# Finiteness of Gorenstein indexes and volumes for varieties in a bounded family

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?

• Assuming that all S you consider is normal and with rational singularities, the you can use Theorem B.1 in the paper "Log canonical thresholds on varieties with bounded singularities". Because by normalizing the family $\mathcal{X}$, you can get a new family with all fibers normal. Commented Nov 12, 2016 at 11:51
• Sorry, but I don't understand. I can assume elements in $S$ is normal and have rational singularities. Since I can always resolve $\mathcal X$ and shrink $T$ such that the generic fibre is smooth. But don't have the property that for any $X \in S$, $X$ is isomorphic to a fibre of this family. So how to I apply Thm B.1? Commented Nov 13, 2016 at 3:07
• Can we assume something about the singularities of $\mathcal X_t$? eg. canonical? log terminal? Can I at least assume that the $\mathcal X_t$ are normal? BTW there is no harm in assuming that $T$ is normal, so if we assume $\mathcal X_t$ is normal, then after shrinking $T$ (which is ok by Noetherian induction) we can assume that $\mathcal X$ is normal (EGA IV, cor. 5.12.7). If moreover $\mathcal X_t$ has rational sings. Then we apply Thm b.1 as Chen says. Are you really interested in the case that the fibers are not normal? Commented Nov 14, 2016 at 17:01
• Yes, we can assume that over a dense set of the base $T$, $\mathcal{X}_t$ is normal and log terminal. But do you think under these assumptions, we can shrink $T$ further, such that $\mathcal{X}_T$ becomes normal? I only know that over a dense set of $T$, the fibre is normal. It seems that in order to apply EGA IV, cor. 5.12.7, one needs to know that for an open set, the fibre is normal... Commented Nov 15, 2016 at 2:18
• I',m confused. It seems to me that EGA IV, cor. 5.12.7 says that if $(A,m)$ is local, $A/tA$ is integrally closed then so is $A$. Since $f$ is projective, doesn't it follow that if a fiber $\mathcal X_t$ is normal, then there is an open subset $T^0\subset T$ such that $\mathcal X\times _T T^0$ is normal? The proof is sort of obvious: since $\mathcal X_t$ is $R_1$ then so is $\mathcal X$ (near $\mathcal X _t$), since $\mathcal X _t$ is $S_2$, then so is $\mathcal X$ (near $\mathcal X _t$). By serre's Criterion, $\mathcal X$ is normal (near $\mathcal X _t$). What did I miss? Commented Nov 15, 2016 at 3:51

By Noetherian induction, it suffices to show that indicies and volumes are bounded over an open subset of any irreducible component of $T$. We may assume that $T$ is smooth and there is a dense set $\{t_i\}\subset T$ such that the corresponding fibers $\mathcal X_{t_i}$ are normal. By EGA IV Theorem 12.2.4(iv), after shrinking $T$, we may assume that $\mathcal X$ is normal. Since $\mathcal X_{t_i}$ is log terminal, it is $\mathbb Q$-Gorenstein (by definition) and has rational singularities (see eg. ́Koll\'ar-Mori). By Theorem B.1 in the paper Log canonical thresholds on varieties with bounded singularities https://arxiv.org/pdf/1004.3336.pdf, $X$ is $\mathbb Q$-Gorenstein. Thus (1) holds. Let $\nu:\mathcal X'\to \mathcal X$ be a log resolution and write $K_{\mathcal X'}+B'=\nu ^* K_{\mathcal X}+E$ where $B'$ and $E$ are effective with no common components. Shrinking $T$ we may assume that $(\mathcal X',B')$ is log smooth over $T$ (so that every stratum of the support of $B'$ is smooth over $T$) and that $\mathcal X '_t\to \mathcal X _t$ is a log resolution for every $t\in T$. Note that since $\lfloor B'_{t_i}\rfloor =0$ then also $\lfloor B'\rfloor =0$. By Theorem 4.2 in https://arxiv.org/pdf/1412.1186.pdf, $h^0(mK_{\mathcal X _t})=h^0(m(K_{\mathcal X_{t}}+B_t))$ is independent of $t\in T$. In particular ${\rm vol}(K_{\mathcal X _t})$ is independent of $t\in T$. Thus (2) also holds.
• Thank you for your terrific answer!! May be the "$X$" in the fifth and sixth lines should be $\mathcal X$? Commented Nov 16, 2016 at 4:23