Just to add a comment on "why this idea is cool" from my perspective (I'm a mathematician).<p>The situation being studied is: C is a curve in the plane (as another commenter pointed out, the z variable can essentially be ignored and set to z=1), described by a horrendous equation f(x,y) = 0 with very few rational solutions.<p>Well, thinking abstractly, if there are only finitely many rational solutions, then there certainly <i>exists</i> a second equation, g(x,y) = 0, giving another curve C' that intersects C at only the rational points. (Because any finite set of points can be interpolated by a curve, e.g. by Newton interpolation. [shrug] Nothing deep about this!)<p>But, it seems completely hopeless to try to find the equation g(x,y) in practice, other than by <i>first</i> finding all the rational points on C by other means, and then just writing down a different curve passing through them.<p>So what's special here is that this "Selmer variety" approach provides a method, partly conjectural, for constructing C' <i>directly</i> from C. And the paper being described has successfully applied this method to <i>prove</i> that, at least in this one case, C' intersects C at precisely the rational points. (And once you have the two equations, it's easy to solve for the intersection points -- we now have two equations in two variables).<p>PS: Part of what's special here is the connection between number theory and geometry. A Diophantine equation has infinitely-many solutions if you allow x and y to be real numbers -- there's the entire curve. It's usually an <i>extremely</i> delicate number theory question to analyze which solutions are rational. But here, we're converting the problem to geometry -- intersecting two curves (much easier).