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randomIsotropicSubspace -- choose a random 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

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/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/PencilsOfQuadrics.m2:3545:0.