Blowing things up isn't all bad!

In fact, in Algebraic Geometry it is rather useful. We always want to normalise algebraic curves, to get rid of those pesky singularities and letting C be an irreducible algebraic curve in P^2 and S denote the set of singular points of C we have the normalisation theorem to do this for us. Here we obtain a compact Riemann surface C' and a holomorphic map

f: C' ---> P^2

where f(C')=C, the number of points in the preimage of S under f (denote this S') is finite and

f: C' \ S' ---> C\S

is injective. But what are we doing to the singularities? We are blowing them up. This is, informally where each singularity is replaced by the space of tangent directions at the point.

We consider blowing up a points in complex space. Let z1,...,zn be the coordinates of n-dimensional complex space C^{n} and let w1,...,wn be the homogeneous coordinates of (n-1)-dimensional complex projective space P^{n-1} . We consider embedding

g: D={zi wj=wi zj} ---> C^{n} x P^{n-1}.

Composing with projection we obtain the holomorphic map

g: D ---> C^{n}.

Formally D is the blowup and g the blow up map. Defining E as the inverse image of the blow up locus Z under g we obtain an isomorphism,

g: D\ E ---> C^{n}\Z.

This is seen in the normalisation theorem. E is defined to be the exceptional divisor.

When have I used this? When calculating monodromy of certain multiple polylogarithms I considered the picture














and it was necessary to blow up to obtain














This told me around where monodromy has to be calculated. So blowing things up is useful!