Blowing up at a reduced subscheme is geometrically clear to me: it is just like blowing up at a reduced point, and point on the exceptional divisor corresponds to different directions oftangent lines through the base point. However, I always have problem understanding blowing up at a nonreduced subscheme. Is it a geometric way to see it as the reduced cases?
As examples, I would like to ask the following questions:
- Consider the blow upblowup of $\mathbb A^2$ at a double point $x={\rm Spec}(k[x,y]/(x^2,y))$. What is the exceptional divisor? Is it isomorphic to $\mathbb P^1$? Is the blowup smooth? (I guess both answers are negativeBy the comment of mtrying46, the exceptional divisor is $\mathbb P^1$ but the blow up is not smooth)
- Suppose $Y$ is theConsider that first blow up of $X$$\mathbb A^2$ at $Z\subset X$one point and both $X,Y$ are smooth. If we know every fiberthen blow up at one point in the exceptional divisor of $Z$ is isomorphic to some $\mathbb P^m$,the blowup. The composition can we conclude thatbe seen as blow up once $Z$(at a nonreduced center). Then what is reducedthe center?
From the two examples above, it seems that blowup at nonreduced center will either result in singularity or central fiber not being projective space. I would like to know whether this is true in general. That is:
- Suppose $Y$ is the blow up of $X$ at $Z\subset X$ and both $X,Y$ are smooth. If we know every fiber of $Z$ is isomorphic to some $\mathbb P^m$, can we conclude that $Z$ is reduced?
Everything is over $\mathbb C$. Any comments will be very helpful!