I'm considering the codimension-1-in-P5-object defined by

x^6+y^6+z^6 = u^6+v^6+w^6

with the obvious solutions-by-permutation removed and GCD(x,y,z,u,v,w)=1. I've searched for z,w<2000, and am finding that the number of these points with z,w<N seems to be slightly more than linear in N, so I suspect there are some families of points lurking. Suspiciously many of the points I've found have x+y+u=z+v+w, for example.

http://www.jstor.org/pss/2005335 gives a set of homogeneous quartics parameterising some solutions, but clearly not all solutions since the quartics happen also to satisfy x^2+y^2+z^2=u^2+v^2+w^2.

What are the right sort of questions to ask about the set of points on a high-dimensional variety?