Birational Invariants of Regular Surfaces

Let $ X,Y$ surfaces (so $ 2$ -dimensional proper $ k$ -schemes) which are regular and birational and denote by $ f: X – \ – \to 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 … \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)

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

My goal would be that after taking closure $ \Gamma := \overline{\Gamma_f} \subset X \times Y$ the rational morphism $ \tilde{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$ .