC.

Publisher Copyright:
© Foundation Compositio Mathematica 2020.


  • Deformation theory
  • Free resolutions
  • Toric varieties
  • Vector bundles


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