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

- $Y$ is an excellent normal Cohen–Macaulay scheme that admits a dualizing complex; and
- 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.

regularfor you meanssmooth, right? Because usually, in surface theory, the termregularmeanswith vanishing $q(X)$. $\endgroup$ – Francesco Polizzi May 28 '19 at 9:34regularhere 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 thansmooth. $\endgroup$ – Bort May 28 '19 at 10:294more comments