Singular

D.15.24.5 Grassmannian

Usage:

Grassmannian(k,n); k int, n int

Return:

variety

Theory:

create a Grassmannian G(k,n) as an abstract variety. This abstract variety has diemnsion k(n-k) and its Chow ring is the quotient ring of a polynomial ring in n-k variables q(1),...,q(n-k), which are the Chern classes of tautological quotient bundle on G(k,n), modulo some ideal generated by n-k polynomials which come from the Giambelli formula. The monomial ordering of this Chow ring is 'wp' with vector (1..k,1..n-k). Moreover, we export the Chern characters of tautological subbundle and quotient bundle on G(k,n)
(say 'subBundle' and 'quotientBundle').

LIB "schubert.lib";
variety G24 = Grassmannian(2,4);
G24;
==> A variety of dimension 4
==> 
def r = G24.baseRing;
setring r;
subBundle;
==> 1/6*q(1)*q(2)-1/2*q(1)^2+q(2)-q(1)+2
quotientBundle;
==> -1/6*q(1)*q(2)+1/2*q(1)^2-q(2)+q(1)+2
G24.dimension;
==> 4
G24.relations;
==> _[1]=q(1)^3-2*q(1)*q(2)
==> _[2]=q(1)^4-3*q(1)^2*q(2)+q(2)^2
ChowRing(G24);
==> //   characteristic : 0
==> //   number of vars : 2
==> //        block   1 : ordering wp
==> //                  : names    q(1) q(2)
==> //                  : weights     1    2
==> //        block   2 : ordering C
==> // quotient ring from ideal ...