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# Toric

## A GAP4 Package for computing with toric varieties

Version 1.3

March 7, 2006

David Joyner
e-mail: wdj@usna.edu
WWW: http://cadigweb.ew.usna.edu/~wdj/toric/
Address:
Mathematics Department,
U. S. Naval Academy,
Annapolis, MD,
21402 USA.

### Copyright

(C) 2004-2005 David Joyner.

### Acknowledgements

The code for the toric package was written during the summer of 2002. It was put into GAP package format in the summer of 2004. toric is released under the GNU General Public License (GPL). This file is part of toric, though as documentation it is released under the GNU Free Documentation License (see http://www.gnu.org/licenses/licenses.html#FDL).

toric is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

toric is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with toric; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA

For more details, see http://www.fsf.org/licenses/gpl.html.

This documentation was prepared with the GAPDoc package of Frank L\"ubeck and Max Neunh\"offer.

### Contents

1. Introduction
1.1 Introduction to the toric package
1.2 Introduction to constructing toric varieties
1.2-1 Generalities
1.2-2 Basic combinatorial geometry constructions
1.2-3 Basic affine toric variety constructions
1.2-4 Riemann-Roch spaces and related constructions
2. Cones and semigroups
2.1 Cones
2.1-1 InsideCone
2.1-2 InDualCone
2.1-3 PolytopeLatticePoints
2.1-4 Faces
2.1-5 ConesOfFan
2.1-6 NumberOfConesOfFan
2.1-7 ToricStar
2.2 Semigroups
2.2-1 DualSemigroupGenerators
3. Affine toric varieties
3.1 Ideals defining affine toric varieties
3.1-1 IdealAffineToricVariety
3.1-2 EmbeddingAffineToricVariety
4. Toric varieties X(Delta)
4.1 Riemann-Roch spaces
4.1-1 DivisorPolytope
4.1-2 DivisorPolytopeLatticePoints
4.1-3 RiemannRochBasis
4.2 Topological invariants
4.2-1 EulerCharacteristic
4.2-2 BettiNumberToric
4.3 Points over a finite field
4.3-1 CardinalityOfToricVariety

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