Mathematics 535: Introduction to Algebraic Geometry, Fall 2008

Movie of the week - Week 2

The moving image at the top of the mathematics 535 home page during the second week shows an affine real locus of the a surface projectively equivalent to the Segre surface {[su,sv,tu,tv]} = P^1 x P^1 in P^3 which is the projective hypersurface z_0z_3=z_1z_2. This real locus is the oriented real surface x^2 +y^2 -z^2=25 in affine 3-space, with gold interior and blue exterior. The points [au,av,bu,bv] for fixed [a,b] give a family of lines on the surface which do not intersect one another. A different family is formed by points [sc,sd,tc,td] for fixed [c,d]. Lines from different families intersect once. Every point on the surface is on two lines, which are in fact the intersection of the surface with the tangent line at that point. The Segre varieties are clearly rational varieties, but are not isomorphic to any projective space, since in projective space any two linear varieties with dimension at least half the ambient dimension must intersect. Every Segre variety contains families of disjoint linear spaces of dimension at least half that of the variety.