Moduli of $\mathbb{Q}$-Gorenstein pairs and applications

The first part of this work is devoted to developing the notion of $\mathbb{Q}$-stable pair and extending several statements from pairs to $\mathbb{Q}$-stable pairs. In particular, in § 2 we set the key definitions and properties, while in § 3 we extend the boundedness results of [HMX18] to the context of $\mathbb{Q}$-stable pairs.

In § 4 and § 5, we analyze the moduli functor $\mathscr{F}_{n,p,I}$. In particular, in § 4 we show that $\mathscr{F}_{n,p,I}$, which a priori is only a category fibered in groupoids over $\mathrm{Sch}/k$, is a Deligne–Mumford stack. Then, in § 5 we show that $\mathscr{F}_{n,p,I}$ is proper. Thus, § 4 and §  5 settle Theorem 1.2, except for the projectivity part.

In § 6, we analyze the notion of family of $\mathbb{Q}$-stable pairs when the base is reduced. Under these assumptions, we show that the existence of a relative good minimal model is determined by a fiberwise condition.

In § 7 we conclude our work. We use the ampleness lemma to show that the coarse space of $\mathscr{F}_{n,p,I}$ is projective, and we use the results from § 6 to descend the analogous statement to the moduli space of stable pairs.

One of the main difficulties to extend the moduli theory from stable varieties to stable pairs lays in the behavior of the boundary divisor in families. Indeed, given a pair $(X,D)$, $K_{X}$ and $D$ need not be $\mathbb{Q}$-Cartier. In this case, we cannot expect a deformation of $(X,D)$ to induce a flat deformation of $D$, so one cannot define the moduli problem by simply requiring that the divisor varies flatly in families. A final and satisfactory notion for defining families of divisors has been achieved in [kol19s] with the notion of K-flatness.

Over the years, there have been attempts to overcome these difficulties in some special situations where the deformations of $(X,D)$ do induce flat deformations of $D$. In [Ale06, Ale15], Alexeev considered pairs $(X,\Delta=\sum_{i=1}^{l}a_{i}D_{i})$, where each $D_{i}$ is prime and the coefficients $a_{i}$ are irrational and $\mathbb{Q}$-linearly independent. This assumption can be thought as a “general choice” of the coefficients of $D$, as opposed to having rational coefficients. Indeed, many of the examples, it can be showed that, as we vary the coefficients of the boundary, the behavior of the moduli space is locally constant, leading to a chamber decomposition of some appropriate polytope contained in $[0,1]^{l}$. These chambers have rational vertices, determined by the conditions of $(X,D)$ being semi-log canonical and of $K_{X}+D$ being ample. For more details on this topic, we refer to [ABIP], and some examples of this phenomena are illustrated in [Ale15, inc20, AB21].

In a different direction, Kollár showed in [Kol14] that, if the coefficients of $D$ are strictly greater than $\frac{1}{2}$, deformations of $(X,D)$ lead to flat deformations of $D$. Thus, even though $D$ is not necessarily $\mathbb{Q}$-Cartier, this allows for a satisfactory moduli theory that does not rely on the latest developments in [kol19s].

In this paper, we work with rational coefficients and impose that the boundary $D$ is $\mathbb{Q}$-Cartier. As we do not require the coefficients to be irrational, in order to retain the properness of the moduli problem, we trade the ampleness of $K_{X}+D$ for the flatness of the deformations of $D$. Thus, while the deformations of the boundary are easier to understand than the general framework of [kol19s], $K_{X}+D$ is in general only big and semi-ample.

In [kol_new]*§ 8.3, Kollár presents a more general version of the approach pursued in this work. In particular, he considers arbitrary coefficients of $D$ and allows for independent perturbations of different components of $D$. In this way, he reconciles Alexeev’s work with our work, thus showing that the case of “$D$ with rational coefficients and $K_{X}+D$ big and semi-ample” we consider can be thought of as a limit case of the “general coefficients with $K_{X}+D$ ample” considered by Alexeev.

Similarly, we remark that many of the ideas in the current paper fit in the general formalism of stable minimal models, developed by Birkar (see for example [birkar2021boundedness]), culminating in [birkar2022moduli], where he develops a moduli theory of varieties and pairs of non-negative Kodaira dimension. We refer the reader to Definition 2.7 and [birkar2021boundedness]*Definition 1.8 for the similarities between the definitions of $\mathbb{Q}$-stable pair and stable minimal models. In particular, we observe that both definitions entail a stability condition prescribed by a polynomial, which guarantees the boundedness of the moduli problem, thus taking inspiration from ideas of Viehweg [Vie95].

We thank János Kollár for pointing to us a mistake in an earlier version of this work, for sharing the latest draft of his book on moduli, for providing helpful comments to improve our work, and for providing feedback on an eralier version of this work. We thank Jarod Alper, Dori Bejleri, Christopher Hacon, Sándor Kovács, Joaquín Moraga, and Zsolt Patakfalvi for helpful discussions. We thank the anonymous referees for helpful suggestions.

Throughout this paper, we will work over the field of complex numbers. For the standard notions in the Minimal Model Program (MMP) that are not addressed explicitly, we direct the reader to the terminology and the conventions of [KM98]. Similarly, for the relevant notions regarding non-normal varieties, we direct the reader to [Kol13]. A variety will be an integral separated scheme of finite type over $\mathbb{C}$. A birational morphism between non-normal schemes will be a morphism $f\colon X\to Y$ with a dense open subset $U\subseteq Y$ such that $f^{-1}(U)$ is dense and $f^{-1}(U)\to U$ is an isomorphism.

A contraction is a projective morphism $f\colon X\rightarrow Z$ of quasi-projective varieties with $f_{\ast}\mathcal{O}_{X}=\mathcal{O}_{Z}$. If $X$ is normal, then so is $Z$.

Let $\mathbb{K}$ denote $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$. We say that $D$ is a $\mathbb{K}$-divisor on a variety $X$ if we can write $D=\sum_{i=1}^{n}d_{i}P_{i}$ where $d_{i}\in\mathbb{K}\setminus\{0\}$, $n\in\mathbb{N}$ and the $P_{i}$ are prime Weil divisors on $X$ for all $i=1,\ldots,n$. We say that $D$ is $\mathbb{K}$-Cartier if it can be written as a $\mathbb{K}$-linear combination of $\mathbb{Z}$-divisors that are Cartier. The support of a $\mathbb{K}$-divisor $D=\sum_{i=1}^{n}d_{i}P_{i}$ is the union of the prime divisors appearing in the formal sum $\mathrm{Supp}(D)=\sum_{i=1}^{n}P_{i}$.

In all of the above, if $\mathbb{K}=\mathbb{Z}$, we will systematically drop it from the notation.

Given a prime divisor $P$ in the support of $D$, we will denote by $\mu_{P}(D)$ the coefficient of $P$ in $D$. Given a divisor $D=\sum\mu_{P_{i}}(D)P_{i}$ on a normal variety $X$, and a morphism $\pi\colon X\to Z$, we define

Let $D_{1}$ and $D_{2}$ be divisors on $X$. We write $D_{1}\sim_{\mathbb{K},Z}D_{2}$ if there is a $\mathbb{K}$-Cartier divisor $L$ on $Z$ such that $D_{1}-D_{2}\sim_{\mathbb{K}}f^{\ast}L$. Equivalently, we may also write $D_{1}\sim_{\mathbb{K}}D_{2}/Z$, or $D_{1}\sim_{\mathbb{K}}D_{2}$ over $Z$. If $\mathbb{K}=\mathbb{Z}$, we omit it from the notation. Similarly, if $Z=\mathrm{Spec}(k)$, where $k$ is the ground field, we omit $Z$ from the notation.

Let $\pi\colon X\rightarrow Z$ be a projective morphism of normal varieties. Let $D_{1}$ and $D_{2}$ be two $\mathbb{K}$-divisors on $X$. We say that $D_{1}$ and $D_{2}$ are numerically equivalent over $Z$, and write $D_{1}\equiv D_{2}/Z$, if $D_{1}.C=D_{2}.C$ for every curve $C\subset X$ such that $\pi(C)$ is a point. In case the setup is clear, we just write $D_{1}\equiv D_{2}$, omitting the notation $/Z$.

There are two important generalizations of the notion of normal variety.

Roughly speaking, the notion of demi-normal schemes allows extending the notion of log canonical singularities to non-normal varieties, allowing for a generalization of the notion of stable curve to higher dimensions. We refer to [kol_new]*§ 10.8 for more details.

This is an auxiliary lemma, it is probably well known. We include it for completeness. We refer to [kol_new]*§ 10.8 for the details about seminormality.

Throughout this section, $X$ will be $S_{2}$ and reduced. We begin this subsection with the following definition:

Now, let $X$ be a demi-normal scheme, and let $\mathrm{Weil}^{*}(X)$ denote the subgroup of $\mathrm{Weil}(X)$ generated by the prime divisors that are not contained in the conductor of $X$. Then, there is an identification between $\mathrm{Weil}^{*}(X)/\sim$ and the group of divisorial sheaves, where $\sim$ denotes linear equivalence. The identification is defined as follows. By the definition of $\mathrm{Weil}^{*}(X)$ and the demi-normality of $X$, for every element $B\in\mathrm{Weil}^{*}(X)$, there is a closed subset $Z\subset X$ of codimension at least 2 such that $B|_{X\setminus Z}$ is a Cartier divisor. Then, the corresponding divisorial sheaf is defined as $j_{*}\mathcal{O}_{X\setminus Z}(B|_{X\setminus Z})$, where we have $j\colon X\setminus Z\rightarrow X$.

Consider a flat family $f\colon X\to B$ with $S_{2}$ fibers, and assume that there is an open subset $i\colon U\subseteq X$ such that $U_{b}$ is big for every $b\in B$. Then, for every locally free sheaf $\mathcal{G}$ on $U$, we can consider $i_{*}\mathcal{G}$ on $X$. From [HK04]*Corollary 3.7, this is a reflexive sheaf on $X$ (a priori, it is not $S_{2}$ relatively to $B$, see § 2.11).

If $X$ is demi-normal, its canonical sheaf is a divisorial sheaf, as $X$ is Gorenstein in codimension 1. By the above identification, we can then choose a canonical divisor $K_{X}$ such that $\mathcal{O}_{X}(K_{X})\cong\omega_{X}$. Observe that this construction can be carried out in families. Indeed, if $f\colon X\to B$ is a flat morphism with demi-normal fibers of dimension $n$, there is an open locus $i\colon U\subseteq X$ that has codimension 2 along each fiber, on which $f$ is Gorenstein. Then we can define $\omega_{X/B}\coloneqq i_{*}\omega_{U/B}$. Observe that this agrees with the $(-n)$-th cohomology of the relative dualizing complex. Indeed, the latter is $S_{2}$ (see [LN18]*§ 5).

Let $X$ be demi-normal, and consider two divisorial sheaves $L_{1}$ and $L_{2}$. Then, their reflexive tensor product is defined as $L_{1}\hat{\otimes}L_{2}\coloneqq(L_{1}\otimes L_{2})^{**}$ and it is a divisorial sheaf itself. If we have $L_{1}\cong\mathcal{O}_{X}(D_{1})$ and $L_{2}\cong\mathcal{O}_{X}(D_{2})$, we have $L_{1}\hat{\otimes}L_{2}\cong\mathcal{O}_{X}(D_{1}+D_{2})$. The $m$-fold reflexive power $L^{[m]}$ is defined as the $m$-fold self reflexive tensor product.

For more details, we refer to [kol_new]*§ 3.3, [Kol13]*5.6 and [HK04].

Let $\mathfrak{D}$ be a set of projective pairs. Then, we say that $\mathfrak{D}$ is log bounded (resp. log birationally bounded) if there exist a variety $\mathcal{X}$, a reduced divisor $\mathcal{B}$ on $\mathcal{X}$, and a projective morphism $\pi\colon\mathcal{X}\rightarrow T$, where $T$ is of finite type, such that $\mathcal{B}$ does not contain any fiber of $\pi$, and, for every $(X,B)\in\mathfrak{D}$, there are a closed point $t\in T$ and a morphism (resp. a birational map) $f_{t}\colon\mathcal{X}_{t}\rightarrow X$ inducing an isomorphism $(X,\operatorname{Supp}(B))\cong(\mathcal{X}_{t},\mathcal{B}_{t})$ (resp. such that $\operatorname{Supp}(\mathcal{B}_{t})$ contains the strict transform of $\operatorname{Supp}(B)$ and all the $f_{t}$ exceptional divisors).

A set of projective pairs $\mathfrak{D}$ is said to be strongly log bounded if there is a quasi-projective pair $(\mathcal{X},\mathcal{B})$ and a projective morphism $\pi\colon\mathcal{X}\rightarrow T$, where $T$ is of finite type, such that $\operatorname{Supp}(\mathcal{B})$ does not contain any fiber of $\pi$, and for every $(X,B)\in\mathfrak{D}$, there is a closed point $t\in T$ and an isomorphism $f\colon X\rightarrow\mathcal{X}_{t}$ such that $f_{*}B=\mathcal{B}_{t}$.

A set of projective pairs $\mathfrak{D}$ is effectively log bounded if it is strongly log bounded and we may choose a bounding pair $\pi\colon(\mathcal{X},\mathcal{B})\rightarrow T$ such that, for every closed point $t\in T$, we have $(\mathcal{X}_{t},\mathcal{B}_{t})\in\mathfrak{D}$.

Given a finite subset $I\subseteq\mathbb{Q}^{n}$, we define the index of $I$ to be the smallest positive rational number $r$ such that $rI\subseteq\mathbb{Z}^{n}$.

Let $(X,D)$ denote a projective semi-log canonical pair, where $D$ has coefficients in $\mathbb{Q}$. We say that $(X,D)$ is a stable pair if $K_{X}+D$ is ample.

For brevity, we denote the datum of a $\mathbb{Q}$-pair by $(X,\Delta;D)$, where the polynomial $p(t)$ and the set of coefficients $I$ are omitted in the notation. In case $\Delta=0$, we then write $(X;D)$.

The following lemma is the main technical tool that we will use in § 3 and § 5:

Observe in particular that from Lemma 2.13, the morphism $\pi\colon X\to Y$ induces an isomorphism at the generic point of the codimension one singular locus of $X$ and $Y$, as those are points not contained in $D$, and $\pi$ is an isomorphism away from the exceptional locus. So $X$ is normal if and only if its codimension one singular locus is empty and if and only if the one of $Y$ is empty.

We recall the main definitions of families of pairs from [kol_new]*Ch. 4 and [kol19s]. A family of pairs $f\colon(X,D)\rightarrow S$ over a reduced base $S$ is the datum of a morphism $f\colon X\rightarrow S$ and an effective $\mathbb{Q}$-divisor $D$ on $X$, such that the following conditions hold:

• •

  $f$ is flat with reduced fibers of pure dimension $n$;

• •

  the fibers of $\operatorname{Supp}(D)\rightarrow S$ are either empty or of pure dimension $n-1$; and

• •

  $f$ is smooth at the generic points of $X_{s}\cap\operatorname{Supp}(D)$ for every $s\in S$.

Furthermore, we say that a family of pairs is well defined if it also satisfies the following property:

• •

  $mD$ is Cartier locally around the generic point of each irreducible component of $X_{s}\cap\operatorname{Supp}(D)$ for every $s\in S$, where $m\in\mathbb{N}$ is a sufficiently divisible natural number clearing the denominators of $D$.

This latter condition guarantees that $mD$ is Cartier on a big open set $U\subset X$ with the property that $U\cap X_{s}$ is a big open set of $X_{s}$ for every $s\in S$. This guarantees that we have a well-defined notion of pull-back of $D$ under any possible base change $S^{\prime}\rightarrow S$, as we can pull back $mD|_{U}$, take its closure in $X\times_{S}S^{\prime}$, and then divide the coefficients by $m$.

There is a more general definition of families of divisors, over possibly non-reduced bases due to Kollár in [kol19s]. We will not report it here since we will not need it, we refer the interested reader to loc. cit.

A well defined family of pairs $f\colon(X,D)\rightarrow S$ over a reduced base $S$ is called locally stable if, for every base change $g\colon(X_{C},D_{C})\rightarrow C$ where $C$ is the spectrum of a DVR with closed point $0$, $(X_{C},D_{C}+X_{0})$ is a semi-log canonical pair. Then, a family $f\colon(X,D)\rightarrow S$ is called stable if it is locally stable, $f$ is proper, and $K_{X/S}+D$ is $f$-ample.

Let us fix a positive integer $n$, a polynomial $p(t)\in\mathbb{Q}[t]$, and a finite set of coefficients $I\subset(0,1]\cap\mathbb{Q}$. Let $r$ be the index of $I$, see § 2.7. Before we introduce our main functor, we discuss some applications of the abundance conjecture that are relevant for this paper.

The condition (K) on commutativity with base change in Definition 2.19 is usually referred to as Kollár’s condition.

We need an additional condition to prove that $\mathscr{F}_{n,p,I}^{\prime}$ is representable in full generality. Indeed, while proving representability if one restricts itself to the category of reduced schemes (i.e., if one is only interested in families over a reduced base) essentially follows from [HX13], to prove that $\mathscr{F}_{n,p,I}^{\prime}$ is representable in general we will need some version of assumption (A) to hold in families. If one works with moduli of surfaces or threefolds, then the Abundance Conjecture is known and there are no issues. Otherwise one possible solution, which was suggested to us by Kollár, is to only consider families $(\mathcal{X};\mathcal{D})\to B$ such that “assumption (A) holds in families”. The advantage of this approach is that the functor can be defined in any dimension unconditionally to the Abundance Conjecture.

The main advantage of dealing with assumption (A) in this way is proved [kol_new]*Proposition 8.36, which he kindly shared with us, where he proves that imposing assumption (A) in families is a constructible condition:

We will use this result for constructing $\mathscr{F}_{n,p,I}$, and to show that it is bounded.

Now, we add a series of remarks and technical statements that are relevant in this context. We keep the notation of Definition 2.19.