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 cubic surface S: x_0x_1x_2=x_3^3 in P^3. In fact this is the affine surface (1-x-y-z)xz=y^3, which is projectively equivalent to S, with the affine open subset chosen as x_0=1. The surface S has lines only on the plane x_3=0, where there are the lines x_0=0,x_1=0,x_2=0. The affine picture is chosen to show all three lines in affine 3 space. The surface S is singular at points where the gradient (x_1x_2,x_0x_2,x_0,x_1,3x_3^2) vanishes, that is exactly along the three lines on the surface. The existence of a cubic surface with a finite number of lines is used in the proof of the fact that a general cubic surface in P^3 has a finite number of lines. There exist cubic surfaces with infinitely many lines, for example the projection of a rational scroll in P^4 to three space, which yields the Cayley ruled cubic appearing in week 4. Cubic surfaces such as x_0^3=0 have a two dimensional space of lines on them.