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# randomIsotropicSubspace -- choose a random isotropic subspace

- 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 : setRandomSeed 0
     -- setting random seed to 0

    o2 = 0

    i3 : g=2;

    i4 : (S,qq,R,u, M1,M2, Mu1,Mu2) = randomNicePencil(kk,g);

    i5 : M=cliffordModule (Mu1, Mu2, R);

    i6 : ru=randomIsotropicSubspace(M,S)

    o6 = | y_1+28z_1+48z_2 y_0+16z_1-21z_2 x_1-18z_1+25z_2 x_0-15z_2 |

                 1      4
    o6 : Matrix S  <-- S

    i7 : assert (betti ru == betti u)

## Caveat

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

## See also

- translateIsotropicSubspace -- choose a random isotropic subspace
- randomLineBundle -- a random balanced line bundle on the hyperelliptic curve
- CliffordModule -- hash table holding details of a Clifford module

## Ways to use randomIsotropicSubspace:

- `randomIsotropicSubspace(CliffordModule,PolynomialRing)`

## For the programmer

The object randomIsotropicSubspace is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PencilsOfQuadrics.m2:3545:0.