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Suppose I am given an algebraic variety $X$ and a (closed) point $x \in X$. I know of two descriptions of the blow-up of $X$ at $x$. One is intrinsic but not geometric: if $\mathcal{I}_x$ denotes the ideal sheaf of $x$ then we can define the blow-up to be $\text{Proj} \ \mathcal{O}_X \oplus \mathcal{I}_x \oplus \mathcal{I}_x^2 \oplus \cdots$. The other, which works only when $X$ is quasi-projective, is geometric but not intrinsic: one chooses an embedding $X \to \mathbb{P}^n$, defines the blow-up of $\mathbb{P}^n$ at a point explicitly in coordinates, and then takes the proper transform (or whatever the terminology is) of $X$.

I hope there is a construction which is both intrinsic and geometric (it may be nothing but a repackaging of the $\text{Proj}$ definition). Here is a starting point to indicate what I'm looking for. Suppose $x \in X$ is a nonsingular point for simplicity, so the tangent space $T_xX$ is well-behaved. As I understand it, the blow-up is set-theoretically $X \setminus \{ x \} \sqcup \mathbb{P}(T_xX)$, but obviously this is not a disjoint union in the sense of varieties. How does one make this into a variety?

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up vote 4 down vote accepted

Eisenbud and Harris' The Geoemtry of Schemes is a good place to learn the connection between these two constructions. Briefly, the Proj construction really is geometric, though it is not so easy to see until you get some familiarity with it. The exceptional divisor $E = \mathbb{P}(T_x X)$ is given as $\text{Proj } \mathcal{O}_X / \mathcal{I}_x \oplus \mathcal{I}_x / \mathcal{I}_x^2 \oplus \cdots$, with the inclusion $E \hookrightarrow \text{Bl}_x(X)$ corresponding to the natural qoutient map $\mathcal{O}_X \oplus \mathcal{I}_x \oplus \cdots \rightarrow \mathcal{O}_x / \mathcal{I}_x \oplus \mathcal{I}_x / \mathcal{I}_x^2 \oplus \cdots$.

The other useful way to think about blowups is via the universal property. If $f: Z \rightarrow X$ is any morphism such that $f^{-1}(x)$ is a Cartier divisor, then there exists a unique morphism $g : Z \rightarrow \text{Bl}_x(X)$ such that $f$ is the composition of $g$ with the blowdown map $\text{Bl}_x(X) \rightarrow X$. This too is covered in Eisenbud and Harris' book.

Another useful source is Chapter 19 of Ravi Vakil's Foundations of Algebaric Geometry.

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Thanks, I happen to have a copy of that book lying around. – Justin Campbell May 11 '12 at 21:01

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