2 ms·

I understand the 2d-form is the real set of solutions for real parameters. What is the 3d-form?

• ykonstant 5y ago
  The 3D form is the real surface of the homogenization of the curve. We begin with an algebraic curve; in this case, say y^2 = x^3 - x. The equation defining this curve is degree 3 but inhomogeneous, meaning that not all monomials have the same total degree. In order to reveal hidden "structure at infinity" (essential for defining and studying the addition law on elliptic curves), we perform so-called homogenization. In our example, we take the inhomogeneous equation y^2 = x^3 - x and introduce a third variable z to balance out the degrees on all monomials leading to the equation y^2 z = x^3 - x z^2; now each of the three monomials has total degree 3. This equation can be thought of as defining a homogeneous surface in real 3D space, or from a more advanced perspective, a homogeneous curve in projective space. The original curve lies at the slice z=1 of this homogeneous surface, and the entire surface can be built by emitting all possible lines from the origin connecting to points on that sliced curve. Geometrically, if you wiggle the model around you can see this structure: the plane z=1 slicing through the surface, cutting out the curve, and lines through the points of the curve tracing out the entire surface. Algebraically, if you have a point (a,b,c) on the surface xz=y^2, that is, satisfying ac=b^2, the entire parametrized line (ra,rb,rc) satisfies rarb=(rc)^2, and so lies on the surface.