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Francesco Polizzi
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Some examples of $\mathbb Q$-Gorenstein smoothing

I am trying to understand $\mathbb Q$-Gorenstein smoothings, and especially the third condition in the following definition.

Definition. For a normal projective surface $X$ with quotient singularities, a $\mathbb Q$-Gorenstein smoothing is a one-parameter flat family of projective surfaces $\psi \colon \mathcal X \rightarrow \Delta$ over a small disk $\Delta$, which satisfies the following three conditions:

  1. the general fibre $X_t$ is a smooth projective surface,
  2. the central fibre $X_0$ is isomorphic to $X$,
  3. the canonical divisor $K_{\mathcal X / \Delta}$ is $\mathbb Q$-Cartier.

We say that $X'$ is a $\mathbb Q$-Gorenstein smoothing of $X$ if there exists such a family $\psi \colon \mathcal X \to \Delta $ as above such that $X'=\psi^{-1}(t)$ for some $t\in \Delta$.

For example, I am trying to understand the following.

Question 1. Assume that there exists a fixed integer $m$ such $mK_{X_t}$ is Cartier for every $t$. Is there any example where 3rd condition is not satisfied?

Question 2. If the total space of the family is $\mathbb Q$-Gorenstein, is it true that the 3rd condition is satisfied?

References with examples would be appreciated.

SAG
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