Convex lattice polytope

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A convex lattice polytope (also called Z-polyhedron or Z-polytope) is a geometric object playing an important role in discrete geometry and combinatorial commutative algebra. It is a polytope in a Euclidean space Rn which is a convex hull of finitely many points in the integer lattice ZnRn. Such objects are prominently featured in the theory of toric varieties, where they correspond to polarized projective toric varieties.

Examples[edit]

  • An n-dimensional simplex Δ in Rn+1 is the convex hull of n+1 points that do not lie on a single affine hyperplane. The simplex is a convex lattice polytope if (and only if) the vertices have integral coordinates. The corresponding toric variety is the n-dimensional projective space Pn.
  • The unit cube in Rn, whose vertices are the 2n points all of whose coordinates are 0 or 1, is a convex lattice polytope. The corresponding toric variety is the Segre embedding of the n-fold product of the projective line P1.
  • In the special case of two-dimensional convex lattice polytopes in R2, they are also known as convex lattice polygons.
  • In algebraic geometry, an important instance of lattice polytopes called the Newton polytopes are the convex hulls of the set A which consists of all the exponent vectors appearing in a collection of monomials. For example, consider the polynomial of the form axy+bx^2+cy^5+d with a,b,c,d \neq 0 has a lattice equal to the triangle
{\rm conv}(\{(1,1),(2,0),(0,5),(0,0)\}).\

See also[edit]

References[edit]

  • Ezra Miller, Bernd Sturmfels, Combinatorial commutative algebra. Graduate Texts in Mathematics, 227. Springer-Verlag, New York, 2005. xiv+417 pp. ISBN 0-387-22356-8