6. Resolution of singularities [039G]

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6. Resolution of singularities

For our applications, we need that regular projective models are cofinal in the categroy of models which makes it necessary to assume resolution of singularities in a certain dimension.

Definition 6.1.

Let $k$ be a field. We say that resolution of singularities holds over $k$ in dimension $n$ if for every quasi-projective variety $Y$ over $k$ of dimension $n$ there exists a regular variety $\tilde{Y}$ over $k$ and a projective morphism $\tilde{Y}\to Y$ which is an isomorphism over the regular locus of $Y$ .

To transfer results from [BFJ16a, BFJ15] to our context, it is essential to show that projective models are dominated by SNC-models. In order to this we are going to use the following assumption.

Definition 6.2.

We say that embedded resolution of singularities in dimension $m$ holds over a field $k$ if for every quasi-projective regular variety $Y$ over $k$ of dimension $m$ and every proper closed subset $Z$ of $Y$ , there is a projective morphism $\pi:Y^{\prime}\to Y$ of quasi-projective regular varieties over $k$ such that the set $\pi^{-1}(Z)$ is the support of a normal crossing divisor and such that $\pi$ is an isomorphism over $Y\setminus Z$ .

Hironaka has shown that resolution of singularities and embedded resolution of singularities holds over a field of characteristic zero in any dimension. Resolution of singularities holds over arbitrary fields in dimension one (Dedekind, M. Noether, Riemann) and in dimension two (Abhyankar, Lipman). Cossart and Piltant have proven that resolution of singularities and embedded resolution of singularities hold in dimension three over perfect fields.

Theorem 6.3 (Cossart-Piltant).

Resolution of singularities and embedded resolution of singularities hold in dimension three over any perfect field.

Proof.

This is shown in [CP09, Thm. on p. 1839] and [CP08, Prop. 4.1]. ∎