Introduction

Content

The toric geometry part of OSCAR comprises algorithms addressing normal toric varieties and objects from commutative algebra and polyhedral geometry derived thereof. In particular, we provide support for the following:

• torus-invariant divisor (classes),
• line bundles,
• line bundle cohomology via cohomCalg (cf. Ralph Blumenhagen, Benjamin Jurke, Thorsten Rahn, Helmut Roschy (2010)),
• vanishing sets of line bundle cohomology (cf. Appendix B of Martin Bies (2018)),
• cohomology ring and cohomology classes,
• Chow ring, algebraic cycles and intersection theory,
• elementary support for closed subvarieties.

Status

This project is work-in-progress.

Tutorial

We provide a tutorial for toric geometry in OSCAR.

Long term goals

We follow David A. Cox, John B. Little, Henry K. Schenck (2011). Our long term goals include the following:

• Ensure that one can perform all computations of Appendix B in David A. Cox, John B. Little, Henry K. Schenck (2011).
• Provide support for coherent sheaves and their sheaf cohomologies. In particular, the existing algorithms in ToricVarieties_project (based on Martin Bies (2018)) should eventually be available in OSCAR.

Contact

Please direct questions about this part of OSCAR to the following people:

• Martin Bies,
• Lars Kastner.

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.