Bounds on the Torsion Subgroups of N\'eron-Severi Groups

TL;DR

This paper provides explicit upper bounds on the torsion subgroup of the Néron-Severi group for smooth projective varieties defined by polynomials of bounded degree, based on bounds for certain Hilbert and divisor schemes.

Contribution

It introduces explicit bounds on the torsion of the Néron-Severi group using bounds on the components of Hilbert and divisor schemes.

Findings

Explicit upper bounds on the torsion subgroup size.

Bounds on the number of generators of the torsion subgroup for primes not dividing the characteristic.

Quantitative estimates for the structure of the Néron-Severi group.

Abstract

Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. We give explicit upper bounds on the order of the torsion subgroup $(\mathrm{NS} \, X)_{\mathrm{tor}}$ of the N\'eron-Severi group of $X$. The bounds are derived from an explicit upper bound on the number of irreducible components of either the Hilbert scheme $\mathbf{Hilb}_Q X$ or the scheme $\mathbf{CDiv}_n X $parametrizing the effective Cartier divisors of degree $n$ on $X$. We also give an upper bound on the number of generators of $(\mathrm{NS} \, X)[\ell^\infty]$ uniform as $\ell \neq \mathrm{char}\, k$ varies.

Equations

$$\#(\mathrm{NS}X{)}_{\mathrm{tor}}\le 2^{d^{r^2+2r{\log }_{2}r}}.$$

$$\prod _{\begin{array}{c}\ell \ne \mathrm{char}k\\ \ell \text{ is prime}\end{array}}\#H_{\acute{\mathrm{e}}\mathrm{t}}^2(X,{\mathbb{Z}}_{\ell }{)}_{\mathrm{tor}}\le 2^{d^{r^2+2r{\log }_{2}r}}.$$

$$N=\{\begin{array}{ll}(\mathrm{NS}X{)}_{\mathrm{tor}},& \text{mbox}ifp=0\\ (\mathrm{NS}X{)}_{\mathrm{tor}}/(\mathrm{NS}X)[p^{\infty }],& \text{mbox}ifp>0.\end{array}$$

$${\mathrm{HP}}_D={\mathrm{HP}}_{{\mathcal{O}}_X}-{\mathrm{HP}}_{\mathcal{O}(-D)}.$$

$$\#\mathrm{conn}({\mathrm{Hilb}}_{{\mathrm{HP}}_{mH}}X)\ge \#(\mathrm{NS}X{)}_{\mathrm{tor}}.$$

$$\#\mathrm{conn}({\mathrm{Hilb}}_{Q}X\cap \mathrm{CDiv}X)\ge \#(\mathrm{NS}X{)}_{\mathrm{tor}}.$$

$$\#\mathrm{conn}({\mathrm{Hilb}}_{Q}X\cap \mathrm{CDiv}X)$$

$$\#\mathrm{conn}({\pi }^{-1}({\mathrm{Pic}}^{c}X))$$

$$=(\mathrm{NS}X{)}_{\mathrm{tor}}.\text{qed}$$

$$\frac{p_0}{q_0}\wedge \frac{p_1}{q_1}\wedge \cdots \wedge \frac{p_{c-1}}{q_{c-1}}$$

$$p_0\wedge p_1\wedge \cdots \wedge p_{c-1}$$

$$g_0\wedge g_1\wedge \cdots \wedge g_{c-1}$$

$$f_0\wedge f_1\wedge \cdots \wedge f_{c-1}$$

$$0\to {\mathcal{I}}_X/{\mathcal{I}}_X^2\to {\Omega }_{{\mathbb{P}}^r}\otimes {\mathcal{O}}_X\to {\Omega }_X\to 0,$$

$${\omega }_{{\mathbb{P}}^r}{|}_X$$

$${\omega }_X^{-1}(-r-1)$$

$$f_0(x_0/x_i,\cdots ,x_r/x_i)\wedge \cdots \wedge f_{c-1}(x_0/x_i,\cdots ,x_r/x_i)$$

$$x_i^d{f}_0(x_0/x_i,\cdots ,x_r/x_0)\wedge \cdots \wedge x_i^d{f}_{c-1}(x_0/x_i,\cdots ,x_r/x_i)$$

$$x_j^d{f}_0(x_0/x_j,\cdots ,x_r/x_j)\wedge \cdots \wedge x_j^d{f}_{c-1}(x_0/x_j,\cdots ,x_r/x_j)$$

$${\omega }_X^{-1}(d\cdot \mathrm{codim}X-r-1)\simeq {\bigwedge }^c\left({\mathcal{I}}_X/{\mathcal{I}}_X^2(d)\right).\text{qed}$$

$$(K+(\dim X)H+(D+H))+(-K+(d\cdot \mathrm{codim}X-r-1)H)$$

$$=$$

$$\#\mathrm{conn}({\mathrm{Hilb}}_{{\mathrm{HP}}_{mH}}X)\ge \#(\mathrm{NS}X{)}_{\mathrm{tor}}.$$

$$H^i\left({\mathbb{P}}^r,\mathcal{F}(m-i)\right)=0$$

$$\phi (P)=\inf \{m|$$

$$\text{closed subvariety Z⊂Pr with Hilbert polynomial P}\}.$$

$${ı}_t:{\mathrm{Hilb}}_{\left(\genfrac{}{}{0ex}{}{t+r}{r}\right)-P(t)}{\mathbb{P}}^r$$

$$[Y]$$

$${\mathrm{Hilb}}_{Q}X\hookrightarrow {\mathrm{Hilb}}_Q{\mathbb{P}}^r.$$

$${\mathrm{Hilb}}_{\left(\genfrac{}{}{0ex}{}{t+r}{r}\right)-P(t)}X$$

$$\phi ({\mathrm{HP}}_I)\le {\left(\frac{3}{2}{d}^{r+1-a}+d\right)}^{a{2}^{a-1}}.$$

$$\#\mathrm{irr}(X)\le d^r.$$

$$\#\mathrm{conn}(\mathbf{X})\le \max \{p+1,q+1{\}}^{q(n-q)}.$$

Full text

This research was partially supported by Samsung Scholarship and National Science Foundation grant DMS-1601946.

1. Introduction

The Néron–Severi group $\operatorname{NS}X$ of a smooth proper variety $X$ over a field $k$ is the group of divisors modulo algebraic equivalence. If $k$ is algebraically closed, $\operatorname{NS}X$ also equals the group of connected components of the Picard scheme of $X$. Néron [17, p. 145, Théorème 2] and Severi [20] proved that $\operatorname{NS}X$ is finitely generated. The aim of this paper is to give an explicit upper bound on the order of $(\operatorname{NS}X)_{\mathrm{tor}}$. To the best of the author’s knowledge, this is the first explicit bound on the order of $(\operatorname{NS}X)_{\mathrm{tor}}$. {restatable*}thmNSbound Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. Then

[TABLE]

For any prime number $\ell\neq\operatorname{char}k$, there is a natural isomorphism $(\operatorname{NS}X)[\ell^{\infty}]\simeq H^{2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})_{\mathrm{tor}}$ [21, 2.2]. Thus, Section 1 implies the corollary below.

{restatable*}

corSHbound Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. Let $\ell\neq\operatorname{char}k$ be a prime number. Then

[TABLE]

We also give a uniform upper bound on the number of generators of $(\operatorname{NS}X)[\ell^{\infty}]$ as described below. This bound and a sketch of its proof were suggested by János Kollár after seeing an earlier draft of our paper containing only Section 1.

{restatable*}

thmNSgen Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth connected projective variety of degree $d$ over $k$. Let $p=\operatorname{char}k$, and

[TABLE]

Then $N$ is generated by less than or equal to $(d-1)(d-2)$ elements.

{restatable*}

corSHgen Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth connected projective variety of degree $d$ over $k$. Let $\ell\neq\operatorname{char}k$ be a prime number. Then $H^{2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})_{\mathrm{tor}}$ is generated by less than or equal to $(d-1)(d-2)$ elements.

The torsion-free quotient $(\operatorname{NS}X)/(\operatorname{NS}X)_{\mathrm{tor}}$ is the group of divisors modulo numerical equivalence. Its rank is bounded by the second Betti number of $X$. Katz found an upper bound on the sum of all Betti numbers of $X$ [10, Theorem 1]; this gives a rough bound on the rank of $\operatorname{NS}X$.

The torsion subgroup $(\operatorname{NS}X)_{\mathrm{tor}}$ is the group of numerically zero divisors modulo algebraic equivalence. This is a birational invariant [1, p. 177]. Recently, Poonen, Testa and van Luijk gave an algorithm to compute it.111They also found an algorithm computing the torsion free quotient assuming the Tate conjecture. The algorithm is based on their theorem that $(\operatorname{NS}X)_{\mathrm{tor}}$ injects into the set of connected components of $\operatorname{\mathbf{Hilb}}_{Q}X$ for some polynomial $Q$ [19, Lemma 8.29]. The order of $(\operatorname{NS}X)_{\mathrm{tor}}$ will be bounded by finding an explicit $Q$ and bounding the number of connected components of $\operatorname{\mathbf{Hilb}}_{Q}X$. We will show that an upper bound on the number of connected components of $\operatorname{\mathbf{CDiv}}_{n}X$ for an integer $n$ depending on $Q$ also gives an upper bound on the order of $(\operatorname{NS}X)_{\mathrm{tor}}$.

Section 3 shows that $Q$ may be taken to be the Hilbert polynomial of $mH$, where $H$ is a hyperplane section of $X$ and $m$ is an explicit integer. Section 4 bounds the number of irreducible components of $\operatorname{\mathbf{Hilb}}_{Q}X$ by using its embedding in a Grassmannian. Section 5 bounds the number of irreducible component of $\operatorname{\mathbf{CDiv}}_{n}X$ by using Kollár’s technique [13, Exercise I.3.28]. Finally, Section 6 gives a uniform upper bound on the number of generators of $(\operatorname{NS}X)[\ell^{\infty}]$.

From now on, the base field $k$ is assumed to be algebraically closed, since a base change makes the Néron–Severi group only larger [19, Proposition 6.1]. However, no assumption is made on the characteristic of $k$.

2. Notation

Given a scheme $X$ over $k$, let $\operatorname{conn}(X)$ and $\operatorname{irr}(X)$ be the set of connected components of $X$ and the set of irreducible components of $X$, respective. Let $X_{\mathrm{red}}$ be the reduced closed subscheme associated to $X$. If $X$ is smooth and proper, then $\operatorname{NS}X$ denote the Néron–Severi group of $X$. Let $H^{i}_{\mathrm{\acute{e}t}}(X,\mathscr{F})$ be the $i$-th étale cohomology group of $X$ corresponding to an étale sheaf $\mathscr{F}$. If $k=\mathbb{C}$, then let $X^{\mathrm{an}}$ be the analytic space of $X$. If $M$ is a topological manifold, then $H_{\mathrm{sing}}^{i}(M,\mathbb{Z})$ (resp. $H_{i}^{\mathrm{sing}}(M,\mathbb{Z})$) denotes the $i$-th singular cohomology (resp. homology) group with integer coefficients.

A projective variety is a closed subscheme of $\mathbb{P}^{r}=\operatorname{Proj}k[x_{0},\cdots,x_{r}]$ for some $r$. Suppose that $X\hookrightarrow\mathbb{P}^{r}$ is a projective variety. Let $I_{X}\subset k[x_{0},\cdots,x_{r}]$ be the saturated ideal defining $X$. Given $f_{0},\cdots,f_{t-1}\in k[x_{0},\cdots,x_{r}]$, let $V_{X}({f_{0},\cdots,f_{t-1}})$ be the subscheme of $X$ defined by the ideal $(f_{0},\cdots,f_{t-1})+I_{X}$. Let $\mathcal{O}_{X}$, $\mathscr{I}_{X}$, $\Omega_{X}$ and $\omega_{X}$ be the sheaf of regular functions, the ideal sheaf, the sheaf of differentials and the canonical sheaf of $X$, respectively.

Given a coherent sheaf $\mathscr{F}$ on $X$, let $\operatorname{HP}_{\mathscr{F}}$ be the Hilbert polynomial of $\mathscr{F}$, and $\Gamma(\mathscr{F})$ be the global section of $\mathscr{F}$. Given a graded module $M$ over $k$, let $\operatorname{HP}_{M}$ be the Hilbert polynomial of $M$, and $M_{t}$ be the degree $t$ part of $M$. Take an effective divisor $D$ on $X$. Let $\operatorname{HP}_{D}$ be the Hilbert polynomial of $D$ as a subscheme of $\mathbb{P}^{r}$. Then $\mathcal{O}(-D)\subset\mathcal{O}_{X}$ is the ideal sheaf corresponding to $D$, and

[TABLE]

Let $\operatorname{\mathbf{Hilb}}X$ be the Hilbert scheme of $X$. Given a polynomial $Q(t)$, let $\operatorname{\mathbf{Hilb}}_{Q}X$ be the Hilbert scheme of $X$ parametrizing closed subschemes of $X$ with Hilbert polynomial $Q$. Let $\operatorname{\mathbf{Chow}}_{\delta,n}X$ be the Chow variety of dimension $\delta$ and degree $n$ algebraic cycles on $X$. Let $\operatorname{\mathbf{CDiv}}X$ be the scheme parametrizing the effective Cartier divisors on $X$, and let $\operatorname{\mathbf{CDiv}}_{n}X$ be the open and closed subscheme of $\operatorname{\mathbf{CDiv}}X$ corresponding to the divisors of degree $n$. Let $\operatorname{\mathbf{Pic}}X$ be the Picard scheme of $X$. Let $\operatorname{\mathbf{Alb}}X$ be the Albanese variety of $X$. Given a vector space $V$ and nonnegative number $t$, let $\operatorname{\mathbf{Gr}}(t,V)$ be the Grassmannian parametrizing $t$-dimensional subspaces of $V$. Let $\operatorname{\mathbf{Gr}}(t,n)=\operatorname{\mathbf{Gr}}(t,k^{n})$.

Given a set $S$, let $\#S$ be the number of elements in $S$. Give a group $A$, let $A^{\mathrm{ab}}$ be the abelianization of $A$, and $A^{(\ell)}=\varprojlim A^{\mathrm{ab}}/\ell^{n}A^{\mathrm{ab}}$ be the maximal pro-$\ell$ abelian quotient of $A$. If $A$ is abelian, let $A_{\mathrm{tor}}$, $A[n]$ and $A[\ell^{\infty}]$ be the set of torsion elements, $n$-torsion elements and $\ell$-power torsion elements, respectively. If $A$ is a finite abelian group, then $A^{*}$ is the Pontryagin dual of $A$.

3. Numerical Conditions

Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth projective variety defined by polynomials of degree $\leq d$. Let $K$ and $H$ be a canonical divisor and a hyperplane section of $X$, respectively. The goal of this section is to give an explicit $m$ such that

[TABLE]

Poonen, Testa and van Luijk proved that an order of a Néron–Severi group is bounded by a number of connected components of a certain Hilbert scheme. The theorem and the proof below is a reformulation of their work in [19, Section 8.4].

Theorem 3.1** (Poonen, Testa and van Luijk).**

Let $F$ be a divisor on $X$ and $Q=\operatorname{HP}_{\mathcal{O}_{X}}-\operatorname{HP}_{\mathcal{O}(-F)}$. If $\mathcal{O}_{X}(F+D)$ has a global section for every numerically zero divisor $D$, then

[TABLE]

Proof.

Recall that $\operatorname{\mathbf{CDiv}}X$ is an open and closed subscheme of $\operatorname{\mathbf{Hilb}}X$ [13, Exercise I.3.28], and there is a natural proper morphism $\pi:\operatorname{\mathbf{CDiv}}X\rightarrow\operatorname{\mathbf{Pic}}X$ sending a divisor to the corresponding class [3, p. 214].

Let $\operatorname{\mathbf{Pic}}^{c}X$ be the finite union of connected components of $\operatorname{\mathbf{Pic}}X$ parametrizing the divisors numerically equivalent to $F$. Since the Hilbert polynomial of a divisor is a numerical invariant, $Q$ is the Hilbert polynomial of each divisor corresponding to a closed point of $\pi^{-1}(\operatorname{\mathbf{Pic}}^{c}X)$. Thus, $\pi^{-1}(\operatorname{\mathbf{Pic}}^{c}X)\subset\operatorname{\mathbf{Hilb}}_{Q}X$. Since $\operatorname{\mathbf{Pic}}^{c}X$ is open and closed in $\operatorname{\mathbf{Pic}}X$, and $\operatorname{\mathbf{CDiv}}X$ is open and closed in $\operatorname{\mathbf{Hilb}}X$, the scheme $\pi^{-1}(\operatorname{\mathbf{Pic}}^{c}X)$ is open and closed in $\operatorname{\mathbf{Hilb}}_{Q}X\cap\operatorname{\mathbf{CDiv}}X$. Thus,

[TABLE]

Since $F+D$ is linearly equivalent to an effective divisor for every numerically zero divisor $D$, the morphism $\pi$ restricts to a surjection $\pi^{-1}(\operatorname{\mathbf{Pic}}^{c}X)\rightarrow\operatorname{\mathbf{Pic}}^{c}X$. Hence,

[TABLE]

The authors of [19] chose $F=K+(\dim X+2)H$ because of the following partial result towards Fujita’s conjecture.

Theorem 3.2** (Keeler [11, Theorem 1.1]).**

Let $L$ be an ample divisor on X. Then

1. (a)

$\mathcal{O}_{X}(K+(\dim X)H+L)$* is generated by global sections, and* 2. (b)

$\mathcal{O}_{X}(K+(\dim X+1)H+L)$* is very ample.*

However, computing the Hilbert polynomial of $\mathcal{O}(-K)$ is somehow difficult. Therefore, we will show that $F=((d-1)\cdot\operatorname{codim}X)H$ is another choice.

Lemma 3.3**.**

Let $Y$ be a nonempty smooth closed subscheme of an affine space $\mathbb{A}^{r}=\operatorname{Spec}k[x_{0},\cdots,x_{r-1}]$. Suppose that the ideal $I$ defining $Y$ is generated by polynomials of degree $\leq d$. Let $c=\operatorname{codim}Y$. Then there are polynomials $f_{0},\cdots,f_{c-1}\in I$ such that

1. (a)

$\deg f_{i}=d$* for all $i$, and* 2. (b)

$f_{0}\wedge\cdots\wedge f_{c-1}$* represents a nonzero element in $\bigwedge^{c}(I/I^{2})$.*

Proof.

Let $R=k[x_{0},\cdots,x_{c-1}]/I$. Then $I/I^{2}$ is a locally free $R$-module of rank $\operatorname{codim}Y$ by [8, Theorem 8.17]. Take any prime ideal $\mathfrak{p}\subset R$. Then $(I/I^{2})_{\mathfrak{p}}$ is a free $R_{\mathfrak{p}}$-module. Thus, there are $p_{0},\cdots,p_{c-1}\in I$ and $q_{0},\cdots,q_{c-1}\in R\setminus\mathfrak{p}$ such that

[TABLE]

represents a nonzero element of $\bigwedge^{c}(I/I^{2})_{\mathfrak{p}}$. Hence,

[TABLE]

represents a nonzero element of $\bigwedge^{c}(I/I^{2})$.

Let $g_{0},\cdots,g_{b-1}$ be polynomials of degree $\leq d$ which generate $I$. Then each $p_{i}$ can be written as a $R$-linear combination of $g_{i}$’s. If we expand (3), at least one term should be nonzero in $\bigwedge^{c}(I/I^{2})$. Therefore, we may assume that

[TABLE]

represents a nonzero element of $\bigwedge^{c}(I/I^{2})$. Let $\ell\not\in I$ be a polynomial of degree 1, and let $f_{i}=\ell^{d-\deg g_{i}}g_{i}$. Then

[TABLE]

represents a nonzero element of $\bigwedge^{c}(I/I^{2})$ and $\deg f_{i}=d$ for every $i$. ∎

Lemma 3.4**.**

The sheaf $\mathcal{O}_{X}\left(-K+(d\cdot\operatorname{codim}X-r-1)H\right)$ has a global section.

Proof.

Let $c=\operatorname{codim}X$. Since $X$ is smooth, there is an exact sequence

[TABLE]

and $\mathscr{I}_{X}/\mathscr{I}_{X}^{2}$ is locally free of rank $c$ [8, Theorem 8.17]. Taking the highest exterior power gives

[TABLE]

Let $U_{i}\subset X$ be the affine open set given by $x_{i}\neq 0$. Then there exist polynomials $f_{0},\cdots,f_{c-1}$ of degree $d$ such that

[TABLE]

represents a nonzero section of $\bigwedge^{c}(\mathscr{I}_{X}/\mathscr{I}_{X}^{2})|_{U_{i}}$ by 3.3. Take another $U_{j}\subset X$ given by $x_{j}\neq 0$. Then two sections

[TABLE]

and

[TABLE]

give same restrictions in $\bigwedge^{c}(\mathscr{I}_{X}/\mathscr{I}_{X}^{2}(d))|_{U_{i}\cap U_{j}}$. Because $i$ and $j$ are arbitrary, the sections above extend to a global section of

[TABLE]

Lemma 3.5**.**

Let $D$ be a divisor on $X$ numerically equivalent to 0.222The condition ‘numerically equivalent to 0’ can be replaced by ‘numerically effective’ due to Kleiman’s criterion of ampleness [12, Chapter IV §2 Theorem 2]. Then

1. (a)

$\mathcal{O}_{X}(D+((d-1)\operatorname{codim}X)H)$* is generated by global sections, and* 2. (b)

$\mathcal{O}_{X}(D+((d-1)\operatorname{codim}X+1)H)$* is very ample.*

Proof.

The divisor $D+H$ is ample, since ampleness is a numerical property. Then $K+(\dim X)H+(D+H)$ is generated by global section by 3.2. Thus, 3.4 implies that

[TABLE]

is generated by global sections. Similarly, $D+((d-1)\operatorname{codim}X+1)H$ is very ample. ∎

Theorem 3.6**.**

Let $m=(d-1)\operatorname{codim}X$. Then

[TABLE]

Proof.

By 3.5(a), we may apply 3.1 to $F=mH$. ∎

4. Irreducible Components of Hilbert Schemes

The aim of this section is to give an explicit upper bound on $\#(\operatorname{NS}X)_{\mathrm{tor}}$ for a smooth projective variety $X$. 3.6 implies that it suffices to give an upper bound on the number of connected components of some Hilbert scheme. Recall the definition of Castelnuovo–Mumford regularity and Gotzmann numbers.

Definition 4.1**.**

A coherent sheaf $\mathscr{F}$ over $\mathbb{P}^{r}$ is $m$-regular if and only if

[TABLE]

for every integer $i>0$. The smallest such $m$ is called the Castelnuovo–Mumford regularity of $\mathscr{F}$.

Definition 4.2**.**

Let $P$ be the Hilbert polynomial of some ideal $I\subset k[x_{0},\cdots,x_{r}]$. The Gotzmann number $\varphi(P)$ of $P$ is defined as

[TABLE]

Hilbert schemes can be explicitly described as a closed subscheme of a Grassmannian, by Gotzmann [5].

Theorem 4.3** (Gotzmann).**

Let $P$ be the Hilbert polynomial of some ideal $I\subset k[x_{0},\cdots,x_{r}]$. Assume that $t\geq\varphi(P)$. Then

[TABLE]

gives a well-defined closed immersion. Moerover, the image is the collection of linear spaces $T\subset k[x_{0},\cdots,x_{r}]_{t}$ such that

1. (a)

$\dim\left(x_{0}T+\cdots+x_{r}T\right)\leq P(t+1)$.

Proof.

See [5, Section 3]. ∎

Let $X\hookrightarrow\mathbb{P}^{r}$ be a projective variety and $Q$ be a polynomial. Then there is the natural closed embedding

[TABLE]

Theorem 4.4**.**

Use the notation in 4.3. Let $X\hookrightarrow\mathbb{P}^{r}$ be a projective variety defined by polynomials of degree $\leq d$. Assume that $t\geq\max\{\varphi(P),d\}$. Then the image of

[TABLE]

under $\imath_{t}$ is the collection of linear spaces $T\subset k[x_{0},\cdots,x_{r}]_{t}$ such that

1. (a)

$\dim\left(x_{0}T+\cdots+x_{r}T\right)\leq P(t+1)$* and* 2. (b)

$\Gamma(\mathscr{I}_{X}(t))\subset T$.

Proof.

See the proof of [19, Lemma 8.23] ∎

Therefore, an upper bound on Gotzmann numbers will give an explicit construction of a Hilbert scheme. Such a bound is given by Hoa [9, Theorem 6.4(i)].

Theorem 4.5** (Hoa).**

Let $I\subset k[x_{0},\cdots,x_{r}]$ be an nonzero ideal generated by homogeneous polynomials of degree at most $d\geq 2$. Let $a$ be the Krull dimension of $k[x_{0},\cdots,x_{r}]/I$. Then

[TABLE]

Once a Hilbert scheme is explicitly constructed, we can bound the number of the irreducible components by the lemma below.

Lemma 4.6**.**

If $X\hookrightarrow\mathbb{A}^{r}$ is an affine scheme defined by polynomials of degree $\leq d$, then

[TABLE]

Proof.

This is a special case of the Andreotti-Bézout inequality [4, Lemma 1.28]. ∎

The Grassmannian $\operatorname{\mathbf{Gr}}(k,n)$ is covered by open sets isomorphic to $\mathbb{A}^{k(n-k)}$. The conditions in 4.4 can be translated into explicit equations in such an affine space, giving the lemma below.

Lemma 4.7**.**

Let $V$ and $W$ be vector spaces. Let $n=\dim V$. Let $\varphi_{i}:V\rightarrow W$ be a linear map for each $i=0,\cdots,r$. Let $U\subset V$ be a subspace. Let $\mathbf{X}$ be the collection of $T\in\operatorname{\mathbf{Gr}}(q,V)$ satisfying

1. (a)

$\dim(\varphi_{0}(T)+\cdots+\varphi_{r}(T))\leq p$, and 2. (b)

$U\subset T$.

Then $\mathbf{X}$ is a closed subscheme of $\operatorname{\mathbf{Gr}}(q,V)$, and with $n=\dim V$,

[TABLE]

Proof.

Given a subspace $S\subset V$ of dimension $n-q$, there is an open set

[TABLE]

Choose a basis of $V$ such that $S$ is spanned by the first $n-q$ entries. Then every $T\in U_{S}$ is uniquely represented as a column space of a block matrix

[TABLE]

where $X_{n-q,q}$ is an $(n-q)\times q$ matrix, and $I_{q}$ is the $q\times q$ identity matrix. The entries of $X_{n-q,q}$ can be regarded as indeterminates, giving an isomorphism $U_{S}\simeq\mathbb{A}^{(n-q)q}$. Let $\varphi_{i}^{\oplus q}(N_{T})$ be the $(\dim W)\times q$ matrix obtained by applying $\varphi_{i}$ to every column of $N_{T}$. Then

[TABLE]

if and only if every $(p+1)\times(p+1)$ minor of the block matrix

[TABLE]

is zero.

Let $M_{U}$ be a matrix whose columns form a basis of $U$. Then $U\subset T$ if and only if every $(q+1)\times(q+1)$ minor of

[TABLE]

is zero. Thus, $\mathbf{X}\cap U_{S}$ is defined by the minors in $U_{S}$, meaning that $\mathbf{X}$ is a closed subscheme of $\operatorname{\mathbf{Gr}}(q,V)$.

Now, take one point from each irreducible component of $\mathbf{X}$, and let them be $T_{0},\cdots,T_{\ell-1}\subset V$. Then $(n-k)$-dimensional subspace $S\subset V$ can be chosen such that

[TABLE]

implying that every irreducible component intersects with $U_{S}$. Since $\mathbf{X}\cap U_{S}$ is defined by polynomials of degree $\leq\max\{p+1,q+1\}$, 4.6 implies

[TABLE]

Lemma 4.8**.**

Let $X\hookrightarrow\mathbb{P}^{r}$ be a projective variety defined by polynomials of degree $\leq d$. Let $P$ be the Hilbert polynomial of an ideal. If $t\geq\max\{\varphi(P),d,8r\}$ and $r\geq 2$, then

[TABLE]

Proof.

4.4 and 4.7 implies that

[TABLE]

Since $r\geq 2$ and $t\geq 8r$,

[TABLE]

Therefore,

[TABLE]

Now, we are ready to give an upper bound:

Theorem 4.9**.**

Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. Then

[TABLE]

Proof.

If $X$ is a curve or a projective space, then $(\operatorname{NS}X)_{\mathrm{tor}}=0$. Thus, we may assume that $r\geq 3$, $d\geq 2$ and $\operatorname{codim}X\geq 1$. Moreover, $X$ may be assumed to be not contained in any hyperplane.

Let $I$ be the ideal defining $X$. Let $H$ be a hyperplane section of $X$ cut by $x_{r}=0$. Let $m=(d-1)\operatorname{codim}X\geq 1$. Then $I+(x_{r}^{m})$ is the ideal defining $mH$ where $m=(d-1)\operatorname{codim}X$. Let $t=(2rd)^{(r+1)2^{r-2}}$ and $a=\dim X$. Then $t\geq d$ and $t\geq 8r$. Moreover,

[TABLE]

Thus, 3.6 and 4.8 implies that

[TABLE]

Notice that

[TABLE]

As a result,

[TABLE]

5. Irreducible Components of Chow Varieties

In this section, $X\hookrightarrow\mathbb{P}^{r}$ is a smooth projective variety defined by homogeneous polynomials of degree $\leq d$. The goal of this section is to give a better upper bound on the order of $(\operatorname{NS}X)_{\mathrm{tor}}$. János Kollár pointed out that the bound may be also derived form an upper bound on the number of connected components of $\operatorname{\mathbf{Chow}}_{\delta,n}X$. Our goal only requires a bound for $\operatorname{\mathbf{CDiv}}_{n}X$ instead of Chow varieties of arbitrary dimensions.

Theorem 5.1**.**

Let $n=(d-1)\operatorname{codim}X\cdot\deg X$. Then

[TABLE]

Proof.

Let $H\subset X$ be a hyperplane section, $m=(d-1)\operatorname{codim}X$, and $Q=\operatorname{HP}_{mH}$. If $D$ is a closed subscheme with Hilbert polynomial $Q$, then $\deg D=\deg mH=n$. Hence,

[TABLE]

Since $\operatorname{\mathbf{Hilb}}_{Q}X$, $\operatorname{\mathbf{CDiv}}X$ and $\operatorname{\mathbf{CDiv}}_{n}X$ are open and closed in $\operatorname{\mathbf{Hilb}}X$, 3.5 and 3.1 implies that

[TABLE]

In [13, Exercise I.3.28], Kollár gives an explicit upper bound on $\#\operatorname{irr}(\operatorname{\mathbf{Chow}}_{\delta,n}\mathbb{P}^{r})$ and an outline of the proof. Moreover, [13, Exercise I.3.28.13] suggests an exercise to find an explicit upper bound on $\#\operatorname{irr}(\operatorname{\mathbf{Chow}}_{\delta,n}X)$, and the proof can be found in [7, Section 2]. However, the proof works only if $\operatorname{char}k=0$, and it does not give a bound in a closed form.

Therefore, we will give another complete proof, but most of the proof up to 5.11 is just a modification of Kollár’s technique and [7, Section 2]. Moreover, we will only bound $\#\operatorname{irr}(\operatorname{\mathbf{CDiv}}_{n}X)$, because this restriction avoids the bad behavior of Chow varieties in positive characteristic, simplies the proof and slightly improves the result.

Lemma 5.2**.**

Let $D\subset X$ be a nonzero effective divisor on $X$ of degree $n$. Then there are $f$ and $g$ in $\Gamma(X,\mathscr{I}_{D}(n))$ such that $D$ is the largest effective divisor contained in $V_{X}(f,g)$.

Proof.

Take any point $x\in X$ and a generic linear projection $\rho_{0}\colon\mathbb{P}^{r}\dashrightarrow\mathbb{P}^{\dim X}$. Then $\rho_{0}(D)$ is a hypersurface and $\rho_{0}|_{X}$ is étale at $x$. Let $f_{0}$ be a homogeneous polynomial of degree $n$ defining $\rho_{0}(D)$, and $f\in\Gamma(X,\mathscr{I}_{D}(n))$ be its pullback. Then

[TABLE]

for some irreducible closed subschemes $E_{i}\subset X$ not set theoretically contained in $D$.

Take $e_{i}\in E_{i}\setminus D$ for each $i$, and let $\rho_{1}\colon\mathbb{P}^{r}\dashrightarrow\mathbb{P}^{\dim X}$ be another generic linear projection. Then $e_{i}\not\in\rho_{1}(D)$ for every $i$. Let $g_{0}$ be a homogeneous polynomial of degree $n$ defining $\rho_{1}(D)$, and $g\in\Gamma(X,\mathscr{I}_{D}(n))$ be its pullback. Then $V_{X}({f,g})$ contains $D$ but not $E_{i}$ for all $i$. Thus, $D$ is the largest divisor contained in $V_{X}({f,g})$. ∎

Definition 5.3**.**

Let

[TABLE]

Then $\mathbb{P}^{N_{n}}$ parameterizes nonzero homogeneous polynomials of degree $n$ with $r+1$ variables up to constant factors. Let

[TABLE]

Let $p\colon T_{n}\rightarrow S_{n}$ and $q\colon T_{n}\rightarrow\operatorname{\mathbf{CDiv}}X$ be the natural projections.

Lemma 5.4**.**

Let $X\hookrightarrow{\left(\mathbb{P}^{r}\right)}^{2}$ be a closed subscheme defined by bihomogeneous polynomials of total degree $\leq d$. Then

[TABLE]

Proof.

Let $L_{0}$ and $L_{1}$ be generic hyperplanes of $\mathbb{P}^{r}$. Then $L_{0}\times\mathbb{P}^{r}$ and $\mathbb{P}^{r}\times L_{1}$ do not contain any irreducible component of $X$. Let

[TABLE]

Then

[TABLE]

and $X\cap U\hookrightarrow\mathbb{A}^{2r}$ is defined by polynomials of degree $\leq d$. Consequently, 4.6 proves the inequality. ∎

Lemma 5.5**.**

Let $n$ be a positive integer. Then $S_{n}$ is closed in $\left(\mathbb{P}^{N_{n}}\right)^{2}$ and

[TABLE]

Proof.

Take $(f,g)\in(\mathbb{P}^{N_{n}})^{2}$ as in 5.3. Then $\operatorname{codim}_{X}(V_{X}({f,g}))=1$, if and only if the intersection of $V_{X}({f,g})$ with $t=\dim X-1$ number of generic hyperplane sections is nonempty. Let $h_{i}=\sum_{j=0}^{t-1}\xi_{i,j}x_{j}$ be a generic hyperplane section for each $i$, where $\xi_{i,j}$ are indeterminates. Take a base extension to $k(\{\xi_{i,j}\}_{i,j})$. Then

[TABLE]

The last condition can be translated into bihomogeneous polynomials in the coefficients of $f$ and $g$ of total degree ${2\max\{n,d\}+(r-1)d\choose r}$. 5.4 proves the inequality. ∎

Lemma 5.6**.**

The set $T_{n}$ is a closed subset of $S_{n}\times\operatorname{\mathbf{CDiv}}X$.

Proof.

Since $\operatorname{\mathbf{CDiv}}X$ is open and closed in $\operatorname{\mathbf{Hilb}}X$, it suffices to show that

[TABLE]

is closed in $S_{n}\times\operatorname{\mathbf{Hilb}}_{Q}X$ for every Hilbert polynomial $Q$. Recall that 4.3 gives a closed embedding

[TABLE]

for some polynomial $P$ and every large $t$. This gives a closed embedding

[TABLE]

We may assume that $t\geq n$. Notice that $Z\subset V(f,g)$ if and only if the saturation of $(f,g)$ is contained in the saturated ideal defining $Z$. Thus,

[TABLE]

Note that $\mathbb{P}^{N}\times\operatorname{\mathbf{Gr}}(P(t),k[x_{0},\cdots,x_{r}]_{t})$ is covered by the standard affine open spaces. In such affine open spaces, the last condition is expressed as $(P(t)+1)\times(P(t)+1)$ minors of some matrix. Consequently, $T_{n}^{Q}$ is identified with a closed subset of $\mathbb{P}^{N}\times\operatorname{\mathbf{Gr}}(P(t),k[x_{0},\cdots,x_{r}]_{t})$. ∎

Definition 5.7**.**

Let $\operatorname{\mathbf{PDiv}}_{n}X$ be the union of the irreducible components of $\operatorname{\mathbf{CDiv}}_{n}X$ which contains at least one closed point corresponding to a reduced and irreducible divisor.

Lemma 5.8**.**

Let $F$ be an irreducible component of $\operatorname{\mathbf{PDiv}}_{n}X$. Then there is a unique irreducible component $E$ of $T_{n}$ such that $q(E)=F$.

Proof.

5.2 implies that $\operatorname{\mathbf{CDiv}}_{n}X$ is contained in the image of $q:T_{n}\rightarrow\operatorname{\mathbf{CDiv}}X$. Because $q$ is proper, there is an irreducible component $E$ of $T_{n}$ such that $q(E)=F$. Moreover, for any $[D]\in F$,

[TABLE]

Then $q^{-1}([D])$ is irreducible, because it is a product of two projective spaces. Thus, such an $E$ is unique. ∎

Lemma 5.9**.**

Let $F$ and $E$ be as in 5.8. Then there is $(f,g)\in S_{n}$ such that $E$ is the only irreducible component of $T_{n}$ satisfying $(f,g)\in p(E)$.

Proof.

Let $W\subset\operatorname{\mathbf{CDiv}}_{n}$ be the complement of the image of the proper morphism

[TABLE]

Then $W$ parametrizes the reduced and irreducible divisors of degree $n$ on X. Notice that $W\cap F\neq\emptyset$, because $F\subset\operatorname{\mathbf{PDiv}}_{n}X$. The uniqueness of $E$ implies that there is a dense open set $U\subset F$ such that $q^{-1}(U)$ does not intersect with any irreducible component of $T_{n}$ other than $E$. Thus, we can take $[D]\in W\cap U$. Then $D$ is a reduced and irreducible divisor, since $[D]\in W$. 5.2 implies that there is $(f,g)\in S_{n}$ such that

[TABLE]

Then $E$ is the only irreducible component of $T_{n}$ containing $((f,g),[D])$, because $[D]\in U$. ∎

Lemma 5.10**.**

Let $F$ and $E$ be as in 5.8. Then $p(E)$ is an irreducible component of $S_{n}$.

Proof.

Let

[TABLE]

Since $R_{n}$ is proper, the restriction $p|_{R_{n}}:R_{n}\rightarrow S_{n}$ is also proper. Moreover, $p|_{R_{n}}$ is surjective, because $\deg V_{X}(f,g)\leq n^{2}\deg X$ for every $(f,g)\in S_{n}$. Thus, every irreducible component of $S_{n}$ is the image of an irreducible component of $R_{n}$ under $p|_{R_{n}}$. Let $B\subset S_{n}$ be the irreducible component containing $p(E)$. Then 5.9 implies that $p(E)=B$. ∎

Lemma 5.11**.**

If $n$ is a positive integer, then

[TABLE]

Proof.

5.8 and 5.10 define a map

[TABLE]

where $E$ is determined by $F$ as in 5.8. Then 5.8 and 5.9 implies that this map is injective. ∎

Lemma 5.12**.**

Let $n$ be a positive integer. Then

[TABLE]

Proof.

Notice that

[TABLE]

is surjective, where the disjoint union runs over all integer partitions of $n$. Thus, the left-hand side has more irreducible components. Take any integer partition $n_{0}+\cdots+n_{t-1}=n$. Then by 5.5 and 5.11,

[TABLE]

Let

[TABLE]

Then $D(0)=0$ and $D$ is concave above in $[0,\infty)$. Therefore,

[TABLE]

Furthermore, the number of integer partitions of $n$ is less than or equal to $2^{n}$. This proves the inequality. ∎

Now, we are ready to give a new upper bound:

\NSbound

Proof.

If $X$ is a curve or a projective space, then $(\operatorname{NS}X)_{\mathrm{tor}}=0$. Thus, we may assume that $r\geq 3$, $d\geq 2$, $\deg X\geq 2$ and $r-2\geq\operatorname{codim}X\geq 1$. Let $n=(d-1)\operatorname{codim}X\cdot\deg X$. Then

[TABLE]

5.1 and 5.12 implies that

[TABLE]

Since

[TABLE]

we have

[TABLE]

We now give an application to the torsion subgroups of second cohomology groups.

Lemma 5.13**.**

Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth projective variety of degree $d$. Let $\ell\neq\operatorname{char}k$ be a prime number. The embedding $(\operatorname{NS}X)\otimes\mathbb{Z}_{\ell}\hookrightarrow H^{2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})$ [15, Remark V.3.29] induces by the Kummer sequence restricts to an isomorphism.

[TABLE]

Proof.

See [21, 2.2]. ∎

Therefore, the bound of Section 1 implies the corollaries below.

\SHbound

Proof.

This follows from Section 1 and 5.13. ∎

Corollary 5.14**.**

Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth projective variety over $\mathbb{C}$ defined by homogeneous polynomials of degree $\leq d$. Then

[TABLE]

Proof.

This follows from Section 1 and the fact that

[TABLE]

6. The Number of Generators of $(\operatorname{NS}X)_{\mathrm{tor}}$

In this section, $X\hookrightarrow\mathbb{P}^{r}$ is a smooth connected projective variety, and $\ell$ is a prime number not equal to $\operatorname{char}k$. The goal is to give a uniform upper bound on the number of generators of $(\operatorname{NS}X)[\ell^{\infty}]$. Our approach is a simplification of a brief sketch suggested by János Kollár.

Lemma 6.1**.**

Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth connected projective variety of degree $d$, and $x_{0}$ be a geometric point of $X$. Let $\ell\neq\operatorname{char}k$ be a prime number. Then

[TABLE]

Proof.

The exact sequence in [22, Proposition 69] gives an exact sequence

[TABLE]

by taking the maximal pro-$\ell$ abelian quotient. Since $\pi^{\mathrm{\acute{e}t}}_{1}(\operatorname{\mathbf{Alb}}X,0)^{(\ell)}$ is a free $\mathbb{Z}_{\ell}$-module,

[TABLE]

If $M$ is a topological manifold, the linking form implies that $H^{2}_{\mathrm{sing}}(M,\mathbb{Z})_{\mathrm{tor}}^{*}\simeq H^{\mathrm{sing}}_{1}(M,\mathbb{Z})_{\mathrm{tor}}.$ 5.13 and 6.1 imply the étale analogy that $H^{2}_{\mathrm{\acute{e}t}}(X,\mathbb{Z}_{\ell})_{\mathrm{tor}}^{*}\simeq\pi^{\mathrm{\acute{e}t}}_{1}(X,x_{0})^{\mathrm{ab}}[\ell^{\infty}].$

Lemma 6.2**.**

Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth connected projective variety of degree $d$. Then $(\operatorname{NS}X)[\ell^{\infty}]$ is generated by less than or equal to $(d-1)(d-2)$ elements.

Proof.

For a general linear space $L\subset\mathbb{P}^{r}$ of dimension $r-\dim X+1$, the intersection $C\coloneqq X\cap L$ is a connected smooth curve of degree $d$. Take a geometric point $x_{0}\in C$. Then the natural map

[TABLE]

is surjective, by repeatedly applying the Lefschetz hyperplane theorem for étale fundamental groups [6, XII. Corollaire 3.5]. Let $g$ be the genus of $C$. Then 6.1 implies that $(\operatorname{NS}X)[\ell^{\infty}]^{*}$ is isomorphic to a subquotient of

[TABLE]

Notice that $2g\leq(d-1)(d-2)$ because $\deg C=d$. As a result, $(\operatorname{NS}X)[\ell^{\infty}]$ is generated by $\leq(d-1)(d-2)$ elements. ∎

\NSgen

Proof.

Since a product of finitely many finite cyclic groups of coprime orders is again cyclic,

[TABLE]

is a product of $\leq(d-1)(d-2)$ cyclic groups by 6.2. ∎

\SHgen

Proof.

This follows from 6.2 and 5.13. ∎

Corollary 6.3**.**

Let $X\hookrightarrow\mathbb{P}^{r}$ be a smooth connected projective variety of degree $d$ over $\mathbb{C}$. Then $H^{2}_{\mathrm{sing}}(X^{\mathrm{an}},\mathbb{Z})$ is generated by less than or equal to $(d-1)(d-2)$ elements.

Proof.

This follows from Section 1 and the fact that

[TABLE]

The bounds in this section exclude the case $\ell=\operatorname{char}k$, because of the bad behavior of the étale cohomology at $p$. One may try to overcome this by using Nori’s fundamental group scheme [18]. However, the Lefschetz hyperplane theorem for Nori’s fundamental group scheme is no longer true [2, Remark 2.4].

Question 6.4**.**

Can one use another cohomology to prove the analogue of 6.2 for $(\operatorname{NS}X)[p^{\infty}]$ in characteristic $p$?

Acknowledgement

The author thanks his advisor, Bjorn Poonen, for suggesting the problem, insightful conversation and careful guidance. The author thanks János Kollár, whose comments led to Section 5 and Section 6. The author thanks Wei Zhang for helpful conversation regarding étale homology. The author thanks Chenyang Xu for helpful conversation regarding Chow varieties.