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I guess in general the answer is NO. For instance if you take $X$ to be an $n$ dimensional variety which is $Y\times E$, where $E$ is an elliptic curve and $Y$ is an $n-1$ dimensional variety of gen …

The answer is surely no. If $D$ itself does not have a minimal log smooth resolution, then certainly $(V,D)$ couldn't have such a log resolution you need. On the other hand, there are bunch of isolat …

I always believed the following statement: if $X$ is a smooth variety over an algebraically closed field of positive characteristic, assuming we know that the general member of a base point free linea …

The answer to the first question: if $X$ has klt singularities, then there exists a $\mathbb{Q}$-factorial variety $Y$ with a birational morphism $\pi: Y\to X$, such that if you write $\pi^*K_X=K_Y+\ …

I think this is true. Taking a point $x\in X$, and base change your setting up to its formal neighborhood, we can assume $K(X)\subset K(Y)$ is a Galois extension with finite abelian group. Now we can …

Let me add my answer. In characteristic 0. Abundance conjecture. This is probably universally accepted as the most important question for the current minimal model program after BHCM's proof of finit …

If you further assume $X$ only has canonical singularity, and $-K_X$ is ample, then for any dimension, this is proved in ACC for log canonical thresholds Corollary 1.8. If you only assume $-K_X$ is …

Under the assumption that $X$ is $\mathbb{Q}$-factorial, section 5 of the paper https://arxiv.org/pdf/math/0606666 addressed this issue, which was also proved earlier in Ambro's paper. Basically, if y …

In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log resolu …

In the surface case, MMP in char p is known. See Koll'ar-Kov'ac's preprint on Koll'ar's webpage. In dimensional 3, the existence of divisorial contractions and flipping contractions is known as EWM ( …

See the proof ON ISOLATED RATIONAL SINGULARITIES OF SURFACES (Artin) Propostiion 2 (i).