We say that a set of varieties $S$ lives in a bounded family if there exists a projective morphism $\mathcal{X} \to T$ between varieties of finite type, such that for any $X \in S$, there exists a closed point $t \in T$, such that its fibre $\mathcal{X}_t$ is isomorphic to $X$.

It seems that the following two facts related to bounded families used quite often without proof in the literature of birational geometry:

(1) If $S$ is a bounded family of $\mathbb{Q}$-Gorenstein varieties, then there exists a universal $m$, such that for any $X \in S$, $mK_X$ is a Cartier divisor, i.e. the Gorenstein index is bounded.

(2) Granted (1) is true, then the volume (suppose $n=\dim X$) $${\rm vol}(mK_X): = \lim_{k \to +\infty}\frac{n! h^0(X, kmK_X)}{k^n}$$ is bounded for any $X \in S$.

If I know the universal family $\mathcal{X}$ is $\mathbb Q$-Gorenstein, then the above two results follows easily. But I don't know if we can assume this because such universal families typically come from Hilbert schemes (I don't know if Hilbert schemes are $\mathbb Q$-Gorenstein or not). The closest result related (1) which I can find is Theorem B.1 in the paper Log canonical thresholds on varieties with bounded singularities. But it requires more than what I have: the fibres are required to be normal.

However, I do think one needs something extra which natural comes from the construction of bounded family in order to make (1) and (2) holds. Any suggestion?

We say that a set of varieties $S$ lives in a bounded family if there exists a projective morphism $\mathcal{X} \to T$ between varieties of finite type, such that for any $X \in S$, there exists a closed point $t \in T$, such that its fibre $\mathcal{X}_t$ is isomorphic to $X$.

It seems that the following two facts related to bounded families used quite often without proof in the literature of birational geometry:

(1) If $S$ is a bounded family of $\mathbb{Q}$-Gorenstein varieties (we can assume the elements in $S$ are normal, have log terminal singularities), then there exists a universal $m$, such that for any $X \in S$, $mK_X$ is a Cartier divisor, i.e. the Gorenstein index is bounded.

(2) Granted (1) is true, then the volume (suppose $n=\dim X$) $${\rm vol}(mK_X): = \lim_{k \to +\infty}\frac{n! h^0(X, kmK_X)}{k^n}$$ is bounded for any $X \in S$.

If I know the universal family $\mathcal{X}$ is $\mathbb Q$-Gorenstein, then the above two results follows easily. But I don't know if we can assume this because such universal families typically come from Hilbert schemes (I don't know if Hilbert schemes are $\mathbb Q$-Gorenstein or not). The closest result related (1) which I can find is Theorem B.1 in the paper Log canonical thresholds on varieties with bounded singularities. But it requires more than what I have: the fibres are required to be normal.

However, I do think one needs something extra which natural comes from the construction of bounded family in order to make (1) and (2) holds. Any suggestion?

We say that a set of varieties $S$ lives in a bounded family if there exists a projective morphism $\mathcal{X} \to T$ between varieties of finite type, such that for any $X \in S$, there exists a closed point $t \in T$, such that its fibre $\mathcal{X}_t$ is isomorphic to $X$.

It seems that the following two facts related to bounded families used quite often without proof in the literature of birational geometry:

(1) If $S$ is a bounded family of $\mathbb{Q}$-Gorenstein varieties, then there exists a universal $m$, such that for any $X \in S$, $mK_X$ is a Cartier divisor, i.e. the Gorenstein index is bounded.

(2) Granted (1) is true, then the volume (suppose $n=\dim X$) $${\rm vol}(mK_X): = \lim_{k \to +\infty}\frac{n! h^0(X, kmK_X)}{k^n}$$ is bounded for any $X \in S$.

If I know the universal family $\mathcal{X}$ is $\mathbb Q$-Gorenstein, then the above two results follows easily. But I don't know if we can assume this because such universal families typically come from Hilbert schemes (I don't know if Hilbert schemes are $\mathbb Q$-Gorenstein or not). The closest result related (1) which I can find is Theorem B.1 in the paper Log canonical thresholds on varieties with bounded singularities. But it requires more than what I have: the fibres are required to be normal and the general fibre to have rational singularities.

However, I do think one needs something extra which natural comes from the construction of bounded family in order to make (1) and (2) holds. Any suggestion?

We say that a set of varieties $S$ lives in a bounded family if there exists a projective morphism $\mathcal{X} \to T$ between varieties of finite type, such that for any $X \in S$, there exists a closed point $t \in T$, such that its fibre $\mathcal{X}_t$ is isomorphic to $X$.

It seems that the following two facts related to bounded families used quite often without proof in the literature of birational geometry:

(1) If $S$ is a bounded family of $\mathbb{Q}$-Gorenstein varieties, then there exists a universal $m$, such that for any $X \in S$, $mK_X$ is a Cartier divisor, i.e. the Gorenstein index is bounded.

(2) Granted (1) is true, then the volume (suppose $n=\dim X$) $${\rm vol}(mK_X): = \lim_{k \to +\infty}\frac{n! h^0(X, kmK_X)}{k^n}$$ is bounded for any $X \in S$.

If I know the universal family $\mathcal{X}$ is $\mathbb Q$-Gorenstein, then the above two results follows easily. But I don't know if we can assume this because such universal families typically come from Hilbert schemes (I don't know if Hilbert schemes are $\mathbb Q$-Gorenstein or not). The closest result related (1) which I can find is Theorem B.1 in the paper Log canonical thresholds on varieties with bounded singularities. But it requires more than what I have: the fibres are required to be normal.

However, I do think one needs something extra which natural comes from the construction of bounded family in order to make (1) and (2) holds. Any suggestion?