What is a blow-up?

Solution 1:

The basic idea of blowups in algebraic geometry is to remove a point from an algebraic variety and replace it by the projectivized tangent cone at that point. The result is a new space, with a projective map down to the old, such that the fibre over the "centre" (the point we blow up) is an effective Cartier divisor. This generalizes in a couple directions: we can blow-up on closed subvarieties, and the construction extends also to schemes (and probably to more general spaces also).

Blowups satisfy a universal property with respect to replacing the centre by an effective Cartier divisor (i.e., a subvariety that is locally defined by a single nonzerodivisor): Let $Y\subseteq X$ be a closed subvariety. A morphism $\pi:\widetilde X\to X$ is called a blowup of X with centre Y if the two following properties hold:

(a) $E:=\pi^{-1}(Y)$ is an effective Cartier divisor on $\widetilde X,$

(b) $\pi$ satisfies a universal property with respect to (a), namely for every morphism $\tau:Z\to X$ such that $\tau^{-1}(Y)$ is an effective Cartier divisor, there is a unique morphism $\varphi:Z\to \widetilde X$ such that $\pi\circ\varphi=\tau.$

Blowups can be easily constructed in a few ways. One way is by computing charts that can be glued. For example, let $X$ be an affine variety over the field $k$ with coordinate ring $k[X],$ and let $I=(g_1,\dots, g_l)\subseteq k[X]$ be an ideal cutting out the subvariety $Y.$ Suppose for simplicity that $X$ is irreducible, so that $k[X]$ is a domain. Then $\widetilde X$ is defined by the collection of charts $U_i=\operatorname{Spec}\left(k[X][g_1/g_i,\ldots, g_l/g_i]\right),$ where $k[X][g_1/g_i,\ldots, g_l/g_i]\subseteq k(X)$ can be viewed as a subring of the fraction field of $k[X].$

Here is an easy example. We compute $\widetilde X$ where $X=\Bbb A^2$ has centre $Y=\{(0,0)\}.$ So $I_Y=(x,y),$ and $\widetilde X$ has two charts: $U_x=\operatorname{Spec}(k[x,y,y/x])=\operatorname{Spec}(k[x,y/x]),$ and $U_y=\operatorname{Spec}(k[x,y,x/y])=\operatorname{Spec}(k[y,x/y]).$ That was easy.

Here is another example, which generalizes. Knowing $\widetilde X=\widetilde{\Bbb A^2}$, we can compute the blowup of say $V=V(y^2-x^3)\subseteq X$ as the strict transform under the map $\pi:\widetilde X\to X.$ What I mean is that in the $x$-chart of $\widetilde X,$ the map $\pi$ has a dual description given by $k[x,y]\to k[x,y,y/x],x\mapsto x,y\mapsto x\cdot y/x,$ and we can use this to get $\widetilde V.$ In this chart, the equation for $V$ maps $y^2-x^3\mapsto (x\cdot y/x)^2-x^3= x^2((y/x)^2-x).$ Saying that $\widetilde V$ is the strict transform means that it has a chart defined by $(y/x)^2-x$ inside $\Bbb A^2=\operatorname{Spec}(k[x,y/x]).$ There is a second chart which you can compute in exactly the same way.

There is much, much more that can be said about blowups, they are a central technique in algebraic geometry in my view. I think the best you can do to learn them is to compute as many examples as possible. There is a great section in Eisenbud and Harris' book on blow-ups. You should try to compute every example there.