Can somebody give me a nice example of blow-up of a smooth algebraic variety along a singular subvariety? Something I can do some exercise on and check the differences with a smooth blow-up. Thanks!



blow-up along singular variety
Perhaps the easiest is to blow up a plane along a fat point:


Blow up $\mathbb A^2\_k=\mathrm{Spec} k[x,y]$ at the ideal $(x^2,y^2)$. You should get a pinch point (Whitney's umbrella). This itself is an interesting singularity. It's simple normal crossing away from the pinch point, but not so simple there. If you want to practice blow-ups, then as a second step blow up the pinch point and marvel at the fact that you get a new pinch point. The only way to resolve or even make a pinch point any better is to normalize it.


Not all fat points lead to singular varieties. Blowing up $(x^2,xy,y^2)$ is the same as blowing up the simple point $(x,y)$.


