randomIsotropicSubspace -- choose a random isotropic subspace

Synopsis

Usage:

ru=randomIsotropicSubspace(M,S)
Inputs:
- M, an instance of the type CliffordModule, corresponding to a maximal isotropic subspace u
- S, a ring, a polynomial ring kk[x_0..y_{(g-1)},z_0,z_1,s,t] in (2g+4) variables
Outputs:
- ru, a matrix, a matrix presenting a random maximal isotropic subspace

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 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)

Caveat

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

Ways to use randomIsotropicSubspace :

randomIsotropicSubspace(CliffordModule,PolynomialRing)

For the programmer

The object randomIsotropicSubspace is a method function.