# Birational Invariants of regular surfaces

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 $$X \times Y$$.

• Certainly $\Gamma$ is proper and birational to $X$ and $Y$. There is no reason for it to be regular any more, but you can resolve its singularities and then replace it by the resolution. – Bort May 28 '19 at 9:03
• Just to be clear: regular for you means smooth, right? Because usually, in surface theory, the term regular means with vanishing $q(X)$. – Francesco Polizzi May 28 '19 at 9:34
• @FrancescoPolizzi: I assume from context that regular here means that all local rings are regular (not anything to do with irregularity). Since $k$ is not assumed to be algebraically closed, this is a bit weaker than smooth. – Bort May 28 '19 at 10:29
• yes in this thread I'm working with definition of regularity as Bort explained above. will update this in my question – KarlPeter May 28 '19 at 11:11
• $\Gamma$ is proper because it is a closed subset of the proper variety $X \times Y$. It is birational to $X$ because if $U$ is the open subset where $f$ is defined, the map $U \mapsto \Gamma_f$ given by $x \mapsto (x,f(x))$ is an isomorphism. – Bort May 28 '19 at 12:41

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.