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PencilsOfQuadrics :: translateIsotropicSubspace

translateIsotropicSubspace -- choose a random isotropic subspace

Synopsis

Description

Reid's theorem says that the set of maximal isotropic subspaces on a complete intersection of two quadrics in (2g+2) variables is isomorphic to the set of degree 0 line bundles on the associated hyperelliptic curve E of genus g. The method computes the maximal isotropic subspace uL corresponding to the translation of u by L.

i1 : kk=ZZ/101;
i2 : g=2;
i3 : (S,qq,R,u, M1,M2, Mu1,Mu2) = randomNicePencil(kk,g);
i4 : M=cliffordModule (Mu1, Mu2, R);
i5 : f=M.hyperellipticBranchEquation

        5      4 2      3 3      2 4        5      6
o5 = 15s t + 6s t  + 33s t  + 30s t  - 48s*t  + 14t

o5 : R
i6 : L=randomLineBundle(0,f);
i7 : uL=translateIsotropicSubspace(M,L,S)

o7 = | y_1-36z_1-2z_2 y_0+20z_1+17z_2 x_1+25z_1-12z_2 x_0+40z_1-43z_2 |

             1       4
o7 : Matrix S  <--- S
i8 : assert (betti uL == betti u)

See also

Ways to use translateIsotropicSubspace :

For the programmer

The object translateIsotropicSubspace is a method function.