Mathematics 535: Introduction to Algebraic Geometry, Fall 2008

Movie of the week - Week 5

The moving image at the top of the mathematics 535 home page during the fifth week shows a portion of an affine real locus of the blowup of the affine plane at the origin. The blowup is the graph of the rational map F: (x,y) -> [x,y]. We can compute this graph as the set of points [a,b] x [x0,x1] such that ax1-bx0=0. In the affine open set where x0=-1 this takes the form of the affine surface z+xy=0 in affine 3 space. It is a portion of that surface which is pictured here above a disk in the affine plane. The first projection of the graph to A^2 is just the vertical projection down to the disk.

The blowup maps isomorphically to the affine plane away from the exceptional divisor [0,0] x P^1 which maps to (0,0). The proper transform of each line through the origin given by [t,bt] is the set [t,bt] x [-1,-b], a line on the blowup which maps onto a line through the origin in the affine plane. Each line in the disk representing a portion of the affine plane has the same color as its proper transform on the blowup above it. Note that the family of lines intersecting at the origin in the affine plane is blown up to a family of disjoint lines, each intersecting the exceptional divisor.

The blowup is birational, but not isomorphic to the affine plane. The rational map of A^2 to P^1 can be realized as the morphism [a,b] x [x0,x1] -> [x0,x1] on the blowup.