# Again, Blow up and Direct Image

In the question (Direct Image by a Blow up), follows the following statements

1) $$\text{Sym}(A^{r}) \longrightarrow \bigoplus_{m \geq 0}I_{Y}^{m}$$ corresponding to the closed immersion $$\widetilde{X} \longrightarrow \mathbb{P}(A^{r})$$.

2) The exceptional divisor $$E$$, correponds to the line bundle $$\mathcal{O}_{\mathbb{P}}(A^{r})(-1)|_{\widetilde{X}}$$

3) There is a canonical isomorphism $$\mathcal{O}_{\widetilde{X}}(1) \simeq \mathcal{O}_{\widetilde{X}}(-E)$$.

4) Where was Serre vanishing used?

5) Why can we identify $$\pi_{*}\mathcal{O}_{\mathbb{P}(A^{r})} = \text{Sym}^{r}(A)$$ and $$\pi_{*}\mathcal{O}_{\widetilde{X}}(-nE) = I^{n}$$?

I would like to understand such statements, so thank you in advance for your suggestions and references.

## 1 Answer

1) The blowup of a sheaf of ideals $$\mathcal{I}\subset \mathcal{O}_X$$ is the relative Proj of the blowup algebra $$\bigoplus_{m\geq 0} \mathcal{I}^m$$. This is a definition, and we're just applying it to the affine case here where we replace $$\mathcal{O}_X$$ by $$A$$ and $$\mathcal{I}$$ by $$I$$. Choosing a set of generators $$f_1,\cdots,f_r$$ for $$I$$ gives you a surjection from $$A^r\to I$$, which turns in to a surjection of graded rings $$\operatorname{Sym}(A^r)\to \bigoplus_{m\geq 0} I^m$$, which corresponds to a closed immersion of their Projs by the general properties of that construction.

2) Every algebraic geometry book which covers blowing up should have a proof. For instance, Griffiths and Harris pg 184, Stacks 02OS, or this MSE question for a low-dimensional example (which generalizes).

3) This is just dualizing the statement of 2). Saying "$$E$$ corresponds to $$\mathcal{O}(-1)$$" means that $$\mathcal{O}(E)\cong\mathcal{O}(-1)$$, and dualizing gives $$\mathcal{O}(-E)\cong\mathcal{O}(1)$$ as requested.

4) Serre vanishing is not actually used in the proof: it's only meant to point out that the claim $$\mathcal{O}_{\widetilde{X}}(-nE)\cong I_Y^n$$ holds for any $$X,Y$$ once $$n\gg 0$$. For the specific situation of the proof, we are able to use the argument involving the exact sequence from the (now doubly) linked post to show this for $$n\geq 1$$ directly.

5) You've missed a $$(n)$$ here: the correct statement should be that global sections of $$\pi_*\mathcal{O}_{\Bbb P(A^r)}(n)$$ are exactly $$\operatorname{Sym}^n(A^r)$$.

Both of these claims follow from the property that for nice rings $$R$$, the sheaf $$\mathcal{O}(n)$$ on $$\operatorname{Proj} R$$ has global sections exactly $$R_n$$, the degree-$$n$$ portion of $$R$$. See for instance the discussion in Stacks 01QG.

In the first case, the ring $$R$$ is $$\operatorname{Sym}(A^r)$$, with graded components $$\operatorname{Sym}^n(A^r)$$, and in the second, the ring is $$\bigoplus_{m\geq 0} I^m$$ with graded component $$I^n$$, and we use the isomorphism $$\mathcal{O}_{\widetilde{X}}(n)\cong\mathcal{O}_{\widetilde{X}}(-nE)$$.

• Hi, @KReiser. Thank you for your answer. See if I'm right : I think it is not necessary for  to be a projective scheme, because I didn't see this hypothesis being used in the prove. Jan 18, 2020 at 21:58
• Again, @KReiser. The statement could be like this: Let $X$ be a smooth normal noetherian scheme and $Y \subset X$ a smooth closed subscheme and $$\pi : \widetilde{X} \longrightarrow X$$ be the blow up along $Y$. Then.... Jan 18, 2020 at 22:04
• 1. You don't need to tag folks if you're commenting on their post. 2. You're correct that projectiveness is not a necessary hypothesis. 3. Smooth implies normal, so there's no need to assume something is "smooth and normal". Jan 18, 2020 at 23:28
• Truth. Smooth $\rightarrow$ regular $\rightarrow$ normal. Another doubt. Is the push forward of line bundles a line bundle? Why the homomorphism of sheaves $$\pi_{*}\mathcal{O}_{\mathbb{P}(A^{r})} \longrightarrow \mathcal{O}_{\widetilde{X}}(-nE)$$ is an isomorphism? Thank you so much. Jan 19, 2020 at 2:24
• In general the pushforwards of a line bundle is not a line bundle: indeed, in the context of the original question, if $Y$ is not a divisor, $I_Y$ is not a line bundle, but $\pi_*\mathcal{O}_\widetilde{X}(-nE)=I_Y^n$. You're still missing the $(n)$ on the $\pi_*\mathcal{O}_{\Bbb P(A^r)}$. The claim you're curious is a general instance of the following fact: any surjective morphism of line bundles is an isomorphism. Proof: look on the level of stalks and use Nakayama's lemma. Jan 19, 2020 at 2:44