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Polyhedra: A package for computations with convex polyhedral objects

Convex polyhedral objects play an important role in numerous areas of algebraic geometry. The package Polyhedra enables one to work with such convex polyhedral objects in Macaulay 2. In particular, Polyhedra implements objects representing polyhedra, cones, and fans. Many standard operations, such as intersections, duality, convex hulls, and calculation of Hilbert bases are included. The package also provides functions to create convex polyhedral objects from some algebraic objects. THE PACKAGE AND ITS APPLICATIONS. Many objects in algebraic geometry can be described by convex polyhedral objects. For example, the Grobner degenerations of a homogeneous ideal are in one-to-one correspondence with the faces of its state polytope; see (St, Chapter 2). Geometric invariant theory (GIT) quotients coming from linearizations of the trivial bundle provide a second class of examples. For an effective action of the torus (C ) k on affine space A n , there is an associated exact sequence of lattices 0! K! Z n ! Z k ! 0. The GIT-quotients of this action are given by the normal fans of certain polyhedra in the associated Q-vector space KQ; for more information see (B, §III.2). The package Polyhedra provides methods within Macaulay 2 (M2) for creating and manipulating convex polyhedral objects such as polyhedra and cones. Some objects from algebraic geometry can also be translated into their corresponding convex polyhedral objects.