Birational Invariants of regular surfaces

Question

Let $X,Y$ surfaces (so $2$-dimensional proper $k$-schemes) which are regular (so the stalks are regular) and birational and denote by $f: X \dashrightarrow Y$ the corresponding rational birational morphism between $X$ and $Y$.

My aim is to show that for regular surfaces the dimensions of cohomology groups are birational invariants; therefore $$\dim_k H^i(X,O_X)= \dim_k H^i(Y,O_Y)$$ for $i=0,1$.

In order to do it I tried following consideration:

It is well know that if $b: B:=Bl_z(Z) \to Z$ is the blowing up of regular surface $Z$ at $z \in Z$, then the dimensions of cohomology groups are conserved; i.e. invariant.

If $f$ would be a "classical" morphism (so not only rational) then according to a factorization theorem (which one I don't know more; does anybody know it's name?) then $f$ factorize into a finite sequence of successively blowing ups

$$f: X= Y_n \to Y_{n-1} \to \dots \to Y_0=Y$$

where $Y_{k+1}= Bl_{Z_i}(Y_k)$ is the blow up of the previous one.

The problem here is that our $f$ is just a rational map.

My idea was to try to construct following diagram (D)

where $\Gamma := \overline{\Gamma_f} \subset X \times Y$ where $\Gamma_f$ also rational defined graph morphism via the pull back on $\operatorname{id}_Y:Y \to Y$ along $f$. This induces rational projection $\widetilde{pr_X}: \Gamma_f \dashrightarrow X$.

My goal would be that after taking closure $\Gamma := \overline{\Gamma_f} \subset X \times Y$ the rational morphism $\widetilde{pr_X}$ would induce "classical" map $pr_X:\Gamma \to X$ making the diagram (D) commutative. Then - if $pr_X,pr_Y$ are birational and $\Gamma$ regular - I can apply the factorization theorem to $pr_X,pr_Y$ and ontain the desired result.

And exactly this is the point: Which properties does this closure $\Gamma$ inherit? Stays it regular, proper and birational to $X,Y$? Why?

The problem is that I'm not sure what control over $\Gamma_f$ I have after taking the closure in $X \times Y$.

Answer 1

This is true pretty generally. We recall that a quasi-compact scheme $Y$ of finite Krull dimension has pseudo-rational singularities if

1. $Y$ is an excellent normal Cohen–Macaulay scheme that admits a dualizing complex; and
2. for every normal scheme $X$ and every projective birational morphism $f\colon X \to Y$, the trace map $f_*\omega_X \to \omega_Y$ is an isomorphism.

We then have the following:

Proposition. Let $k$ be field, and let $X$ and $Y$ be two proper $k$-schemes of pure dimension $d$ with pseudo-rational singularities. Suppose $X$ and $Y$ are birational over $k$. Then, $$\dim_k H^i(X,\mathcal{O}_X) = \dim_k H^i(Y,\mathcal{O}_Y)$$ for all $i$.

Proof. Let $f\colon X \dashrightarrow Y$ be the given birational map over $k$, and let $U$ be the subset of $X$ inducing a dense open embedding $f\rvert_U\colon U \hookrightarrow Y$. Denote the image of $f\rvert_U$ by $V$. We have two projection morphisms $$X \overset{\mathrm{pr}_1}{\longleftarrow} X \times_k Y \overset{\mathrm{pr}_2}{\longrightarrow} Y.$$ By the universal property of the fiber product, $f\rvert_U$ induces an open embedding $U \hookrightarrow X \times_k Y$. We denote by $\Gamma_f$ the closure of $U$ under this open embedding to $X \times_k Y$; note that $\Gamma_f$ is proper over $k$ since it is a closed subscheme of the proper scheme $X \times_k Y$. The projection morphisms $\mathrm{pr}_1\colon \Gamma_f \to X$ and $\mathrm{pr}_2\colon \Gamma_f \to Y$ are proper and birational, since they are morphisms of proper schemes over $k$, and since for $i \in \{1,2\}$, the morphism $\mathrm{pr}_i$ induces an isomorphism from the image of $U$ in $\Gamma_f$ onto $U$ and $V$, respectively.

We now consider a projective Macaulayfication [Kov, Cor. 4.5] $g\colon \widetilde{\Gamma}_f \to \Gamma_f$ of $\Gamma_f$, i.e., $g$ is a projective birational morphism and $\widetilde{\Gamma}_f$ is a Cohen–Macaulay projective $k$-scheme. Now [Kov, Thm. 8.6] implies the canonical morphisms $$\mathcal{O}_X \longrightarrow \mathbf{R}(\mathrm{pr}_1 \circ g)_* \mathcal{O}_{\widetilde{\Gamma}_f} \qquad\text{and}\qquad \mathcal{O}_Y \longrightarrow \mathbf{R}(\mathrm{pr}_2 \circ g)_* \mathcal{O}_{\widetilde{\Gamma}_f}\tag{1}\label{eq:qis}$$ are quasi-isomorphisms, which induce isomorphisms $$H^i(X,\mathcal{O}_X) \overset{\sim}{\longrightarrow} H^i(\widetilde{\Gamma}_f,\mathcal{O}_{\widetilde{\Gamma}_f}) \overset{\sim}{\longleftarrow} H^i(Y,\mathcal{O}_Y)$$ as required. $\blacksquare$

Note that regular schemes have pseudo-rational singularities [Kov, Lem. 7.7], hence the proposition above implies the special case you are interested in. One can also avoid using [Kov] when both $X$ and $Y$ are regular $k$-schemes of pure dimension $2$ as follows: Fix notation as in the first paragraph of the proof above. Instead of constructing $g \colon \widetilde{\Gamma}_f \to \Gamma_f$ as a projective Macaulayfication, we may define $g$ to be a resolution of singularities, in which case $\widetilde{\Gamma}_f$ is a regular projective $k$-scheme of pure dimension $2$, and $g$ is a projective birational morphism. Now [Stacks, Tag 0C5R] implies $g$ is a composition of blowups at closed points. Moreover, combining [Stacks, Tag 0AGS] and the strategy of [Har77, Prop. V.3.4] implies the quasi-isomorphisms \eqref{eq:qis} hold.

References

[Har77] Robin Hartshorne. Algebraic geometry. Grad. Texts in Math., Vol. 52. New York-Heidelberg: Springer-Verlag, 1977. DOI: 10.1007/978-1-4757-3849-0. MR: 463157.

[Kov] Sándor J. Kovács. "Rational singularities." May 11, 2018. arXiv:1703.02269v6 [math.AG].

[Stacks] The Stacks project authors. The Stacks project. 2019. https://stacks.math.columbia.edu.