# Rational singularities for fibered surfaces

This question consists of two parts. I will try to be as short and clear as possible.

Let $S$ be a Dedekind scheme of characteristic zero. The main examples are $\mathbf{P}^1_k$, with $k$ a field of characteristic 0 and $\textrm{Spec} \ \mathbf{Z}$.

A fibered surface is a projective flat morphism $X\longrightarrow S$ with $X$ an integral scheme of dimension 2.

Suppose $S=\mathbf{P}^1_k$ with $k$ a field of characteristic 0. Then $X$ is an algebraic surface over a field of characteristic zero. The theory of rational singularities for $X$ is then explained in Chapter 5 of Kollar and Mori: Birational geometry of algebraic varieties.

Now, I would like to know if there are analogous results for when $S=\textrm{Spec} \ \mathbf{Z}$ as in loc. cit.. Let me make this more precise.

Assume that $X$ is normal. (This will suffice for my applications.)

Call a resolution of singularities $\rho:Y\longrightarrow X$ rational if it satisfies

$$R^i\rho_\ast \mathcal{O}_Y =0 \ \textrm{for} \ i>0.$$

We say that $X$ has rational singularities if all resolutions of $X$ are rational.

To have a good theory, we should probably show that $R^i \rho_\ast \omega_{Y/S} = 0$ for $i>0$.

Question 1: Is it true that $R^i \rho_\ast \omega_{Y/S} = 0$ for $i>0$ for any resolution of singularities?

Compare my last question with Theorem 5.10 of Kollar-Mori.

Question 2: Is it true that the following properties

A. $X$ has a rational resolution

B. $X$ has rational singularities

C. For every resolution of singularities $\rho:Y\longrightarrow X$, we have that $\rho_\ast \omega_{Y/S} = \omega_{X/S}$.

are equivalent?

I guess one could try to see if some of the arguments given in Kollar-Mori carry over to this setting. For example, it probably suffices to show that a resolution is rational if and only if it satisfies C.

Here is a possible application: Assume $X$ to be regular from now on. Let $S$ be $\mathbf{P}^1_{\mathbf{C}}$. Let $\pi:Y\longrightarrow X$ be a finite surjective morphism with branch locus a normal crossings divisor. Then, it is easy to see that the singularities of $Y$ are quotient. (Consider the analytic topology or the etale topology and use Abhyankar's lemma). It is well-known that quotient singularities are rational.

I guess Abhyankar's lemma is still valid and therefore I was hoping to show that $Y$ in case $S= \textrm{Spec} \mathbf{Z}$ still has rational singularities. (The notion of quotient singularities seems only to work over $\mathbf{C}$...)

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Your $R^i \rho_{\ast} \omega_{X/S}$ must be a typo. Anyway, have you looked at Lipman's "Rational singularities", or considered to email Lipman? As for qt singularities seeming to only work in char. 0, there's a notion of "tame cyclic quotient singularities" valid over any dvr which exhibits many features in comment with char. 0 (e.g., minimal regular resolution related to the usual continued fraction stuff): see Def. 2.3.6 and section 2.4 of my paper with Edixhoven & Stein called "$J_1(p)$ has connected fibers". –  BCnrd Oct 6 '10 at 16:38
Indeed, that was a typo. I fixed it. I never considered email Lipman. I'll definitely consider that. Even if quotient singularities "make sense", does it remain true that they are rational? I will definitely look at your paper. –  Ari Oct 6 '10 at 18:07
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## 1 Answer

Ariyan,

EDIT: This contains some substantial edits and added references.

Lipman has defined the following notion (EDIT: twice):

Definition (Lipman ; Section 9 of "Rational singularities with applications to algebraic surfaces and factorization"): If $X$ is 2-dimensional and normal, $X$ has two pseudo-rational singularities if for every proper birational map $\pi : W \to X$ there exists a proper birational normal $Y$ over $W$ where, $R^1 \pi_* \mathcal{O}_Y = 0$

Definition (Lipman-Teissier ; Section 2 of "Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closures of ideals"): $X$ has pseudo-rational singularities if $X$ is CM (Cohen-Macaulay) and if for every proper birational map $\pi : Y \to X$ with $Y$ normal, $\pi_* \omega_Y = \omega_X$.

If these are the same in dimension 2, this seems pretty close to what you want in dimension 2.

EDIT2: These are the same in dimension 2, I was in Purdue and asked Lipman about question 1, which holds, and certainly implies this.

He also points out that regular schemes are pseudo-rational. In particular, this implies that if $\pi_* \omega_Y = \omega_X$ for one resolution of singularities, it also holds for every resolution of singularities (in fact, for every proper birational map with normal domain).

In dimension 2, he also studies relations between this condition and the local-finiteness of the divisor class group.

On the other hand, I'm pretty sure this is different from the definition of rational singularities you gave above at least in higher dimensions (with the appropriate $R^i$ vanishing instead of just $R^1$).

With regards to your specific questions:

Question #1: That vanishing, called Grauert-Riemenschneider vanishing, is known to fail for $\dim X > 2$ outside of equal characteristic zero. I believe the answer should hold in the two-dimensional case, certainly it should assuming that Lipman's various definitions of pseudo-rational singularities are consistent.

EDIT: This holds in dimension 2, see Theorem 2.4 in Lipman's "Desingularization of two-dimensional schemes".

In any dimension, that vanishing has recently been proven in equal characteristic $p > 0$ over a smooth variety (or a variety with tame quotient singularities), see arXiv:0911.3599.

Question #2: In higher dimensions, I'm pretty confident that the answer is no. In the 2-dimensional case, probably this is done by Lipman? In view of question #1, in order to find such a counter example in higher dimensions, one should look at various cones probably over 3 or 4-dimension schemes with negative Kodaira dimension (probably Fano's) but which fail Kodaira vanishing.

I have some thoughts on some other definitions of rational singularities which might be better in mixed characteristic, but I'm not sure I want to post them on MathOverflow right now. If you email me, I'd be willing discuss it a bit.

Quotient singularities can behave a little different outside of characteristic zero as well (see various papers of Mel Hochster from the 70s for instance). This can also lead one to look at questions like the Direct Summand Conjecture.

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