Let $X$ be a proper smooth algebraic variety (or algebraic space) over an algebraically closed field of characteristic $p.$ By Chow's lemma, there exists a projective variety $Z$ and a projective birational morphism $Z\to X.$ Of course $Z$ may not be smooth anymore. My question is,

Could we make $X$ projective by a finite number of blow-ups?

In the standard proof of Chow's lemma, no blow-up is used. Both Chow's lemma and resolution of singularities apply to much more general situation, and I wonder if blow-ups is enough in this special case (that $X$ proper smooth). Since it would be a little optimal to require that the blow-up $\widetilde{X}$ is both projective and smooth (which would give the resolution of singularities in char. $p$ in this special case), I don't care if the subvarieties in $X$ which we blow-up are smooth or not. And the answer is yes (in a stronger way that $\widetilde{X}$ is also smooth) in characteristic 0, due to Moishezon.

  • $\begingroup$ shenghao, I'm confused. We have $Z \to X \to k$ which we know is projective. $X \to k$ is also proper and so separated. Does this not imply that $Z \to X$ is projective - and so a blow-up of some ideal? See Exercise 4.8(e) in Hartshorne. Do you intend that this blow-up should be a sequence of at some reduced centers or something? $\endgroup$ Jan 29 '11 at 17:07
  • $\begingroup$ The map Z --> X is indeed projective (I also mentioned it in the question). But I don't think every projective map is a sequence of blow-ups: e.g. a projective space bundle; not true even for projective birational maps: at least we need the inverse image of the locus where the map is not an isomorphism to be a divisor. The center being reduced or not is not an issue: as long as two ideals are "cofinal". $\endgroup$
    – shenghao
    Jan 29 '11 at 17:34
  • $\begingroup$ Dear Karl: sorry, I sort of misunderstood your comment. Maybe you were using Hartshorne II 7.17, which, I'm afraid, does not apply since the variety X I am considering is not quasi-projective. It'll be great if we could impose additional conditions on the centers. For instance in char. 0, one may take smooth centers (Moishezon). $\endgroup$
    – shenghao
    Jan 29 '11 at 18:49
  • $\begingroup$ shenghao, thanks, I didn't realize the problem was the quasi-projectivity. I was using II, 7.17 as you said, but I guess I hadn't realized that one needed the quasi-projectiveness (you can't glue the ideals you construct together for instance). $\endgroup$ Jan 29 '11 at 19:05
  • $\begingroup$ Actually, why doesn't the following work: $f : Z \to X$ is projective and birational and $U_i$ is a cover of $X$ with $V_i = f^{-1}(U_i)$. Each $f_i : V_i \to U_i$ is projective and birational and so the blow-up of some ideal $J_i \subset O_{U_i}$. We can find ideal an sheaf $I_i \subset O_{X}$ which restricts to $J_i$. Then consider the blow-up of $I = \prod I_i$. This should be the same as the blow-up of $f^{-1}(I) \cdot \O_Z$ on $Z$. $\endgroup$ Jan 29 '11 at 19:11

So here's what I think should work (it is based on my comments above). I'm going to assume for simplicity that the variety means irreducible (just to avoid some silly complications, it is not really needed).

Start with $f : Z \to X$ projective and birational (as discussed above, $Z$ is contructed via Chow's Lemma). We don't know that $X$ is quasi-projective, so we can't apply Hartshorne II, 7.11 and thus immediately argue that $Z$ is itself a blow-up.

Set $U_i$ to be an open affine (or even quasi-projective) cover of $X$ and fix $V_i = f^{-1}(U_i)$. On each chart $U_i$, set $J_i$ to be an ideal sheaf such that $f_i : V_i \to U_i$ is the blow-up of $J_i$.

For each $i$, fix $I_i$ to be a (generically non-zero) ideal sheaf on $X$ such that $I_i |_{U_i} = J_i$.

Now fix $I = \prod_i I_i$. This is an ideal sheaf on $X$. On each open chart $U_i$, it is equal to $J_i \cdot \prod_{j \neq i} I_j|_{U_i}$.

Set $\pi : Y \to X$ to be the blow-up of $I$. We wish to show that $\pi$ factors through $f$, and is in fact a blow-up of $Z$, and so $Y$ is indeed projective. Set $W_i = \pi^{-1}(U_i)$.

All schemes involved are separated, and so to verify that $Y \to X$ actually factors through $Z \to X$, it is sufficient to work locally, so work on a chart $U_i$.

There we are comparing the blow-up of $J_i$ with the blow-up of $J_i \cdot \prod_{j \neq i} I_j|_{U_i} = J_i \cdot K_i$. $V_i$ is the blow-up of $J_i$ and $W_i$ is the blow-up of $J_i \cdot K_i$. However, I claim it is a straightforward exercise to verify that $W_i$ is the same as the blow-up of the ideal sheaf $(J_i \cdot K_i) \cdot \mathcal{O}_{V_i}$.
Let me give a hint as to how to do this.

Set $R = \Gamma(U_i, \mathcal{O}_X)$, set $J_i = (x_1, \dots, x_n)$ and $K_i = (y_1, \dots, y_m)$. Then the blow-up of $J_i$ is covered by affine charts $U_{i,l} = \text{Spec} R[x_1/x_l, \dots, x_n/x_l]$ where the gluings are the obvious ones. Likewise the blowup of $J_i \cdot K_i$ is covered by charts $U_{i,s,t} = \text{Spec} R[(x_1y_1)/(x_s y_t), \dots, (x_1y_m)/(x_s y_t), \dots, (x_ny_m)/(x_s y_t) ]$. Which are easily checked to be the blow-ups of $U_{i,s}$ at $(J_i \cdot K_i) \cdot R[x_1/x_l, \dots, x_n/x_l]$ (unless I've completely forgotten how to do this).

This sort of computation should be viewed as a generalization of the fact that the blow-up of an ideal is the same as the blow-up of a power of an ideal.

But this proves everything I claimed, right? $Y$ is exactly the blow-up of $Z$ at $I \cdot \mathcal{O}_Z$, and so $Y$ is projective, and is the blow-up of some ideal sheaf.

Of course, writing this as a sequence of blow-ups at subvarieties (and not subschemes/ideals) is probably much much harder.

  • $\begingroup$ The fact that $W_i$ is the blow-up of $(J_iK_i)O_{V_i}$ can also be done with the universal property of blow-ups, see Raynaud-Gruson: <i>Techniques de platification d'un module</i>, Remarque 5.1.2(v). So in fact $Y\to X$ can factor into $Y\to X_i\to X$ where $X_i\to X$ is the blow-up of $X$ along $I_i$. $\endgroup$
    – Qing Liu
    Feb 7 '11 at 10:34

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