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Is the composition of blowing-up of algebraic varieties itself a blowing-up ?

I think this is true but I am surprised not to have found any reference, though it seems to be an interesting property. Of course, I'm not able either to prove it myself...

If one blows up quasi projective varieties, then the result is easy : since the composition is be a projective birationnal morphism, it is a blowing-up [Hartshorne, th 7.17, p. 166]. But what about the general case ?


Edit

In the late but very good answer of Lierre (!), it is claimed that “There exists a $p$ and an ideal sheaf $J\subset \mathcal O_{X_0}$ such that $J\cdot \mathcal O_{X_1} = E_0^p I_1$.” How to prove it ? I not sure we can patch the argument of the short answer.

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1 Answer 1

Short answer

(After Huneke, C. and Swanson, I. (2006). Integral closure of ideals, rings, and modules, exercise 5.10)

Take $R$ a noetherian ring, an ideal $I$ of $R$ and an element $x$ of $I$. Define $S$ to be the ring $R[\frac Ix]$. Let $J$ be an ideal of $S$ and $y$ an element of $J$.

There exists an ideal $K$ of $R$ and $z$ in $K$ such that $R[\frac Ix][\frac Jy] = R[\frac Kz]$.

Indeed, let $J = (y, a_1, \dotsc, a_n)$. We can assume that all the generators of $J$ are in $R$ since $S[\frac Jy] = S[\frac{fJ}{fy}]$ for all $f$ in $R$. Set $K = IJ$ and $z = xy$ and you're done.

How ever, it is not clear how things patch together when working with non affine varieties.

Long answer

Let $X_2 \to X_1 \stackrel{\epsilon}{\longrightarrow} X_0$ a sequence of blowups, with center $I_0\subset \mathcal O_{X_0}$ and $I_1\subset \mathcal O_{X_1}$. Let $E_0\subset \mathcal O_{X_1}$ the exceptional divisor of the first blowup. There exists a $p$ and an ideal sheaf $J\subset \mathcal O_{X_0}$ such that $J\cdot \mathcal O_{X_1} = E_0^p I_1$.

I claim that the morphism $X_2 \to X_0$ is the blowup with center $I_0 J$, and I'll check this by showing the universal property of the blowup.

Let $Z$ be a variety with a morphism $Z\to X$ such that $I_0 J \cdot \mathcal O_Z$ is invertible. In particular, the ideal sheaf $I_0\cdot O_Z$ is invertible, so $Z\to X_0$ factors through $X_1$. I don't know what is $I_1\cdot \mathcal O_Z$ but $E_0^p I_1 \cdot \mathcal O_Z$ is $J\cdot \mathcal O_Z$, which is invertible. So $I_1 \cdot \mathcal O_Z$ is also invertible, and thus $Z\to X_1$ factors through $X_2$. The uniqueness of the factorization is implied by the uniqueness of $Z\to X_2$ given $Z\to X_1$ and the one of $Z\to X_1$ given $Z\to X_0$.

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