## 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.