  We consider a random Gaussian ensemble of Laplace eigenfunctions on the 3D
torus, and investigate the 1-dimensional Hausdorff measure (`length') of nodal
intersections against a smooth 2-dimensional toral sub-manifold (`surface').
The expected length is universally proportional to the area of the reference
surface, times the wavenumber, independent of the geometry.
  For surfaces contained in a plane, we give an upper bound for the nodal
intersection length variance, depending on the arithmetic properties of the
plane. The bound is established via estimates on the number of lattice points
in specific regions of the sphere.
