# 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 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.