The Grassmannian ChowScheme¶

AUTHORS:

• Manfred Lehn (2013)
• Christoph Sorger (2013)

sage.schemes.chow.library.grass.Grass(n, r, chern_class='c', names=None, name=None, latex_name=None)¶

Return either depending respectively whether \(n > r\) or \(n < r\):

• The Grassmannian of quotients of rank r of an n dimensional vector space;
• The Grassmannian of subspaces of rank n of an r dimensional vector space.

EXAMPLES:

sage: G = Grass(6, 4, chern_class='w')
sage: G.dimension()
8
sage: G.rels()
[w2^5, w1*w2^4, w1^2*w2^3 - 4/3*w2^4, w1^6 - 3*w1^2*w2^2 + w2^3, w1^3*w2 - 3/2*w1*w2^2]
sage: G.sheaves["universal_sub"]
Bundle(Grass(6, 4), 2, [1, -w1, w1^2 - w2])
sage: G.sheaves["universal_quotient"]
Bundle(Grass(6, 4), 4, [1, w1, w2, -w1^3 + 2*w1*w2, -w1^4 + w1^2*w2 + w2^2])

sage: H = Grass(6, 2, chern_class='v')
sage: H.dimension()
8
sage: H.rels()
[v2^5, v1*v2^4, v1^2*v2^3 - v2^4, v1^3*v2^2 - 2*v1*v2^3, v1^4*v2 - 3*v1^2*v2^2 + v2^3, v1^5 - 4*v1^3*v2 + 3*v1*v2^2]
sage: H.sheaves["universal_sub"]
Bundle(Grass(6, 2), 4, [1, -v1, v1^2 - v2, -v1^3 + 2*v1*v2, v1^4 - 3*v1^2*v2 + v2^2])
sage: H.sheaves["universal_quotient"]
Bundle(Grass(6, 2), 2, [1, v1, v2])

sage.schemes.chow.library.grass.GrassBundle(A, B, chern_class='c', names=None, name=None, latex_name=None)¶

Return either

• the Grassmannian of quotients of rank B of A if A is a sheaf or
• the Grassmannian of subbundles of B of rank A if B is a bundle.