On the geography of $3$-folds via asymptotic behavior of invariants

When working with slopes, it turns out that we can apply the tool of $n$-th root coverings to investigate their behavior. The main reason is that, as $n$ grows to infinity, the slopes erase the effects of the degree $n$ of the cover. This allows us to control the slopes from the base of the cover, and this is the main technique to get density results for surfaces

Consider the following data. Let $Z$ be a non-singular $d$-fold, and let $D_{1},\ldots,D_{r}$ be distinct prime divisors in $Z$. Assume that $D_{red}:=D_{1}+\ldots+D_{r}$ is a simple normal crossing divisor (SNC). Let $n>1$ be a positive integer, and let $0<\nu_{i}<n$ be a collection of $r$ integers coprime to $n$. Assume that there exists a line bundle ${\mathcal{L}}$ on $Z$ such that

$${\mathcal{L}}^{\otimes n}\simeq{\mathcal{O}}_{Z}\left(\sum_{i=1}^{r}\nu_{i}D_{i}\right).$$

(1)

Then, there exists an $n$-th root cover $h_{n}\colon Y_{n}\to Z$ branched along $D_{red}$, where $Y_{n}$ is a $d$-fold (Section 2.2). These are the $n$-th root covers developed by Esnault and Viehweg (Cf. [EV92]). For curves, the prime divisors $D_{1},\ldots,D_{r}$ are distinct points on $Z$. Since points are isolated, we have $Y_{n}$ as a non-singular curve. By the Riemann-Hurwitz formula, we have

$$\frac{c_{1}(Y_{n})}{n}=\frac{nc_{1}(Z)-(n-1)r}{n}=c_{1}(Z)-r+\frac{r}{n}=\bar{c}_{1}(Z,D)+\frac{r}{n}.$$

Here $\bar{c}_{1}(Z,D)$ is the first logarithmic Chern number of the pair $(Z,D_{red})$ (see Section 2.1). Hence, if we fix the points $D_{red}$ and we consider partitions $\nu_{1}+\ldots+\nu_{r}=n$ with $n\gg 0$ a prime number, then we asymptotically have $c_{1}(Y_{n})\approx n\bar{c}_{1}(Z,D)$.

In Section 3, we prove that this phenomenon occurs in any dimension for logarithmic morphisms of degree $n$ with branch locus as a disjoint collection of non-singular distinct prime divisors $D_{1},\ldots,D_{r}$. As an application, for $n$-root cover we have $Y_{n}$ as a non-singular projective variety, thus in Corollary 3.2 we get.

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However, this theorem is restrictive for us in terms of geography. It is not easy to get the necessary hypothesis to construct a minimal $d$-fold of general type. On the other hand, we do not drop the possibility of having applications in other contexts. Also, this is a cornerstone of our research and opens the following discussion.

For $\dim Z\geq 2$, if the branch divisor $D_{red}=D_{1}+\dots+D_{r}$ has singularities, then $Y_{n}$ have rational singularities [Vie77]. In order to have well-behaved invariants, we can choose a (partial) resolution of singularities. For $\dim Z=2$, the asymptoticity of Chern numbers was proved in [Urz09] for random $n$-th root covers. Let us explain briefly what random means (see Section 2.5 for more details). First, in dimension two, each singularity of $Y_{n}$ is a cyclic surface singularity of type $\frac{1}{n}(q,1)$ for some $0<q<n$. Thus, we use the Hirzebruch-Jung algorithm (see Section 2.4.1) to resolve these singularities in a minimal way. For us, there are two important quantities, (1) the length of the resolution, i.e., the number of steps of the algorithm, and (2) the Dedekind sums,

$$d(q,1,n)=\sum_{i=1}^{n-1}\left(\left(\frac{iq}{n}\right)\right)\left(\left(\frac{i}{n}\right)\right),$$

where $((\cdot)):{\mathbb{R}}\to{\mathbb{R}}$ is the saw-tooth function (see Section 2.4.2). We get a resolution of singularities $X_{n}\to Y_{n}$. However, the Chern numbers $c_{1}^{2}$ and $c_{2}$ depend on the lengths and the Dedekind sums coming from all cyclic singularities resolved. To guarantee asymptoticity, we have to consider asymptotic arrangements. Indeed, for each prime number $n\geq 17$, there exists a set $O_{n}\subset\{1,\ldots,n\}$ (Section 2.5) such that for each $q\in O_{n}$, the lengths and Dedekind sums are bounded by $c\sqrt{n}$ for a constant $c>0$. Thus we say that $D_{red}$ is an asymptotic arrangement if satisfies :

• •

  For prime numbers $n\gg 0$, there exists multiplicities $0<\nu_{j}<n$ such that the singularity over $D_{j}\cap D_{k}$ of $Y_{n}$ is of type $\frac{1}{n}(q_{jk},1)$ with $q_{jk}\in O_{n}$.

• •

  For each $n$ there are line bundles ${\mathcal{L}}\in\mbox{Pic}(Z)$ such that

  $${\mathcal{L}}^{\otimes n}\simeq{\mathcal{O}}_{Z}\left(\sum_{j=1}^{r}\nu_{j}D_{j}\right).$$

The main set-up is when there exists $H\in\mbox{Pic}(Z)$ such that $D_{j}\simeq H$ for each component of $D_{red}$. Observe that the condition (1) is satisfied as

$$D=\nu_{1}D_{1}+\ldots+\nu_{r}D_{r}\sim(\nu_{1}+\ldots+\nu_{r})H\sim nH,$$

thus we can construct $n$-th root covers. In [Urz09] was proved that for a random partition $\nu_{1}+\ldots+\nu_{r}=n$, the probability of $D$ being an asymptotic arrangement tends to $1$ as $n$ grows. In this way, for $n\gg 0$ we take random asymptotic partitions, and we get a family of random surfaces $X_{n}$ with

$$c_{1}^{2}(X_{n})\approx n\bar{c}_{1}^{2}(Z,D),\quad c_{2}(X_{n})\approx n\bar{c}_{2}(Z,D).$$

Now we can discard the SNC property for the branch divisor for those one having only normal crossings. We consider a $\log$ resolution $\gamma\colon Z^{\prime}\to Z$ such that the reduced divisor defined by $\gamma^{*}D_{red}$ is SNC. If such a divisor turns out to be an asymptotic arrangement, then we have the asymptotic result for Chern numbers as above. For this see [TN23, p. 57]), and see [Urz16] to connect this result with minimal models. A direct application is a relation between Chern slopes of simply connected surfaces of general type and Chern slopes of arrangements of lines [EFU22]. In this way, in higher dimensions, we have several issues with achieving an analog asymptotic result. For example, the singularities of $Y_{n}$ are not cyclic, and the choice of a right (partial) resolution of $Y_{n}$ with good behavior as $n$ grows is a challenging problem.

For instance, if this question has a positive answer, then we would be able to study the geography of $3$-folds using arrangements of planes in ${\mathbb{P}}^{3}$. The first result in this direction is in Section 4.1, where we find that the Chern number $c_{1}c_{2}=24\chi$ is asymptotic and independent of the chosen resolution.

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However, the canonical volume and the topological characteristic depend on the chosen resolution. In this way, the first issue is that the singularities of $Y_{n}$ are of order (multiplicity) $n^{2}$, too big. This means that to connect with a non-singular model, at a bad choice of resolution we could have big exceptional data with respect to $n$, and so we would lose the asymptotic property. In Section 4.2, we introduce a prototype of first step, i.e., by toric methods we construct a local cyclic resolution In this way, we get singularities of multiplicity lower than $n$, and of cyclic quotient type. We are interested in cyclic quotient singularities since they are $\log$-terminal and have a well-known algorithm to resolve them: the Fujiki-Oka continuous fraction (Cf. [Ash19]). In Section 4.3 we globalize this local cyclic resolution, and we get a cyclic resolution $X_{n}\to Y_{n}$ having the desired asymptotic property. The following is a summary of Theorem 4.2 (using Corollary 2.16), Theorem 4.18, and Theorem 4.24.

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It is important to note that the above is an embedded ${\mathbb{Q}}$-resolution in the language of [ABMMOG12], introduced as an efficient resolution without useless data. In Section 5.1, as a by-product of the computations to get C, we construct minimal non-singular $3$-folds of general type using as a base $3$-folds $Z\hookrightarrow{\mathbb{P}}^{4}$ with $3$ hyperplane sections. This allows us to prove.

∎

In Section 5.2, we see our resolution in terms of pairs $(X_{n},\tilde{D}_{red})\to(Z,D_{red})$. As a corollary, we prove that for asymptotic arrangements of hyperplane sections on a minimal $3$-fold of general type, the resolution preserves the bigness of the $log$-canonical divisor $K_{X_{n}}+\tilde{D}_{red}$.

We point out that our cyclic resolution preserves the positivity of the canonical volume, i.e., we start with $K_{Z}^{3}>0$ and we get $K_{X_{n}}^{3}>0$. So in future work, if we are able to control the MMP of the chosen resolution, then we will have the asymptotic results with $K_{X_{n}}$ nef, so for $n\gg 0$, the varieties $X_{n}$ will be minimal of general type. For us, the goal is that the asymptotic behavior of the slopes of $X_{n}$ coincides with the slopes of its minimal models. For this, we need a good terminalization of the cyclic singularities obtained. In Section 6 we discuss what means the word good.

For a non-singular projective variety $Z$ of dimension $d$, let $A(Z)=\bigoplus_{e\geq 0}A^{e}(Z)$ be its Chow ring, where $A^{e}(Z)$ are the Chow groups of $e$-cycles. We denote its Chern classes by $c_{e}(Z)\in A^{e}(Z)$, i.e. the Chern classes of its tangent bundle ${\mathcal{T}}_{Z}$ [Har77, Appendix A]. We have that $c_{1}(Z)=-K_{Z}$, where $K_{Z}$ is the canonical class. We define the Chern numbers of $Z$ as the degree of the top-intersection of its Chern classes

$$c_{i_{1}}(Z)\ldots c_{i_{m}}(Z)\in A^{d}(Z),\quad i_{1}+\ldots+i_{m}=d.$$

In the following, if the context is understood, we abuse notation using the symbol $c_{i_{1}}\ldots c_{i_{m}}(Z)$ for Chern numbers or more simply the notation $c_{i_{1}}\ldots c_{i_{m}}$.

We have $c_{1}^{d}=(-1)^{d}K_{Z}^{d}$, and since we work over ${\mathbb{C}}$, it is well-known that $c_{d}=e(Z)$, the topological Euler characteristic. These numbers are codified into the Todd class of ${\mathcal{T}}_{Z}$ by the following formal sum

$$\mbox{td}({\mathcal{T}}_{Z})=1+\frac{c_{1}}{2}+\frac{c_{1}^{2}+c_{2}}{12}+\frac{c_{1}c_{2}}{24}-\frac{c_{1}^{4}-4c_{1}^{2}c_{2}-3c_{2}^{2}-c_{1}c_{3}+c_{4}}{720}+\ldots.$$

As a consequence of the Hirzebruch-Riemann-Roch Theorem, we have the Noether’s identities, i.e., the analytic Euler characteristic are equal to the $d$-th summand of $\mbox{td}({\mathcal{T}}_{Z})$. For example,

$$\chi({\mathcal{O}}_{Z})=\dfrac{c_{1}c_{2}}{24},\quad\text{when }d=3.$$

From [Iit77] we introduce the following sheaf on $Z$.

In the rest of this section, we assume $D=\sum_{j=1}^{r}\nu_{j}D_{j}$ as a divisor with $D_{red}$ a SNC divisor.

We set $\Omega^{e}_{Z}(\log D):=\bigwedge^{e}\Omega_{Z}^{1}(\log D)$ for any $1\leq e\leq d$. In this way, $\Omega^{d}_{Z}(\log D)={\mathcal{O}}_{Z}(K_{Z}+D_{red})$, i.e., ${\bar{c}}_{1}(Z,D)=c_{1}(Z)-D_{red}$, and it is known that,

$${\bar{c}}_{d}=e(Z)-e(D_{red}),$$

see [Iit78, Prop. 2].

In this section, we follow [EV92, Sec. 3]. Consider the following building data $(Z,D,n,{\mathcal{L}})$ where

1. (1)

   $Z$ is a non-singular projective variety of dimension $d$,

2. (2)

   $D=\sum_{j=1}^{r}\nu_{j}D_{j}$ is an effective divisor on $Z$, with $D_{red}=\sum_{j=1}^{r}D_{j}$ a SNC divisor,

3. (3)

   $n\geq 2$ is a prime number, and

4. (4)

   ${\mathcal{L}}$ a line bundle on $Z$ such that ${\mathcal{O}}_{Z}(D)\simeq{\mathcal{L}}^{\otimes n}$.

With this building data, we construct

$$f^{\prime}\colon Y^{\prime}_{n}=\mbox{Spec}_{Z}\bigoplus_{i=0}^{n-1}{\mathcal{L}}^{-i}\to Z,$$

and the normalization $f\colon Y_{n}\to Y^{\prime}_{n}\to Z$ will be called the $n$-th root covering associated to the building data $(Z,D,n,{\mathcal{L}})$. Since the morphisms are finite of degree prime, both $Y^{\prime}_{n}$ and $Y_{n}$ are projective and irreducible. The morphism $f$ is branched at $D$, and $Y_{n}$ has its singularities over the intersection of components of $D$. Thus, $Y_{n}$ is non-singular if $D_{j}\cap D_{k}=\emptyset$ for all $j,k$. On the other hand, if $D_{j_{1}}\cap\ldots\cap D_{j_{e}}\neq\emptyset$, then the intersection is defined by a local equation

$$z_{j_{1}}^{\nu_{j_{1}}}\ldots z^{\nu_{j_{e}}}_{j_{e}}=0,$$

where $z_{1},\ldots,z_{d}$ are local parameters for $Z$ on $p$. Thus the singularity of $Y_{n}$ over $D_{j_{1}}\cap\ldots\cap D_{j_{e}}$ is locally analytically isomorphic to the normalization of

$$\mbox{Spec}\left(\frac{{\mathbb{C}}[z_{1},\ldots,z_{d},t]}{t^{n}-z_{j_{1}}^{\nu_{j_{1}}}\ldots z^{\nu_{j_{e}}}_{j_{e}}}\right).$$

Since the degree $n$ of $f$ is a prime number, we have $f^{*}D_{j}=nD^{\prime}_{j}$, where $D_{j}^{\prime}=(f^{*}D_{j})_{red}$. Thus, for any partial resolution $h\colon X\to Y_{n}\to Z$ we must have a ramification formula

$$h^{*}D_{j}=nD^{\prime}_{j}+\Delta_{j},$$

where $\Delta_{j}$ is a divisor supported in the exceptional divisors of $h$ over $D_{j}$.