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PencilsOfQuadrics :: randomIsotropicSubspace

randomIsotropicSubspace -- choose a random isotropic subspace



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 chooses a random line bundle L of degree 0 on E, and computes the maximal isotropic subspace ru 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 : ru=randomIsotropicSubspace(M,S)

o5 = | 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
o5 : Matrix S  <--- S
i6 : assert (betti ru == betti u)


The ground field kk (=coefficientRing S) has to be finite, since it uses the method randomLineBundle.

See also

Ways to use randomIsotropicSubspace :

For the programmer

The object randomIsotropicSubspace is a method function.