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!
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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 $(x,y)$.
answered Aug 23, 2013 at 16:43
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6$\begingroup$ Yet another example is blowing up $(x^2,y)$. Then you will get an $A_1$ singularity. $\endgroup$– SashaCommented Aug 23, 2013 at 17:15
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2$\begingroup$ In general, blowing up a toric variety along a toric subscheme will get you another toric variety, so that's fun, since you can think about it polytopally. In Sasha's example, start with the first quadrant, and cut off the triangle with vertices $(2,0),(1,0),(0,0)$. The result has two vertices, one of which is smooth and the other an $A_1$ singularity. (Sándor, do you remember us discussing this example at the blackboard around 1995?) In Sándor's second example, we cut off the triangle with vertices $(2,0),(0,2),(0,0)$ which is just a rescaling of $(1,0),(0,1),(0,0)$. Try $xy=z=0$ in $A^3$. $\endgroup$ Commented Aug 23, 2013 at 18:10
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$\begingroup$ @Allen: That's nice. Actually I do remember that conversation (somewhat vaguely though). I still don't think about polytopes instinctively even though again and again I see the usefulness of it. Thanks for reminding me! $\endgroup$ Commented Aug 23, 2013 at 18:25
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$\begingroup$ Can one explain the example of blowing up along a snc divisor? e.g. Blowing up P^3 along union of two planes. $\endgroup$ Commented Oct 3, 2015 at 16:01
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$\begingroup$ @MohammadF.Tehrani: the union of two planes in $\mathbb P^3$ or more generally an snc divisor in a smooth variety is Cartier and hence blowing it up does nothing. It's an isomorphism. $\endgroup$ Commented Oct 19, 2016 at 16:09