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A normal singularity $(X,x)$ over a field $k$ is terminal (resp. canonical) if
$(i)$ it it is $\mathbb{Q}$-Gorenstein. and
$(ii)$For any resolution of singularity $F:Y\rightarrow X$, $K_Y-f^*K_X>0$(resp. $\geqq 0$)

My question is as follows.
Let $\pi:Bl_x(X)\rightarrow X$ be a blow up of $X$ at $x$.

1.Then is $Bl_x(X)$ terminal (canonical)?
2. In the case 1 does not hold, is there exist a normal variety $Y$ which has at most terminal ( canonical) singularity and birational morphism $\pi :Y\rightarrow X$ s.t the pullback of the ideal $m_{X,x}O_Y$ is invertiblle .

2 always hold if characteristic of k is zero because there is always resolution of singularity.

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    $\begingroup$ I assume 1. does not hold but I'd have to think hard to find an example. 2. is open outside of when we know resolution of singularities holds (ie, char 0 or dim $\leq 3$). If you can weaken $\pi$ birational to $\pi$ and alteration (I assume you want $\pi$ to be proper/projective) then you can make $Y$ nonsingular. $\endgroup$
    – Karl Schwede
    Commented Aug 1 at 3:25
  • $\begingroup$ That's the first time I've heard about alteration . Thank you. $\endgroup$
    – George
    Commented Aug 1 at 10:35

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The origin of the hypersurface defined by $$x_0^2 + x_1^4 + x_2^4 + \cdots + x_n^4 = 0$$ is a canonical singularity for $n = 3$, and a terminal singularity for $n \ge 4$ (see e.g. Theorem 2 in this paper). The blowup at the origin is defined by $$x_0^2 + x_1^2(1 + x_2^4 + \cdots x_n^4) = 0$$ in some local chart, which is singular in codimension $1$ and therefore non-normal.

The case $n=3$ is already mentioned in Miles Reid's "Canonical 3-folds".

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