Mathematics 535: Introduction to Algebraic Geometry, Fall 2008

Movie of the week - Week 7

The moving image at the top of the mathematics 535 home page during the seventh week shows a portion of an affine real locus of the projective variety 3*y^2*z^2-4*x*z^3-4*y^3*w+6*x*y*z*w-x^2*w^2. The singular locus of this variety is the twisted cubic curve in P^3. Given a projective variety V in P^n of dimension k there is a rational map from V to the Grassmanniann of k planes in projective n space which sends a smooth point to the projective tangent space at the point. This is called the Gauss map. If the dimension of the image of the Gauss map is less than the dimenesion of V we call V a developable variety. The variety pictured here is developable, in fact the fiber of the map to the Grassmanian is a line on the surface which is tangent to the twisted cubic, and the surface is ruled by these tangent lines. One can parametrically describe the surface in affine space by two parameters t,u as the set of points (t^3,t^2,t)+u(3t^2,2t,1). At each point not on the twisted cubic the surface is smooth and has curvature 0. The twisted nature of the cubic is shown by the large variation in the tangent line as you proceed along the curve.