Apologies if this is an obvious question. Suppose I have a variety which I know is smooth and simply connected and blow-up a smooth point so that the resulting variety is smooth. Does the exact point that I blow-up matter or will the resulting variety be the same regardless of which points I blow-up?
If "the same" is interpreted to mean "isomorphic", then this is not true. I will give a counterexample below, but first let me mention a positive result:
- If the variety is homogeneous, by which I mean that its automorphism group acts transitively, then the isomorphism class is independent of the point you blow up. (This also implies you can't get counterexamples from really simple varieties such as $\mathbf P^n$ or quadrics.)
Now for a counterexample:
- Here's the simplest one I can think of. Let $\pi: X \rightarrow \mathbf P^2$ be the blowup of $\mathbf P^2$ in any 2 distinct points $p_1$, $p_2$. Now let $f: Y \rightarrow X$ be the blowup of a third point $p$. If $\pi(p)$ does not lie on the line joining $p_1$ and $p_2$, then $Y$ will not have any smooth curves of self-intersection $-2$, but if $\pi(p)$ does lie on the line, there will be such a curve. So the isomorphism class of $Y$ changes depending on $p$.
Maybe that is not so satisfying, because I only showed that the isomorphism class changes for "special" positions of the point $p$. But there are examples where the isomorphism class varies continuously with $p$:
- Let $X$ be the blowup of $\mathbf P^2$ in 5 general points, and $f: Y \rightarrow X$ the blowup of a sixth point $p$. Then $Y$ is a cubic surface. There is a 4-dimensional moduli space of cubic surfaces up to automorphism, which implies that the isomorphism class of $Y$ must be nonconstant as we vary $p$. (I can give more details if necessary.)
Finally, in spite of these negative results, there is a weaker sense in which the blowups are "the same", which might suffice for many purposes:
- The family of blowups parametrised by $p \in X$ forms a flat family over $X$. So any property of varieties that is constant in flat families is independent of the choice of blown-up point of $p$. In particular, if $X$ is a complex projective variety then the diffeomorphism class of the blowup is independent of $p$.