  Given the ensemble of random Gaussian Laplace eigenfunctions on the
three-dimensional torus (`3d arithmetic random waves'), we investigate the
$1$-dimensional Hausdorff measure of the nodal intersection curve against a
compact regular toral surface (the `nodal intersection length'). The expected
length is proportional to the square root of the eigenvalue, times the surface
area, independent of the geometry.
  Our main finding is the leading asymptotic of the nodal intersection length
variance, against a surface of nonvanishing Gauss-Kronecker curvature. The
problem is closely related to the theory of lattice points on spheres: by the
equidistribution of the lattice points, the variance asymptotic depends only on
the geometry of the surface.
