# translateIsotropicSubspace -- choose a random isotropic subspace

## Synopsis

• Usage:
uL=translateIsotropicSubspace(M,L,S)
• Inputs:
• M, an instance of the type CliffordModule, corresponding to a maximal isotropic subspace u
• L, an instance of the type VectorBundleOnE, a degree 0 line bundle on the associated hyperelliptic curve E of genus g.
• S, , a polynomial ring kk[x_0..y_{(g-1)},z_0,z_1,s,t] in (2g+4) variables
• Outputs:
• uL, , a matrix presenting a maximal isotropic subspace u translated by L

## 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)