Standard Constructions in Algebraic Geometry

This page documents a collection of standard constructions in algebraic geometry available in OSCAR.

Grassmann Plücker Ideal

grassmann_pluecker_ideal([ring::MPolyRing,] subspace_dimension::Int, ambient_dimension::Int)

Given a ring, an ambient dimension and a subspace dimension return the ideal in the given ring generated by the Plücker relations. If the ring is not specified return the ideal in a multivariate polynomial ring over the rationals.

The Grassmann-Plücker ideal is the homogeneous ideal generated by the relations defined by the Plücker Embedding of the Grassmannian. That is given Gr$(k, n)$ the Moduli space of all $k$-dimensional subspaces of an $n$-dimensional vector space, the relations are given by all $d \times d$ minors of a $d \times n$ matrix. For the algorithm see Bernd Sturmfels (1993).


julia> grassmann_pluecker_ideal(2, 4)
ideal(x[0]*x[5] - x[1]*x[4] + x[2]*x[3])

julia> R, x = PolynomialRing(ResidueRing(ZZ, 7), "x" => (1:2, 1:3), ordering=:degrevlex)
(Multivariate Polynomial Ring in 6 variables x[1, 1], x[2, 1], x[1, 2], x[2, 2], ..., x[2, 3] over Integers modulo 7, nmod_mpoly[x[1, 1] x[1, 2] x[1, 3]; x[2, 1] x[2, 2] x[2, 3]])

julia> grassmann_pluecker_ideal(R, 2, 4)
ideal(x[1, 2]*x[2, 2] + 6*x[2, 1]*x[1, 3] + x[1, 1]*x[2, 3])