## 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.