# What is a blow-up?

Can anyone explain to me what a blow-up is? If would be great if someone could provide a definition and some examples. Any free introductory texts are welcome too. Thanks!

• It's likely there is a question already addressing this. Try our search bar. Sep 10, 2012 at 16:40
• Maybe you should tag this as algebraic-geometry. Do you want to know about varieties? Schemes? I would have a look at Wikipedia and Section 4.3 of Gathmann's notes to begin. Sep 10, 2012 at 16:44
• Im not sure about context at the moment sorry.
– mick
Sep 10, 2012 at 16:51
• @mick This cannot be properly answered without more context. Where did you hear about blow-ups? What do you know about algebraic geometry (because they usually appear in that context)? Sep 10, 2012 at 17:09
• @mick Also, in the same spirit as Graphth's answer (I think), what do you expect from the MSE community that the Wikipedia page (for example) does not already address? Sep 10, 2012 at 17:10

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.