Log resolutions on surfaces and 3-folds in characteristic p

2011-10-24T14:03:17Z

If $X$ is a normal projective variety and $D$ a divisor in it, we say that $\pi\colon (\widetilde X,\widetilde D)\rightarrow (X,D)$ is a log resolution if $\widetilde X$ is a resolution of $X$, the strict transfor $\widetilde D$ of $D$ is non-singular and $\widetilde D \cup { \text{exceptional divisors} }$ have simple normal crossings.

I know little about techniques for resolution of singularities and as far as I am aware, for varieties over algebraically closed fields, the problem of finding resolutions of singularities is open.

However, I was wondering if the following is solved, by whom and if someone can provide me with a 'black-box' reference:

Question: Given a projective variety $X$ over a field of characteristic $p$ and a divisor $D$ on $X$, is there a log resolution of the pair $(X,D)$ in the cases where $X$ is a non-singular variety of small dimension (1,2,3) and/or in the case the $p\neq 2,3,5\ldots$? What if the variety is a product of a projective variety and the affine line?

Of course partial answers are appreciated. However the purpose of this is just to use it in a birational proof for something else, so by no means I intend to prove it myself or get any close to it.

Answer by ulrich for Log resolutions on surfaces and 3-folds in characteristic p

2011-10-25T09:34:08Z

What you want does follow from Cutkosky's paper cited by Donu Arapura (and presumably already from results of Abhyankar, but I have not checked). One just has to combine his Theorems 1.1 and 1.2. More precisely, one can resolve singularities of $X$ using Theorem 1.1 to get $\pi_1: X_1 \to X$ with $X_1$ smooth and then apply Theorem 1.2 to the pair $(V,S) = (X_1,D_1)$, where $D_1$ is the union of $\pi_1^{-1}(D)$ and the exceptional divisors.

This requires that the characteristic be $>5$, but if you use the result of Cossart and Piltant instead of Cutkosky's Theorem 1.1 you can get a log resolution in any characteristic (since Cutkosky's Theorem 1.2 has no characteristic restriction).

Log resolutions on surfaces and 3-folds in characteristic p - MathOverflow

Log resolutions on surfaces and 3-folds in characteristic p

Answer by ulrich for Log resolutions on surfaces and 3-folds in characteristic p