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Let $X$ a variety. Say $X=\operatorname{Spec} A$.

Consider two ideals of $A$, say $I$ and $J$, with equal radical ; and consider the blow-ups of X with centre $I$ and $J$, say $Y_I$ and $Y_J$. How can I decide if the blow-up map $Y_I \to X$ factors through $Y_J$ ?

For example, if $X = \operatorname{Spec}k[x,y]$ is the affine plane. Then we can consider the blow-ups with centre $I_1 = (x,y)$, $I_2 = (x,y^2)$, $I_3=(x^2,y^2)$ and $I_4=(x^2,xy,y^2)=I_1^2$. We have $I_4\subset I_3\subset I_2\subset I_1$. Let $Y_i$ denote the corresponding blow-ups.

It is well-known that $Y_1$ and $Y_4$ are isomorphic. However $Y_2$ and $Y_3$ are two other different varieties. Using the universal property of the blow-up we see that $Y_1\to X$ factors through $Y_3$ but not through $Y_2$.

What kind of property should I look at to predict this kind of factorization, without computing the actual blow-up ?

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2 Answers

up vote 2 down vote accepted

There are two conditions I can think of.

  1. Perhaps $I = J \cdot J'$ for some other ideal $J'$. Indeed, blowing up a product of ideals is the same as blowing up one ideal (say $J$) and then blowing up the total transform of the other (say $J'$).

  2. The other is requiring that $I = \overline{J}$, here this is integral closure of ideals (more generally $I$ and $J$ have the same integral closure and $I \subseteq J$). Indeed, two ideals have the same integral closure if and only if they have the same normalized blow-up and if their total transforms on the common normalized blow-up agree.

Let me give an alternate explanation of 2., suppose for example that $I = \overline{J}$. Then the blow-up of $I$ and the blow-up of $J$ have the same normalization and $Y_I$ is just a partial normalization of $Y_J$.

So this explains some of your examples. Indeed, the integral closure of $I_3$ is just $I_4$. On the other hand, $I_4 = I_1^2$. So by 1., $I_1$ and $I_4$ have the same blowup but by 2., the blow up of $I_4$ factors through the blowup of $I_3$. However, if you consider $I_1 \cdot I_2$ or $I_3 \cdot I_2$, you will find that their blowups factor through $I_2$.

For smooth surfaces, there is a theory of factorization of complete (integrally closed) ideals due to Zariski. This should allow you to answer these sorts of questions even more precisely.

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I think this will help me a lot! The point 2 is really great, I was not aware of it. –  Lierre Feb 14 '12 at 8:19
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The universal property of the blowup $\pi:Bl_I X\to X$ of an ideal $I$ in $X$ is that the sheaf of ideals $\pi^{-1}I\cdot O_{Bl_I X}$ is invertible. So, if you want to find out whether $Bl_I$ factors through $Bl_J$ you just have to check whether $\pi^{-1}J\cdot O_{Bl_I X}$ is invertible.

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