Generic triviality of automorphism groups of complete intersections

Renjie Lyu School of Mathematical Sciences, Xiamen University, Xiamen 361005, China. [email protected] and Dingxin Zhang YMSC, Tsinghua University, Beijing 100086, China. [email protected]

Abstract.

We prove in most cases that a general smooth complete intersection in the projective space has no non-trivial automorphisms.

Introduction

Let $k$ be an algebraically closed field, and $X\subset\mathbf{P}^{n}_{k}$ be a smooth closed subvariety. A linear automorphism is an automorphism of $X$ that extends to an automorphism of $\mathbf{P}^{n}$ . We denote by $\mathrm{Aut}_{L}(X)$ the group of linear automorphisms of $X$ .

The primary aim of this note is to demonstrate that the linear automorphism group of a general smooth complete intersection in a projective space is trivial with a few undetermined cases. For hypersurfaces, Matsumura and Monsky [10] showed that when $n\geq 3$ and $d\geq 3$ , the group of linear automorphisms for a generic degree $d$ hypersurface in $\mathbf{P}^{n}$ is trivial. The cases for $n=2,d\geq 3$ can be found in [8, § 10] and [12]. Benoist [1] proved that the group of linear automorphisms of a smooth complete intersection is finite, except for hyperquadrics. Building upon this result, Chen et al. [2] further derived the following.

Theorem 0.1 ([2, Thm. 1.3]).

Let $(d_{1},\ldots,d_{c};n)$ be a sequence of natural numbers, with $d_{i}\geq 2$ and $n-c\geq 2$ . Let $X$ be a general smooth complete intersection in $\mathbf{P}^{n}_{k}$ of multidegree $(d_{1},\ldots,d_{c})\neq(2,2)$ .

(1)

if $k$ has characteristic $0$ , then $\mathrm{Aut}_{L}(X)=\{1\}$ .
(2)

if $k$ has characteristic $p>0$ , then there exists $r\geq 0$ , such that

$\mathrm{Card}\mathrm{Aut}_{L}(X)=p^{r}.$

The assertion $(2)$ in Theorem 0.1 does not determine the number $r$ explicitly in terms of $(d_{1},\ldots,d_{c};n)$ . In this exposition, we present an alternative method for studying the automorphism group of a general complete intersection, and show that $r=0$ in many cases. The main result is:

Theorem 0.2 (= Theorem 2.1).

Let $k$ be an algebraically closed field of characteristic $p\neq 2$ . Let $X\subset\mathbf{P}^{n}_{k}$ be a general smooth complete intersection of multidegree $(d_{1},\ldots,d_{c})$ with $c<n$ . Assume that $(d_{1},\ldots,d_{c})$ satisfies one of the following conditions

•

$3\leq d_{1}\leq d_{2}\leq\cdots\leq d_{c}$ and $p\nmid d_{1}$ ;
•

$c\geq 3$ and $2=d_{1}=d_{2}=d_{3}\leq\cdots\leq d_{c}$ .

Then we have $\mathrm{Aut}_{L}(X)=\{1\}$ if $(d_{1},\ldots,d_{c};n)\neq(3;2)$ .

We note that Javanpeykar and Loughran [7, Lemma 2.13-2.14] obatined similar conclusion for complex complete intersections of multidegree $(d_{1},\ldots,d_{c})$ being $(2,2,2)$ or satisfying $3\leq d_{1}<d_{2}\leq\cdots$ .

Our proof of Theorem 0.2 is inspired by Katz and Sarnak [8, §10-11]. They showed that the monodromy group of the universal family of smooth curves or smooth projective hypersurfaces is sufficiently big in a sense. As a consequence, it implies that a general member within the family has no automorphisms. We refine their method in general settings, and apply it to complete intersections.

Consider a smooth projective family of subvarieties of $\mathbf{P}^{n}$ parameterized by a smooth $k$ -variety $B$ . The family is represented in the diagram

Let $\mathcal{L}:=i^{*}\mathcal{O}_{\mathbf{P}^{N}}(1)$ be the natural polarization. In Section 1, we demonstrate the following criterion of triviality of linear automorphisms for a general member in the family.

Theorem 0.3 (= Theorem 1.2).

Assume that the following three conditions hold for the family $\pi:\mathcal{X}\to B$ .

•

The relative polarized automorphism group scheme $\operatorname{\mathrm{Aut}}_{B}(\mathcal{X},\mathcal{L})$ is finite and unramifed over $B$ .
•

There exists a closed point $b\in B$ such that $\mathrm{Aut}_{L}(X_{b})$ acts faithfully on the $\ell$ -adic cohomology $\mathrm{H}^{m}_{\mathrm{\acute{e}t}}(X_{b};\mathbf{Q}_{\ell})$ with $\dim X_{b}=m$ .
•

The monodromy group associated to the lisse $\mathbf{Q}_{\ell}$ -sheaf $(R^{m}\pi_{\ast}\mathbf{Q}_{\ell})_{\mathrm{prim}}$ is sufficiently big (see Theorem 1.2(3) for the precise statement).

Then the linear automorphism group $\mathrm{Aut}_{L}(X_{b})$ associated to a general point $b\in B$ is either trivial or isomorphic to $\mathbf{Z}/2\mathbf{Z}$ .

In Section 2, we show that the three conditions in Theorem 0.3 hold for most complete intersections. Suppose that $\pi\colon\mathcal{X}\to B$ is a family of smooth complete intersections of a given type. The group scheme $\operatorname{\mathrm{Aut}}_{B}(\mathcal{X},\mathcal{L})$ is finite over $B$ if the corresponding moduli stack is separated, which was explored by Benoist [1]. In cases the canonical divisor is ample or trivial, the separation isca direct consequence due to Matsusaka and Mumford [11]. The sufficient bigness of the monodromy group for complete intersections will be obtained by considering the vanishing cycles of Lefschetz pencils [3, §4-5]. Verifying the cohomological action by the automorphism group is faithful requires substantial effort and occupies the main part of this section. The advantage is that we need only verify the faithfullness property for a single member in the family. In our circumstance, we look into a special complete intersection of Fermat types (2.4). Lastly, we rule out the possibility of $\mathrm{Aut}(X_{b})\simeq\mathbf{Z}/2\mathbf{Z}$ through a direct argument, given the assumption $\mathrm{char}(k)\neq 2$ .

It is likely that conclusions in Theorem 0.2 can be strengthened. The assumption $p\nmid d_{1}$ is indispensible for considering complete intersections of Fermat type. One may check the faithfulness property for other special complete intersections. For the cases of complete intersections that not covered in Theorem 0.2, see Remark 2.3.

The article [2] also confirmed the faithfulness of the cohomological action. The main difference is: they directly proved that a general complex complete intersection has no non-trivial automorphisms, then use this result to deduce all complex complete intersections have the faithfulness property; In contrast, we gain the generic triviality result by verifying the faithfulness property for a single smooth complete intersection.

The cohomological method may find applications that are not limited to complete intersections in projective spaces. One example is offered below:

Theorem 0.4 (= Theorem 1.4).

Let $P$ be a smooth, complex projective variety. Let $\omega_{P}$ be the canonical sheaf on $P$ , and let $\mathscr{L}$ be a very ample invertible sheaf such that $\mathscr{L}\otimes\omega_{P}$ remains very ample. Then, a general smooth section of $\mathscr{L}$ has no non-trivial automorphisms.

Acknowledgements

We are grateful to Wenfei Liu for communications on this work. We would like to thank Xi Chen for comments on the manuscript.

1. A criterion of generic triviality of automorphisms

Situation 1.1.

Let $k$ be an algebraically closed field. Consider a smooth projective family of algebraic varieties

parameterized by a smooth, irreducible $k$ -variety $B$ . Let $\mathcal{L}:=i^{*}\mathcal{O}_{\mathbf{P}^{N}}(1)$ be the natural polarization. Denote $\operatorname{\mathrm{Aut}}_{B}(\mathcal{X},\mathcal{L})$ as the group scheme of automorphisms relative to the family $\pi:\mathcal{X}\to B$ , preserving $\mathcal{L}$ . Let $\mathcal{G}$ be a closed subgroup scheme of $\operatorname{\mathrm{Aut}}_{B}(\mathcal{X},\mathcal{L})$ . Let $X_{b}$ be the fiber $\pi^{-1}(b)$ over a closed point $b\in B$ , and $G_{b}$ be the fiber of $\mathcal{G}$ over $b$ .

Fix a prime number $\ell$ different from the characteristic of $k$ . Let $\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(X_{b};\mathbf{Q}_{\ell})$ be the $\ell$ -adic cohomology group. The cup-product on $\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(X_{b};\mathbf{Q}_{\ell})$ gives a non-degenerate symmetric (resp. anti-symmetric) form $\psi$ if $m$ is even (resp. odd). In this paper, the primitive cohomology $\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(X_{b};\mathbf{Q}_{\ell})_{\textrm{prim}}$ is defined to be the orthogonal complement of the image of the restriction map $\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(\mathbf{P}^{N};\mathbf{Q}_{\ell})\to\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(X_{b};\mathbf{Q}_{\ell})$ . Let $\pi_{1}(B,b)$ denote the étale fundamental group for of $B$ at the point $b$ .

Theorem 1.2.

Consider the notation in Situation 1.1. Suppose that the following hypotheses hold.

(1)

The group scheme $\mathcal{G}\to B$ is finite and unramifed over $B$ .
(2)

There exists a closed point $b\in B$ such that the map $G_{b}\to\operatorname{Aut}(\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(X_{b};\mathbf{Q}_{\ell}))$ is injective.
(3)

The geometric monodromy group, that is, the Zariski closure of the image of the monodromy representation

$\rho\colon\pi_{1}(B,b)\to\operatorname{Aut}(\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(X_{b};\mathbf{Q}_{\ell})_{\mathrm{prim}},\psi),$

equals the full symplectic group $\mathrm{Sp}(\mathrm{H},\psi)$ if $m$ is odd or the orthogonal group $\mathrm{O}(\mathrm{H},\psi)$ if $m$ is even, where $\mathrm{H}$ represents the $\mathbf{Q}_{\ell}$ -vector space $\mathrm{H}_{\mathrm{\acute{e}t}}^{m}(X_{b};\mathbf{Q}_{\ell})_{\mathrm{prim}}$ .

Then $G_{b}$ is either a trivial group or isomorphic to $\mathbf{Z}/2\mathbf{Z}$ for a general $b\in B$ .

Note that Hypotheses 1.2(1–3) are not sufficient to deduce the triviality of $G_{b}$ . For instance, let $B$ be the space of $6$ distinct ordered points on $\mathbf{P}^{1}$ , $\mathcal{X}\to B$ is the family of genus $2$ curves branching over the given points, and $\mathcal{G}=\operatorname{Aut}_{B}(\mathcal{X})$ . Then every $X_{b}$ has an automorphism of order $2$ , although the finiteness of automorphism groups, the faithfulness of $\operatorname{Aut}(X_{b})$ acting on $\mathrm{H}^{1}(X_{b})$ , and the bigness of the geometric monodromy group are all satisfied.

Proof of Theorem 1.2.

The morphism $p$ is finite and unramified. The fiber group $G_{b}$ over any $b\in B$ is thus finite and discrete, and the order $|G_{b}|$ is equal to the rank of the finite $\mathcal{O}_{B}$ -module $p_{*}\mathcal{O}_{\mathcal{G}}$ at the stalk $b$ . It follows that the function $b\mapsto|G_{b}|$ is upper semi-continuous. Then there is a non-empty open subset on which the number $|G_{b}|$ reaches the minimum. Consider the closed subgroup scheme

\mathcal{Z}\colon=\operatorname{Ker}(\mathcal{G}\to\mathrm{GL}(R^{m}\pi_{*}\mathbf{Q}_{\ell}))

of automorphisms acting trivially on the cohomology. Then $\mathcal{Z}\to B$ remains a finite and unramified morphism. By the previous argument, let $B^{\prime}\subset B$ be the open subset on which the order function $b\mapsto|Z_{b}|$ reaches the minimum. By Hypothesis 1.2(2), the minimum must be $1$ , i.e. $\mathcal{Z}|_{B^{\prime}}$ is the identity group scheme.

By the generic flatness, there exists a non-empty open subset $U\subset B$ such that morphism $p\colon\mathcal{G}\to B$ is flat over $U$ . Then $p$ is a finite étale morphism over $U$ . Since $B$ is irreducible, $U$ must meet with $B^{\prime}$ . In conclusion, we can obtain a Zarski open dense subset $U\subset B$ such that $\mathcal{G}$ is a finite étale cover over $U$ , and the fiber group $G_{b}$ for any $b\in U$ acts faithfully on $\mathrm{H}^{m}_{\mathrm{\acute{e}t}}(X_{b};\mathbf{Q}_{\ell})$ .

Let $b\in U$ be a closed point. The induced map $\pi_{1}(U,b)\to\pi_{1}(B,b)$ shall be surjective. By Hypothesis 1.2(3), the geometric monodromy group of $\pi_{1}(U,b)$ equals to $\mathrm{Sp}(\mathrm{H},\psi)$ or $\mathrm{O}(\mathrm{H},\psi)$ . Let $V\to U$ be a finite étale base change such that the pullback $\mathcal{G}|_{V}$ becomes a constant group scheme over $V$ . Under a finite base change, the identity component of the geometric monodromy group does not change. Since the symplectic group or the orthogonal group is connected, the geometric monodromy group of $\pi_{1}(V,b)$ is still $\mathrm{Sp}(\mathrm{H},\psi)$ or $\mathrm{O}(\mathrm{H},\psi)$ . Here we abusively denote $b$ as a closed point in $V$ lying over $b\in U$ .

As $\mathcal{G}|_{V}$ is a constant group scheme, the monodromy action of $\pi_{1}(V,b)$ on $G_{b}$ is invariant. Viewing $G_{b}$ as a subgroup of $\mathrm{GL}(\mathrm{H})$ , we have

\rho(\gamma)\cdot g\cdot\rho(\gamma)^{-1}=g,~{}\forall g\in G_{b},~{}\forall\gamma\in\pi_{1}(V,b)

where $\rho$ is the monodromy action of $\pi_{1}(V,b)$ . A linear automorphism on $\mathrm{H}$ that commutes with all symplectic or orthogonal matrices must be the scalar action by some $\lambda\in\mathbf{Q}^{*}_{\ell}$ . Assume that the degree of the finite étale covering $\mathcal{G}\to U$ is $r$ , that is, the order of $G_{b}$ is $r$ . Then $\lambda$ is a $r$ -th root of unity in $\mathbf{Q}_{\ell}$ .

By Hensel’s lemma, we know that any root of unity in $\mathbf{Q}_{\ell}$ is an $(\ell-1)$ -th root of unity for $\ell$ odd or $\pm 1$ for $\ell=2$ . In the case of $\mathrm{char}(k)\neq 2$ , let $\ell=2$ . Then $\lambda=\pm 1$ . When $\mathrm{char}(k)=2$ and $r$ is odd, we can select an odd prime $\ell$ such that $\ell-1$ is coprime to $r$ . Then $\lambda^{\ell-1}=1$ implies $\lambda=1$ since $(\ell-1,r)=1$ . When $\mathrm{char}(k)=2$ and $r$ is even, choose the odd prime $\ell$ such that $\frac{\ell-1}{2}$ is coprime to $r$ . Then we have $\lambda^{\frac{\ell-1}{2}}=\pm 1$ , which implies $\lambda=\pm 1$ since $(\frac{\ell-1}{2},r)=1$ . In conclusion, the order of the group $G_{b}$ for $b\in U$ is bounded by $2$ . ∎

As a simple application of Theorem 1.2, we show that a sufficiently ample, sufficiently general hypersurface in any smooth complex projective variety is free of automorphisms. To begin with, we need a lemma.

Lemma 1.3.

Let $X$ be an m–dimensional, smooth, complex projective variety with very ample canonical bundle. Then the natural map $\operatorname{Aut}(X)\to\operatorname{Aut}(\mathrm{H}^{m}(X;\mathbf{Q}))$ is injective.

Proof.

Since the canonical bundle $\omega_{X}$ is very ample, we consider the canonical embedding $X\hookrightarrow\mathbf{P}(\mathrm{H}^{0}(X,\omega_{X})^{\vee})$ . Let $f$ be an automorphism of $X$ such that $f^{\ast}$ is the identity. By the Hodge decomposition $f^{\ast}$ acts identically on $\mathrm{H}^{0}(X,\omega_{X})$ , which induces the idenity on $\mathbf{P}(\mathrm{H}^{0}(X,\omega_{X})^{\vee})$ . Via the canonical embedding, we have $f=\operatorname{Id}_{X}$ . ∎

Theorem 1.4.

Let $P$ be a smooth, complex projective variety. Let $\omega_{P}$ be the canonical sheaf on $P$ , and let $\mathscr{L}$ be a very ample invertible sheaf such that $\mathscr{L}\otimes\omega_{P}$ remains very ample. Then a general smooth section of $\mathscr{L}$ has no non-trivial automorphisms.

Proof.

We need to check Hypotheses 1.2(1–3) are satisfied. Let $X$ be the zero locus of any smooth transverse section of $\mathscr{L}$ . By the adjunction formula and our assumption, the canonical divisor $\omega_{X}\cong\mathscr{L}\otimes\omega_{P}$ is very ample.

Let $\mathrm{H}^{m}(X;\mathbf{C})_{p}$ be the kernel of the cup-product operator

\cup\omega_{X}\colon\mathrm{H}^{m}(X;\mathbf{C})\to\mathrm{H}^{m+2}(X;\mathbf{C})

with $m=\dim X$ . Let $Q$ be the intersection form. The Hodge-Riemann bilinear relation shows that the Hermitian form $h:=\oplus_{p+q=m}i^{p-q}Q(\cdot,\bar{\cdot})$ is positive definite on

\mathrm{H}^{m}(X;\mathbf{C})_{p}=\bigoplus_{p+q=m}\mathrm{H}^{p,q}\cap\mathrm{H}^{m}(X;\mathbf{C})_{p}.

Any automorphism of $X$ preserves $\omega_{X}$ and $h$ . By Lemma 1.3, $\operatorname{Aut}(X)$ can be regarded as a subgroup of the unitary group $\mathrm{U}(\mathrm{H}^{m}(X;\mathbf{C})_{p},h)$ . Additionally, $\operatorname{Aut}(X)$ acts on the integral cohomology $\mathrm{H}^{n}(X;\mathbf{Z})$ . Therefore, $\operatorname{\mathrm{Aut}}(X)$ is a discrete subgroup in a compact group, which is necessarily finite.

Let $\mathcal{X}\to B$ be the family of smooth sections of $\mathscr{L}$ . The above discussion implies that the relative automorphism group scheme

\mathrm{Aut}_{B}(\mathcal{X})\to B

is a quasi-finite morphism. The hypersurface $X$ is not uniruled since $\omega_{X}$ is ample. Matsusaka-Mumford’s theorem [11] ensures that the morphism $\mathrm{Aut}(\mathcal{X}/B)\to B$ is proper and hence finite. It is unramifed since a group scheme over the complex number is always smooth. Thus, Hypothesis 1.2(1) is verified.

The $\ell$ -adic cohomology of a complex variety is isomorphic to the Betti cohomology of the associated analytic space. In the following, when we apply Theorem 1.2, we will use Betti cohomology in place of the $\ell$ -adic cohomology. Then Lemma 1.3 affirms Hypothesis 1.2(2)

To show the monodromy group is as big as possible, we consider a Lefschetz pencil $D$ of hyperplane sections $|\mathscr{L}|$ on $P$ such that $D$ passing through $[X]\in|\mathscr{L}|$ . Vanishing cycles for the Lefschetz pencil $D$ are conjugate under the monodromy action, see [3] or [16, §2] for the details.

If $m$ is odd, Kazhdan-Margulis theorem [3, Théorème 5.10] shows that the geometric monodromy group associated to $D$ is the full sympelctic group $\mathrm{Sp}(\mathrm{H},\psi)$ where $\mathrm{H}\subset\mathrm{H}^{m}_{\textrm{prim}}(X;\mathbf{Q})$ stands for the subspace generated by vanishing cycles. If $m=2p$ is even, the geometric monodromy group is either a finite group or the full orthogonal group $\mathrm{O}(\mathrm{H},\psi)$ , see [4, Théorème 4.4.2]. The case of finite group occurs when the intersection form $\psi$ is definite. By the Hodge index theorem, it deduces that $\mathrm{H}^{p,q}(X)=0$ unless $p=q$ . However, in our case $\mathrm{H}^{m,0}(X)$ is non-trivial since the canonical divisor $\omega_{X}$ is very ample. Hence monodromy group is sufficently big.

As a consequence of Theorem 1.2, a general section $X$ of $\mathscr{L}$ either has no non-trivial automorphism, or admits an involution that acts by $-1$ on $\mathrm{H}^{m}(X)$ . The second possibility is readily ruled out by the same argument in the proof of Lemma 1.3. ∎

2. Automorphism of complete intersection in projective spaces.

Let $k$ be an algebraically closed field. Let $(d_{1},\ldots,d_{c};n)$ be a sequence of positive integers with $2\leq d_{i}\leq d_{i+1}$ , $n>c$ . We say a complete intersection $X\subset\mathbf{P}^{n}_{k}$ is of type $(d_{1},\ldots,d_{c};n)$ if the ideal of $X$ is generated by $c$ homogeneous polynomials with the prescribed degrees $\{d_{1},\ldots,d_{c}\}$ . The goal of this section is proving the following:

Theorem 2.1.

•

$3\leq d_{1}\leq d_{2}\leq\cdots\leq d_{c}$ and $p\nmid d_{1}$ ;
•

$c\geq 3$ and $2=d_{1}=d_{2}=d_{3}\leq\cdots\leq d_{c}$ .

Then we have $\mathrm{Aut}_{L}(X)=\{1\}$ if $(d_{1},\ldots,d_{c};n)\neq(3;2)$ .

We have the following simple observation.

Observation 2.2.

Let $X$ be a general complete intersection of multidegree $(d_{1},\ldots,d_{c})$ , and $F_{1},\ldots,F_{c}$ denote the defining polynomials of $X$ with $\deg F_{i}=d_{i}$ . Suppose that $r$ is the maximal number such that $d_{1}=\cdots=d_{r}<d_{r+1}$ . Then the action of any linear automorphism of $X$ preserves the ideal $(F_{1},\ldots,F_{r})$ . Regard $X$ as a subscheme of the complete intersection $Y$ defined by $(F_{1},\ldots,F_{r})$ , then we have $\operatorname{\mathrm{Aut}}_{L}(X)\subset\operatorname{\mathrm{Aut}}_{L}(Y)$ . Therefore, to prove $\operatorname{\mathrm{Aut}}_{L}(X)=\{1\}$ , it suffices to prove that a general complete intersection with equal multidegrees has no non-trivial automorphisms.

Remark 2.3.

The cases of complete intersections of type $(d_{1},\ldots,d_{c})$ with $2=d_{1}\leq d_{2}<d_{3}\leq\cdots\leq d_{c}$ are not covered in Theorem 2.1. According to Observation 2.2, such cases correspond to general hyperquadrics or complete intersections of two quadrics, which admit non-trivial linear automorphisms, see [13]. Hence, the proof strategy in Observation 2.2 does not work directly. Instead, one may attempt to characterize the autmorphism group for complete intersections of type $(2,d,\ldots,d;n)$ and $(2,2,d,\ldots,d;n)$ with $d>2$ .

To apply Theorem 1.2 to complete intersections with equal multidegrees, we will verify Hypothesis 1.2(2) for complete intersection of Fermat type. In the following, we will describe their automorphism group and cohomology group, see Proposition 2.5 and Theorem 2.10, and prove that the automorphism group acts faithfully on the cohomology group in Proposition 2.11.

Automorphism of complete intersections of Fermat type

Let $k$ be an algebraically closed field. Fix natural numbers $n\geq 3,r\geq 2$ , and $d\geq 2$ , where $d$ is prime to $\mathrm{char}(k)$ . Let $X:=X_{n,r,d}\subset\mathbf{P}^{n}_{k}$ be the complete intersection defined by the following Fermat equations:

(2.4)

\left\{\begin{matrix}x_{0}^{d}+\cdots+x_{n}^{d}=0,\\ \lambda_{0}x_{0}^{d}+\cdots+\lambda_{n}x_{n}^{d}=0,\\ \vdots\\ \lambda_{0}^{r-1}x_{0}^{d}+\cdots+\lambda_{n}^{r-1}x_{n}^{d}=0,\end{matrix}\right.

where $\{\lambda_{0},\ldots,\lambda_{n}\}$ are pairwise distinct elements in $k^{*}$ . It is known that $X$ is non-singular [15, Prop. 2.4.1]. Let $\mu_{d}$ be the group of $d$ -th roots of unity in $k$ , and $\mu_{d}^{n+1}$ be the $(n+1)$ -th product of $\mu_{d}$ . Denote by $G_{n}^{d}$ the quotient group $\mu_{d}^{n+1}/\Delta(\mu_{d})$ , where $\Delta:\mu_{d}\to\mu_{d}^{n+1}$ is the diagonal embedding. The group $G_{n}^{d}$ acts on the variety $X$ as follows:

[\xi_{0},\ldots,\xi_{n}]\cdot(x_{0}:\ldots:x_{n})\mapsto(\xi_{0}x_{0}:\ldots:\xi_{n}x_{n}),~{}\xi_{i}\in\mu_{d}

Proposition 2.5.

Let $X:=X_{n,r,d}\subset\mathbf{P}^{n}_{k}$ be the Fermat complete intersection defined by the equations above, and let $\operatorname{\mathrm{Aut}}_{L}(X)$ be the group of linear automorphisms of $X$ . Then $\operatorname{\mathrm{Aut}}_{L}(X)$ fits into an exact sequence

1\to G^{d}_{n}\to\operatorname{\mathrm{Aut}}_{L}(X)\to\mathfrak{S}_{n+1},

where $\mathfrak{S}_{n+1}$ is the permutation group of $n+1$ elements. Moreover, if the coefficients $\lambda_{0},\ldots,\lambda_{n}$ correspond to a general point in the $k$ -vector space $k^{n+1}$ , then $G^{d}_{n}=\operatorname{\mathrm{Aut}}_{L}(X)$ .

Proof.

Let $(x_{0}\colon\dots\colon x_{n})$ be a coordinate in $\mathbf{P}^{n}$ . The action of a linear automorphism $g\in\operatorname{\mathrm{Aut}}_{L}(X)$ on $x_{i}$ transforms $x_{i}$ to a linear form

g_{i}:=g^{*}(x_{i})=\sum_{j=0}^{n}a_{ij}x_{j},~{}a_{ij}\in k.

Since the ideal of $X$ is generated by the Fermat equations (2.4), we observe

(2.6)

g^{*}(x_{0}^{d}+\cdots+x_{n}^{d})=\sum_{i=0}^{n}g_{i}^{d}=\sum_{i=0}^{n}c_{i}x_{i}^{d},\text{~{}for some~{}}c_{i}\in k.

Differentiating both sides of (2.6) with respect to $\frac{\partial}{\partial x_{j}}$ , we obtain

d\sum_{i=0}^{n}g^{d-1}_{i}\cdot\frac{\partial g_{i}}{\partial x_{j}}=d\sum_{i=0}^{n}g^{d-1}_{i}a_{ij}=d\cdot c_{j}\cdot x_{j}^{d-1}.

Focusing on the coefficients of the monomial $x_{0}^{d-1}$ on both sides. There is the relation

d\cdot\sum_{i=0}^{n}a_{ij}a_{i0}^{d-1}=\begin{cases}0,&j\neq 0;\\ d\cdot c_{0},&j=0.\end{cases}

Let $A$ represent the matrix $(a_{ij})$ . Since $d$ is invertible in the field $k$ , we have

(a_{00}^{d-1},\ldots,a_{n0}^{d-1})\cdot A=(c_{0},0,\ldots,0).

Similarly,

g^{*}(\lambda_{0}x_{0}^{d}+\cdots+\lambda_{n}x_{n}^{d})=\sum_{i=0}^{n}\lambda_{i}g_{i}^{d}=\sum_{i=0}^{n}c_{i}^{\prime}x_{i}^{d},\text{~{}for some~{}}c^{\prime}_{i}\in k.

Following the same discussion as above, we have

(\lambda_{0}a_{00}^{d-1},\ldots,\lambda_{n}a_{n0}^{d-1})\cdot A=(c_{0}^{\prime},0,\ldots,0).

Note that the matrix $A$ is invertible, and $c_{0},c^{\prime}_{0}$ are non-zero. Hence $(a_{00}^{d-1},\ldots,a_{n0}^{d-1})$ is proportional to $(\lambda_{0}a_{00}^{d-1},\ldots,\lambda_{n}a_{n0}^{d-1})$ . Since $\lambda_{0},\lambda_{1},\ldots,\lambda_{n}$ are pairwise distinct, only one element among $\{a_{00},\ldots,a_{n0}\}$ is non-zero. Considering the coefficients of $x_{j}^{d-1}$ for each $j$ , the same conclusion holds for each tuple $(a_{0j},\ldots,a_{nj})$ . Therefore, $g$ determines a permutation $\sigma_{g}\in\mathfrak{S}_{n+1}$ such that $g^{*}(x_{i})=a_{\sigma_{g}(i)i}x_{i}$ . The assignment $g\mapsto\sigma_{g}$ thus establishes a group homomorphism

\sigma:\operatorname{\mathrm{Aut}}_{L}(X)\to\mathfrak{S}_{n+1}.

In the following, we demonstrate that the kernel of $\sigma$ is $G^{d}_{n}$ . Suppose that $g\in\operatorname{\mathrm{Ker}}\sigma$ . Then $g$ is represented by a diagonal matrix $(a_{ii})_{0\leq i\leq n}$ . The action of $g$ on the Fermat polynomials $\{\sum\limits_{i=0}^{n}\lambda_{i}^{j}x_{i}^{d}~{}|~{}0\leq j\leq r-1\}$ yields an $r\times r$ matrix $B$ satisfying the relationship

(2.7)

B\cdot\begin{pmatrix}1&\cdots&1\\ \lambda_{0}&\cdots&\lambda_{n}\\ \vdots&&\vdots\\ \lambda_{0}^{r-1}&\cdots&\lambda_{n}^{r-1}\end{pmatrix}=\begin{pmatrix}1&\cdots&1\\ \lambda_{0}&\cdots&\lambda_{n}\\ \vdots&&\vdots\\ \lambda_{0}^{r-1}&\cdots&\lambda_{n}^{r-1}\end{pmatrix}\cdot\begin{pmatrix}a^{d}_{00}&&\\ &\ddots&\\ &&a^{d}_{nn}\end{pmatrix}

Let us view each column $(1,\lambda_{i},\ldots,\lambda_{i}^{r-1})^{\mathsf{T}}$ as a vector $v_{i}$ in the $r$ -dimensional vector space $k^{r}$ . Then $B$ represents a linear map with eigenvectors $\{v_{0},\ldots,v_{n}\}$ and eigenvalues $\{a_{00}^{d},\ldots,a_{nn}^{d}\}$ . Note that any $r$ elements among $\{v_{0},\ldots,v_{n}\}$ form a basis of $k^{r}$ since the basis corresponds to a non-degenerate Vandermonde matrix. It implies that $a_{ii}^{d}=a_{jj}^{d}$ for all $i,j$ . Consequently, the matrix $(a_{ii})$ can be represented by

\begin{pmatrix}\xi_{0}\cdot a_{00}&&\\ &\ddots&\\ &&\xi_{n}\cdot a_{00}\end{pmatrix},~{}\textrm{for some}~{}\xi_{i}\in\mu_{d}.

Hence, the action of $g$ on $X$ is equivalent to the action of $[\xi_{0},\cdots,\xi_{n}]\in G_{n}^{d}$ .

Now let us prove the final assertion. Given a linear automorphism $g$ , let $\sigma$ denote the permutation $\sigma(g)\in\mathfrak{S}_{n+1}$ , and $(a_{\sigma(i)i})$ be the matrix representing $g$ . Again, the action of $g^{*}$ on the Fermat polynomials (2.4) yields an $r\times r$ matrix $B:=(b_{ij})_{0\leq i,j\leq r-1}$ such that

B\cdot\begin{pmatrix}1&\cdots&1\\ \lambda_{0}&\cdots&\lambda_{n}\\ \vdots&&\vdots\\ \lambda_{0}^{r-1}&\cdots&\lambda_{n}^{r-1}\end{pmatrix}=\begin{pmatrix}1&\cdots&1\\ \lambda_{0}&\cdots&\lambda_{n}\\ \vdots&&\vdots\\ \lambda_{0}^{r-1}&\cdots&\lambda_{n}^{r-1}\end{pmatrix}\cdot(a_{\sigma(i)i}).

By this relation, the first two rows of the matrix $B$ correspond to two polynomials

q(t)=\sum_{j=0}^{r-1}b_{0j}t^{j},~{}p(t)=\sum_{j=0}^{r-1}b_{1j}t^{j}

satisfying the interpolation data

q(\lambda_{i})=a_{\sigma(i)i},~{}p(\lambda_{i})=\lambda_{\sigma(i)}a_{\sigma(i)i},\textrm{~{}for~{}}0\leq i\leq n.

Consider the Lagrangian polynomial

L(t):=\sum_{k=0}^{n}a_{\sigma(k)k}\prod_{0\leq j\leq n,j\neq k}\frac{t-\lambda_{j}}{\lambda_{k}-\lambda_{j}}

which interpolates the given data $(\lambda_{i},\alpha_{\sigma(i)i})$ . Assume that the constants $\{\alpha_{\sigma(i)i}\}_{0\leq i\leq n}$ are not all equal. It is known that, for a generic choice of the tuple $(\lambda_{0},\ldots,\lambda_{n})$ in $k^{n+1}$ , the degree of $L(t)$ is $n$ . Hence $L(t)=q(t)$ since $L(t)-q(t)$ contains $n+1$ distinct roots. However, we have $\deg q(t)=r-1<n$ as a contradiction. Therefore, all $a_{\sigma(i)i}$ must be equal, and $q(t)$ is the constant polynomial $b_{00}$ . It follows that the second interpolation data becomes

p(\lambda_{i})=b_{00}\lambda_{\sigma(i)}.

Applying the Lemma 2.8 below, such a polynomial $p(t)$ exists for a generic choice of $(\lambda_{0},\ldots,\lambda_{n})$ in $k^{n+1}$ only if $\sigma=\operatorname{Id}$ . Thus, the last assertion follows. ∎

Lemma 2.8.

Let $\sigma\in\mathfrak{S}_{n+1}$ be a non-trivial permutation, and let $r$ be an integer with $1<r<n$ . Suppose that $(\lambda_{0},\ldots,\lambda_{n})\in k^{n+1}$ is a general point. Then there exists no polynomial function $p(t)\in k[t]$ with $\deg p(t)\leq r-1$ that interpolates the $n+1$ data points $(\lambda_{0},\lambda_{\sigma(0)}),\ldots,(\lambda_{n},\lambda_{\sigma(n)})$ , i.e.,

p(\lambda_{i})=\lambda_{\sigma(i)},0\leq i\leq n.

Proof.

We identify the space of one-variable polynomials of degree at most $r-1$ with the $r$ -dimensional vector space $k^{r}$ . We consider the incidence subvariety

\Gamma_{\sigma}:=\{(p(t),(y_{0},\ldots,y_{n}))\in k^{r}\times k^{n+1}~{}|~{}p(y_{i})-y_{\sigma(i)}=0,~{}0\leq i\leq n\}.

Let $\pi_{1}:\Gamma_{\sigma}\to k^{r}$ and $\pi_{2}:\Gamma_{\sigma}\to k^{n+1}$ be natural projections. Suppose that $(p(t),(\lambda_{0},\ldots,\lambda_{n}))$ is a point in $\Gamma_{\sigma}$ , and $N$ is the order of $\sigma$ , then

p^{\circ N}(\lambda_{i})=\lambda_{\sigma^{N}(i)}=\lambda_{i},~{}\forall~{}0\leq i\leq n.

Case 1. The polynoimal $p^{\circ N}(t)-t$ is non-zero. Then $p^{\circ N}(t)-t$ has finite roots. Hence the possiblity of $\{\lambda_{0},\ldots,\lambda_{n}\}$ are finitely many. It follows that $\pi_{1}$ is a quasi-finite map, and the dimension of $\Gamma_{\sigma}$ is $r$ . Then $\pi_{2}(\Gamma_{\sigma})$ is a proper subset in $k^{n+1}$ . Therefore, for a generic $(\lambda_{0},\ldots,\lambda_{n})\in k^{n+1}$ , there exists no $p(t)\in k^{r}$ satisfying the condition $p(\lambda_{i})=\lambda_{\sigma(i)},0\leq i\leq n$ .

Case 2. The polynoimal $p^{\circ N}(t)-t$ is zero. This situation occurs only if $p(t)$ is a linear form $\alpha+\beta t$ with $\beta^{N}=1$ . All such linear forms consitutes a one-dimensional subset $\Xi\subset k^{r}$ . We claim that $\dim\pi_{1}^{-1}(\Xi)<n+1$ . The fiber of $\pi_{1}$ over a linear form $\alpha+\beta t$ is a subset in $k^{n+1}$ defined by $n+1$ equations

\alpha+\beta y_{i}=y_{\sigma(i)},0\leq i\leq n.

Since $n>r\geq 2$ and $\sigma\neq\operatorname{Id}$ , these equations impose at least two constraints on $k^{n+1}$ . Therefore, the dimension of $\pi_{1}^{-1}(\Xi)$ is less than $n+1$ . In conclusion, for a generic choice of $(\lambda_{0},\ldots,\lambda_{n})\in k^{n+1}$ , no linear form can interpolate the data $(\lambda_{i},\lambda_{\sigma(i)})_{0\leq i\leq n}$ . ∎

The exposition regarding the cohomology of the complete intersection of Fermat type is attributed to Terasoma [15]. It is characterized by abelian covers of the projective line, as outlined in the following.

Cohomology of the complete intersection of Fermat type

Let $\lambda_{0},\ldots,\lambda_{n}$ be the $n+1$ distinct elements in $k$ . Let $\pi_{1}(\mathbf{P}^{1}-\{\lambda_{0},\ldots,\lambda_{n}\})$ be the fundamental group of the punctured sphere $\mathbf{P}^{1}-\{\lambda_{0},\ldots,\lambda_{n}\}$ . We denote $G^{d}_{n}$ as the quotient group $(\mathbf{Z}/d\mathbf{Z})^{n+1}/\Delta(\mathbf{Z}/d\mathbf{Z})$ , where $\Delta:\mathbf{Z}/d\mathbf{Z}\to(\mathbf{Z}/d\mathbf{Z})^{n+1}$ represents the diagonal embedding.

Let $\gamma_{i}\in\pi_{1}(\mathbf{P}^{1}-\{\lambda_{0},\ldots,\lambda_{n}\})$ be the loop winding counter-clockwise around the point $\lambda_{i}$ . There is a surjective map

\pi_{1}(\mathbf{P}^{1}-\{\lambda_{0},\ldots,\lambda_{n}\})\to G^{d}_{n}

by assigning each $\gamma_{i}$ to the $i$ -th generator $(0,\ldots,1,\ldots,0)\in G^{d}_{n}$ . It corresponds to a nonsingular curve $D$ with function field

k(D):=k(x,\sqrt[d]{f_{1}},\ldots,\sqrt[d]{f_{n}}),~{}f_{i}=\frac{x-\lambda_{i}}{x-\lambda_{0}}.

The covering map $D\to\mathbf{P^{1}}$ is unramified over $\mathbf{P}^{1}-\{\lambda_{0},\ldots,\lambda_{n}\}$ , and the Galois group $\mathrm{Gal}(D/\mathbf{P}^{1})$ is isomorphic to $G^{d}_{n}$ , cf. [14, §2.1].

Let $\mathfrak{S}_{n-r}$ be the permutation group of $(n-r)$ elements. The group $\mathfrak{S}_{n-r}$ acts on the product group $(G^{d}_{n})^{n-r}$ by permuting its components. The product $D^{n-r}$ of the curve $D$ is naturally endowed with a group action by the semi-direct product $(G^{d}_{n})^{n-r}\rtimes\mathfrak{S}_{n-r}$ . Define $N$ as the kernel of the homomorphism

\Sigma:(G^{d}_{n})^{n-r}\rtimes\mathfrak{S}_{n-r}\to G^{d}_{n},~{}(g_{1},\ldots,g_{n-r};\sigma)\mapsto\sum_{i=1}^{n-r}g_{\sigma(i)}.

It has benn shown by Terasoma [15, Thm. 2.4.2] that the complete intersection $X_{n,r,d}$ of Fermat type is isomorphic to the quotient space $D^{n-r}/N$ .

Let $\zeta_{d}\in\mu_{d}$ be a primitive $d$ -th root of unity. A character $\chi:G^{d}_{n}\to\mu_{d}$ can be represented by a tuple $(a_{0},\ldots,a_{n})\in(\mathbf{Z}/d\mathbf{Z})^{\oplus n+1}$ with $\sum a_{i}\equiv 0\ \mathrm{mod}\ d$ , which allows us to express $\chi([k_{0},\ldots,k_{n}])=\zeta_{d}^{a_{0}k_{0}+\cdots+a_{n}k_{n}}$ for any $[k_{0},\ldots,k_{n}]\in G^{d}_{n}$ . By Kummer theory, the character $\chi$ corresponds to a nonsingular curve $C_{\chi}$ as the group quotient of $D$ by the action of $\operatorname{Ker}(\chi)$ . To be more explicit, $C_{\chi}$ is the cyclic cover of $\mathbf{P}^{1}$ determined by the equation

(2.9)

y^{e}=(x-\lambda_{0})^{b_{0}}\cdots(x-\lambda_{n})^{b_{n}},~{}b_{i}=\frac{ea_{i}}{d}

where $e=\#\operatorname{Im}(\chi)$ and $\frac{d}{e}=\textrm{gcd}(a_{0},\ldots,a_{n})$ .

Let $K$ be a field extension of $\mathbf{Q}_{\ell}$ containing $d$ -th roots of unity, such as $K=\mathbf{Q}_{\ell}(\zeta_{d})$ . For a $G^{d}_{n}$ -representation $V$ over $K$ and a character $\chi$ of $G^{d}_{n}$ , the $\chi$ -eigenspace is

V_{\chi}:=\{v\in V~{}|~{}g(v)=\chi(g)v\text{~{}for all~{}}g\in G^{d}_{n}\}.

The group $G^{d}_{n}$ naturally acts on the cohomology space $\mathrm{H}^{1}(D,K)$ . By the description of $C_{\chi}$ , the $\chi$ -eignespace $\mathrm{H}^{1}(D,K)_{\chi}$ is isomorphic to $\mathrm{H}^{1}(C_{\chi},K)$ . Recall that the primitive part $\mathrm{H}^{i}_{\mathrm{prim}}(X_{n,r,d},K)$ is the orthogonal complement of the image of the restriction map $\mathrm{H}^{i}(\mathbf{P}^{n},K)\to\mathrm{H}^{i}(X_{n,r,d},K)$ .

Theorem 2.10 ([15, Thm. 2.5.1]).

Let notaions be as above. The primtivie cohomology of $X_{n,r,d}$ decomposes as follows

\mathrm{H}^{i}_{\mathrm{prim}}(X_{n,r,d},K)\cong\begin{cases}0&i\neq n-r;\\ \bigoplus_{\chi\in\hat{G}^{d}_{n}}\wedge^{n-r}\mathrm{H}^{1}(D,K)_{\chi}&i=n-r.\end{cases}

The cohomology of $X_{n,r,d}$ is therefore expressed in terms of the cyclic covers $C_{\chi}$ . This decomposition is essential for proving the faithfulness of the cohomological action by the automorphism group of $X_{n,r,d}$ . We first examine the cases of complete intersections of quadrics.

Complete intersections of two or more quadrics

Let $X_{n,r,2}\subset\mathbf{P}^{n}_{k}$ be the complete intersection of quadrics of Fermat type defined as in (2.4), and $k$ is an algebraically closed field of $\mathrm{char}(k)\neq 2$ . Let $\chi\in\hat{G}^{2}_{n}$ be the character represented by $(a_{0},\ldots,a_{n})\in(\mathbf{Z}/2\mathbf{Z})^{\oplus n+1}$ , and $g$ be the integer $\frac{1}{2}\#\{a_{i}=1\}$ . The corresponding cyclic cover $C_{\chi}$ is a hyperelliptic curve ramified at $2g$ points. By Hurwitz’s formula, the genus of $C_{\chi}$ is equal to $g-1$ . As $C_{\chi}$ is complete and nonsingular, it follows that $\mathrm{H}^{1}(C_{\chi},\mathbf{Q}_{\ell})\cong\mathbf{Q}_{\ell}^{2g-2}$ for any odd prime $\ell$ .

•

$n+1$ even, $r=2$ . In this case, the space $\wedge^{n-2}\mathrm{H}^{1}(C_{\chi},\mathbf{Q}_{\ell})$ is non-zero if and only if $2g=n+1$ , i.e., $\chi=(1,\ldots,1)$ . Hence $\mathrm{H}^{n-2}_{\mathrm{prim}}(X_{n,r,2},\mathbf{Q}_{\ell})$ is isomorphic to the single $\chi$ -eigenspace $\wedge^{n-2}\mathrm{H}^{1}(C_{\chi},\mathbf{Q}_{\ell})$ . Let $e_{i}\in G^{2}_{n}$ be the action

$(x_{0}:\cdots:x_{i}:\cdots:x_{n})\mapsto(x_{0}:\cdots:-x_{i}:\cdots:x_{n}).$

The eigenvalue of the induced action $e_{i}^{*}$ on $\wedge^{n-2}\mathrm{H}^{1}(C_{\chi})$ is $\chi(e_{i})=-1$ . We can see that $e^{*}_{i}\circ e^{*}_{j}=1$ but $e_{j}\circ e_{i}\neq\operatorname{Id}$ . Thus the action of $G^{2}_{n}$ on the cohomology of $X$ is not faithful.
•

$n+1$ even, $r\geq 3$ . Consider the characters

$\chi_{i,j}:=(1\ldots,a_{i},\ldots,a_{j},\ldots,1),~{}a_{i}=a_{j}=0.$

The genus of $C_{\chi_{i,j}}$ is $\frac{n-3}{2}$ , and $\mathrm{H}^{1}(C_{\chi_{i,j}},\mathbf{Q}_{\ell})\cong\mathbf{Q}_{\ell}^{n-3}$ . Hence $\wedge^{n-r}\mathrm{H}^{1}(C_{\chi_{i,j}},\mathbf{Q}_{\ell})$ is non-zero. For an automorphism $\sigma:=(b_{0},\ldots,b_{n})\in G^{2}_{n}$ , we have

$\frac{\chi_{i,j}(\sigma)}{\chi_{i,j^{\prime}}(\sigma)}=(-1)^{b_{j^{\prime}}-b_{j}}.$

Assuming that $\sigma$ acts trivially on the cohomology, then $\chi_{i,j}(\sigma)=1$ for all $i,j$ . It implies that

$b_{j}\equiv b_{j^{\prime}}\mathrm{~{}mod~{}}2,~{}\forall j^{\prime},j,$

and $\sigma=(b_{0},\ldots,b_{n})$ is a diagonal element in $G^{2}_{n}$ , which represents the identity map on $X$ . Thus the action of $G^{2}_{n}$ on the primitive cohomology of $X_{n,r,2}$ is faithful.
•

$n+1$ odd, $r\geq 2$ . Let us consider the characters

$\chi_{i}=(1,\ldots,a_{i},\ldots,1),~{}a_{i}=0.$

The genus of $C_{\chi_{i}}$ is $\frac{n-2}{2}$ , and $\mathrm{H}^{1}(C_{\chi_{i}},\mathbf{Q}_{\ell})\cong\mathbf{Q}_{\ell}^{n-2}$ . Then $\wedge^{n-r}\mathrm{H}^{1}(C_{\chi_{i}},\mathbf{Q}_{\ell})$ is non-trivial for all $0\leq i\leq n$ . Let $\sigma:=(b_{0},\ldots,b_{n})\in G^{2}_{n}$ . We have

$\frac{\chi_{i}(\sigma)}{\chi_{j}(\sigma)}=(-1)^{-b_{i}+b_{j}}.$

Assuming that $\sigma$ acts trivially on the cohomology, then $\chi_{i}(\sigma)=1$ for all $i$ . It implies that

$b_{i}\equiv b_{j}\mathrm{~{}mod~{}}2,~{}\forall i,j.$

By the same reason as above, $G^{2}_{n}$ acts faithfully on the primitive cohomology of $X_{n,r,2}$ .

Proposition 2.11.

Let $X:=X_{n,r,d}\subset\mathbf{P}^{n}_{k}$ be the complete intersection of Fermat type defined in (2.4). Fix a prime $\ell\neq\mathrm{char}(k)$ . Then the group $\operatorname{\mathrm{Aut}}_{L}(X)$ of linear automorphisms acts faithfully on the primitive cohomology $\mathrm{H}^{n-r}_{\mathrm{prim}}(X,\mathbf{Q}_{\ell})$ unless $d=r=2$ and $n+1$ is even.

Proof.

We begin by proving the faithfulness of the action of the subgroup $G^{d}_{n}$ on $\mathrm{H}^{n-r}_{\mathrm{prim}}(X,\mathbf{Q}_{\ell})$ .

Consider a field extension $K$ of $\mathbf{Q}_{\ell}$ containing the primitive $d$ -th root of unity, e.g., $K=\mathbf{Q}_{\ell}(\zeta_{d})$ . By Theorem 2.10, there is the eigenspace decomposition

\mathrm{H}^{n-r}_{\mathrm{prim}}(X,K)=\bigoplus_{\chi\in\hat{G}^{d}_{n}}\wedge^{n-r}\mathrm{H}^{1}(C_{\chi},K),

where $C_{\chi}$ is the cyclic over of $\mathbf{P}^{1}$ defined by (2.9).

Suppose that a linear automorphism $g\in\operatorname{\mathrm{Aut}}_{L}(X)$ acts trivially on $\mathrm{H}^{n-r}_{\mathrm{prim}}(X,\mathbf{Q}_{\ell})$ . By considering the base change of coefficients and the above decomposition, $g$ also acts trivially on each summand $\wedge^{n-r}\mathrm{H}^{1}(C_{\chi},K)$ . Since

v=g^{*}(v)=\chi(g)v,~{}\forall v\in\wedge^{n-r}\mathrm{H}^{1}(C_{\chi},K),

it follows that $\chi(g)=1$ provided that $\wedge^{n-r}\mathrm{H}^{1}(C_{\chi},K)\neq 0$ . To prove $g=\operatorname{Id}$ , we aim to select a subset $S$ of characters of $G^{d}_{n}$ satisfying

•

$\wedge^{n-r}\mathrm{H}^{1}(C_{\chi},K)\neq 0,\forall\chi\in S$ ;
•

$\{g\in G^{d}_{n}~{}|~{}\chi(g)=1,\forall\chi\in S\}=\{\operatorname{Id}\}.$

Choose integers $k,s,t\in\mathbf{Z}_{\geq 0}$ satisfying

d\mid k+n,~{}d\mid s+t+n-1\text{~{}and~{}}(n+t,d)=1.

Consider the following characters

	$\displaystyle\chi_{k}$	$\displaystyle:=(1,\ldots,1,k);$
	$\displaystyle\chi_{(s,t,i)}$	$\displaystyle:=(1,\ldots,s,\ldots,t),1\leq i\leq n,$

where $\chi_{(s,t,i)}$ means the $i$ -th component is $s$ , the $(n+1)$ -th component is $t$ , and $1$ otherwise. As a result of Lemma 2.11 below, we can assert from

\chi_{k}(g)=\chi_{(s,t,i)}(g)=1,~{}1\leq i\leq n,

that $g$ is the identity element in $G^{d}_{n}$ . The remaining is to verify

\wedge^{n-r}\mathrm{H}^{1}(C_{\chi},K)\neq 0

for all $\chi=\chi_{k}$ or $\chi_{(s,t,i)}$ .

•

For $\chi=\chi_{k}$ , the cyclic covering $\pi:C_{\chi}\to\mathbf{P}^{1}$ is determined by the affine equation

$y^{d}=\prod_{i=0}^{n-1}(x-\lambda_{i})(x-\lambda_{n})^{k}.$

The branch points $\{\lambda_{0},\ldots,\lambda_{n-1}\}$ have a common ramification index $d$ . The ramification index over the point $\lambda_{n}$ is $\frac{d}{\operatorname{gcd}(k,d)}$ . The point at infinity $\infty\in\mathbf{P}^{1}$ is unbranched because $k+n\equiv 0\ \mathrm{mod}\ d$ . By Hurwitz’s formula

$2g(C_{\chi})-2=-2d+\sum_{0\leq i\leq n-1}(d-1)+\frac{d}{\operatorname{gcd}(k,d)},$

we obtain

$\dim_{K}\mathrm{H}^{1}(C_{\chi},K)=2g(C_{\chi})\geq(d-1)(n-2).$

Since $n\geq 3$ and $r\geq 2$ in our case, it follows that $\wedge^{n-r}\mathrm{H}^{1}(C_{\scalebox{0.8}{$\chi$}_{k}},K)\neq 0$ for all possible $k$ .

•

For $\chi=\chi_{(s,t,i)}$ , the cyclic covering $C_{\chi}\to\mathbf{P}^{1}$ is determined by the affine equation

y^{d}=\prod_{0\leq j\leq n-1,j\neq i}(x-\lambda_{j})(x-\lambda_{i})^{s}(x-\lambda_{n})^{t}

Again by Hurwitz’s formula

2g(C_{\chi})-2=-2d+\sum_{0\leq j\neq i\leq n-1}(d-1)+\frac{d}{\operatorname{gcd}(s,d)}+\frac{d}{\operatorname{gcd}(t,d)},

we have

\dim\mathrm{H}^{1}(C_{\chi},K)=2g(C_{\chi})\geq(d-1)(n-3).

Let us discuss following cases.

(1)

$n>3,d>2$ and $r\geq 2$ . Then $(d-1)(n-3)\geq n-r$ . Therefore $\wedge^{n-r}\mathrm{H}^{1}(C_{\chi_{(s,t,i)}})\neq 0$ for all possible $s$ and $t$ .
(2)

$n=3,d>2$ and $r=2$ . Note that $s\cdot t\neq 0$ , otherwise the condition $(\star)$ implies $d\mid 2$ as a contradiction. Hence we can assume $s\neq 0$ and $\frac{d}{\operatorname{gcd}(s,d)}>0$ . Then Hurwitz’s formula gives the inequality $2g(C_{\chi})\geq(d-1)(n-3)+1\geq n-2$ , which implies that $\wedge^{n-2}\mathrm{H}^{1}(C_{\chi_{(s,t,i)}})\neq 0$ for all possible $s$ and $t$ .
(3)

For $d=2$ , this case has been exhibited in the previous discussion for complete intersections of quadrics. The faithfulness only fails when $d=r=2$ and $n+1$ is even.

In conclusion, the action of $G^{d}_{n}$ on $\mathrm{H}^{n-r}_{\mathrm{prim}}(X,K)$ , as well as on $\mathrm{H}^{n-r}_{\mathrm{prim}}(X,\mathbf{Q}_{\ell})$ , is faithful, except for $X$ being an odd-dimensional complete intersection of two quadrics. Then the assertion of this proposition holds for a generic complete intersection $X$ of Fermat type as Proposition 2.5 ensures $\operatorname{\mathrm{Aut}}_{L}(X)=G^{d}_{n}$ .

Now we consider the cases $G^{d}_{n}<\operatorname{\mathrm{Aut}}_{L}(X)$ . For a character $\chi\in\hat{G}^{d}_{n}$ and an automorphism $\sigma\in\operatorname{\mathrm{Aut}}_{L}(X)$ , we define the character $\chi^{\sigma}$ as

\chi^{\sigma}(g):=\chi(\sigma g\sigma^{-1}).

Note that $\sigma g\sigma^{-1}\in G^{d}_{n}$ since $G^{d}_{n}$ is a normal subgroup. The action $\sigma^{*}$ transfers a $\chi$ -eigenspace to a $\chi^{\sigma}$ -eigenspace if the $\chi$ -eigenspace is nontrivial. Specifically, for any $v\in\mathrm{H}^{n-r}_{\mathrm{prim}}(X,K)_{\chi}$

g^{*}\sigma^{*}(v)=\sigma^{*}(\sigma g\sigma^{-1})^{*}(v)=\sigma^{*}(\chi(\sigma g\sigma^{-1})v)=\chi^{\sigma}(g)\sigma^{*}(v).

Suppose that $\sigma^{*}$ acts trivially on $\mathrm{H}^{n-r}_{\mathrm{prim}}(X,K)$ . Then $\chi=\chi^{\sigma}$ for any $\chi\in S$ . Hence

\chi(g)=\chi(\sigma g\sigma^{-1}),~{}\forall g\in G^{d}_{n}.

From the preceding discussion, it follows that $g=\sigma g\sigma^{-1}$ for all $g\in G^{d}_{n}$ , i.e., $\sigma$ is a commutator of the subgroup $G^{d}_{n}$ . Let $g$ be a diagonal matrix with distinct entries. The relation $\sigma g=g\sigma$ implies $\sigma$ must be a diagonal matrix, thus $\sigma$ is contained in $G^{d}_{n}$ . Given that the action of $G^{d}_{n}$ on the cohomology is faithful, we conclude that $\sigma$ is the identity map. ∎

Lemma 2.11.

Let $d$ and $n$ be positive integers. Choose positive integers $k,t,s\in\mathbf{Z}$ such that

d\mid k+n,~{}(n+t,d)=1\textrm{~{}and~{}}s+t\equiv k+1\ \textrm{mod}\ d.

Consider the $(n+1)\times(n+1)$ -matrix

(2.12)

A:=\begin{pmatrix}s&\ldots&1&t\\ \vdots&\ddots&\vdots&\vdots\\ 1&\ldots&s&t\\ 1&\ldots&1&k\end{pmatrix}\in M_{n+1}(\mathbf{Z}/d\mathbf{Z})

where the $i$ -th row $(1,\ldots,s,\ldots,t)$ has $s$ at place $i$ , $t$ at place $n+1$ , and $1$ otherwise. Then the set of solutions $\{\vec{x}~{}|~{}A\cdot\vec{x}=0\}$ in $(\mathbf{Z}/d\mathbf{Z})^{\oplus n+1}$ consists of the diagonal elements $\{(a,\cdots,a)~{}|~{}a\in\mathbf{Z}/d\mathbf{Z}\}$ .

Proof.

Suppose that $\vec{x}=(b_{1},\ldots,b_{n+1})$ is a solution of $A\cdot\vec{x}=0$ . Then we have

\sum_{j\neq i,1\leq j\leq n}b_{j}+sb_{i}+tb_{n+1}\equiv 0\ \textrm{mod}\ d,~{}\sum_{j=1}^{n}b_{j}+kb_{n+1}\equiv 0\ \textrm{mod}\ d.

It follows that

(2.13)

(s-1)b_{i}+(t-k)b_{n+1}\equiv 0\ \textrm{mod}\ d,~{}\forall 1\leq i\leq n.

Given the assumption on the integers $k,t,s$ , we have

(s-1)\equiv(k-t)\ \textrm{mod}\ d,~{}(t-k)\equiv(n+t)\ \textrm{mod}\ d.

Since $n+t$ is coprime to $d$ , it follows that $k-t$ and $s-1$ are invertible in $\mathbf{Z}/d\mathbf{Z}$ . Then the equation (2.13) implies

b_{i}\equiv b_{n+1}\ \textrm{mod}\ d,~{}\forall{1\leq i\leq n}.

Thus our assertion follows. ∎

Now let us prove Theorem 2.1.

Proof of Theorem 2.1.

By Observation 2.2, the complete intersection $X\subset\mathbf{P}^{n}$ of type $(d_{1},\ldots,d_{c})$ , satisfying the assumptions in Theorem 2.1, can be regarded as a subscheme in the complete intersection defined by the first $r$ polynomials with minimal degree $d:=d_{1}$ where $d\geq 3$ , $r\geq 1$ , or $d=2$ , $r\geq 3$ . Thus we shall prove that a general codimension $r$ complete intersection $Y\subset\mathbf{P}^{n}$ of hypersurfaces with equal degree $d$ has no non-trivial linear automorphisms.

(i) Suppose that $d\geq 3$ , $r=1$ . This reduces to the cases of hypersurfaces. A general smooth hypersurface $Y\subset\mathbf{P}^{n}$ of $\deg Y\geq 3$ has no linear automorphisms unless $(d;n)=(3;2)$ , see [10, 12].

(ii) Suppose that $d\geq 3$ , $r\geq 2$ or $d=2$ , $r\geq 3$ . Let us verify Hypotheses (1)–(3) in Theorem 1.2.

(1)

Set $\pi:\mathcal{X}\to B$ as the universal family of smooth complete intersections of type $(d_{1},\ldots,d_{c};n)$ . The relative automorphism schemes $\operatorname{\mathrm{Aut}}_{B}(\mathcal{X})\to B$ is a finite group scheme over $B$ if the moduli stack of smooth complete intersections is a separated and Delgine-Mumford stack, see [9, Lem. 7.7] or [7, Lem. 2.3]. Benoist affirmed that the moduli stack is separated and Delgine-Mumford if $(d_{1},\ldots,d_{c};n)\neq(2;n)$ , see [1]. Hence Hypothsis (1) holds.
(2)

Let $X_{n,r,d}\subset\mathbf{P}^{n}$ be the complete intersection of Fermat type defined as (2.4) with $d\geq 3,r\geq 2$ or $d=2,r\geq 3$ . By Proposition 2.11 $\operatorname{\mathrm{Aut}}_{L}(X_{n,r,d})$ acts faithfully on $\mathrm{H}^{n-r}_{\mathrm{prim}}(X_{n,r,d})$ . Thus the action is also faithful on $\mathrm{H}^{n-r}(X_{n,r,d})$ . Then Hypothesis (2) is satisfied.
(3)

The proof of the bigness of the monodromy group for complete intersections is in line with the proof of Theorem 1.4

Let $X_{b}$ be the smooth complete intersection of multidegree $(d_{1},\ldots,d_{c})$ over a general point $b\in B$ . Let $X_{1},\ldots,X_{c}$ be the hypersurfaces such that $X_{b}=X_{1}\cap\cdots\cap X_{c}$ . We may assume the complete intersection $Y\colon=X_{2}\cap\cdots\cap X_{c-1}$ is smooth. Then $X_{b}$ is a hyperplane section in $Y$ . Let $D$ be a Lefschetz pencil of hyperplane sections in $Y$ passing through a point $[X_{b}]\in D$ . The set of hyperplane sections that admit ordinary double points is a finite subset $S$ in $D$ . Let $U$ be the open complement $D\mathbin{\rule[1.99997pt]{6.69998pt}{1.19995pt}}S$ . The monodromy action of the fundamental group $\pi_{1}(U,0)$ on $\mathrm{H}^{n-c}_{\mathrm{prim}}(X_{b};\mathbf{Q}_{\ell})$ factors through the monodromy action of $\pi_{1}(B,b)$ via the natural inclusion $(U,0)\hookrightarrow(B,b)$ . Therefore it suffices to prove the monodromy group of $\pi_{1}(U,0)$ is as big as possible.

Let $\mathrm{H}$ denote the cohomology space $\mathrm{\mathrm{H}}^{n-c}_{\mathrm{prim}}(X_{b};\mathbf{Q}_{\ell})$ , $\psi$ the intersection form on $\mathrm{H}$ , and $M\subset\operatorname{\mathrm{Aut}}(\mathrm{H},\psi)$ the geometric monodromy group of $\pi_{1}(U,0)$ . If $n-c$ is odd, $M$ is the symplectic group $\mathrm{Sp}(\mathrm{H},\psi)$ [3, Théorèm 5.10]. If $n-c$ is even, then $M$ is either the full orthogonal group $\mathrm{O}(\mathrm{H},\psi)$ or a finite subgroup of $\mathrm{O}(\mathrm{H},\psi)$ [4, Théorèm 4.4.1].
If $M$ is finite, the $p$ -adic Newton polygon for $X_{b}$ is a straight line [8, Thm. 11.4.9]. Illusie [6] proved that the Newton polygon of a general complete intersection conincides with its Hodge polygon. The Hodge polygon is a straight line if and only if the Hodge numbers $h^{i,n-c-i}$ of $X_{b}$ all vanishes except for $i=\frac{n-c}{2}$ . By [5, Exposé XI], such siutation arises only when $(d_{1},\ldots,d_{c};n)$ falls into one of the following
- •
  
  $(2;n)$ , hyperquadrics;
- •
  
  $(3;3)$ , cubic surfaces;
- •
  
  $(2,2;n)$ and $n$ is even, even-dimensional complete intersections of two quadrics.
None of the three cases fits the conditions $d\geq 3$ , $r\geq 2$ , or $d=2$ , $r\geq 3$ .

Therefore, Theorem 1.2 assures that $\operatorname{\mathrm{Aut}}_{L}(Y)$ is isomorphic to $\{1\}$ or $\mathbf{Z}/2\mathbf{Z}$ . The second possibility will be ruled out by Proposition 2.14. This completes the proof. ∎

Proposition 2.14.

Let $k$ be an algebraically closed field of $\mathrm{char}(k)\neq 2$ . Let $X\subset\mathbf{P}^{n}_{k}$ be a complete intersection of multidegree $\underbrace{(d,\ldots,d)}_{r}$ with $r<n$ . Assmuing $d\geq 3$ , or $d=2,r\geq 3$ , then a generic $X$ has no linear automorphisms of order $2$ .

Proof.

Let $\mathbf{P}^{n}_{k}$ be the projective space, associated with a $k$ -linear space $V$ of dimension $n+1$ . The defining polynomials of the codimension $r$ complete intersection $X\subset\mathbf{P}^{n}_{k}$ span an $r$ -dimensional subspace $W_{X}$ in the $k$ -space $\operatorname{\mathrm{Sym}}^{d}V^{*}$ . If $\sigma\in\operatorname{\mathrm{Aut}}_{L}(X)$ is a linear automorphism, then $W_{X}$ is $\sigma$ -invariant. Our goal is to show that a generic $r$ -dimensional subspace in $\operatorname{\mathrm{Sym}}^{d}V^{*}$ is not stabilized by any involution $\sigma\neq\pm\operatorname{Id}$ .

Let $\mathbf{G}(r,\operatorname{\mathrm{Sym}}^{d}V^{*})$ be the Grassmannian of $r$ -dimensional subspaces in $\operatorname{\mathrm{Sym}}^{d}V^{*}$ . For an element $g\in G\colon=\operatorname{GL}(V)$ , we define the subset

G_{r}(g):=\{[U]\in\mathbf{G}(r,\operatorname{\mathrm{Sym}}^{d}V^{*})~{}|~{}g\cdot U=U\}

of the $g$ -stabilized subspaces in $\operatorname{\mathrm{Sym}}^{d}V^{*}$ . If two elements $g,h\in G$ are conjuagte, then $G_{r}(g)$ is isomorphic to $G_{r}(h)$ respectively.

Let $\sigma\in G:=\operatorname{GL}(V)$ be an involution. Denote by $\operatorname{Cl}_{G}(\sigma)$ the conjugacy class of $\sigma$ in $G$ . Define the incidence variety

\mathcal{I}_{\sigma}:=\{(g,[U])\in\operatorname{Cl}_{G}(\sigma)\times\mathbf{G}(r,\operatorname{\mathrm{Sym}}^{d}V^{*})~{}|~{}g\cdot U=U\}.

The projection $\mathcal{I}_{\sigma}\to\operatorname{Cl}_{G}(\sigma)$ is a fibration, with each fiber isomorphic to $G_{r}(\sigma)$ .

Given that $\mathrm{char}(k)\neq 2$ , involutions in $G$ are similar to the diagonal matrices of order $2$ . Therefore, to establish our assertion, it suffices to prove

\dim\mathbf{G}(r,\operatorname{\mathrm{Sym}}^{d}V^{*})>\dim\mathcal{I}_{\sigma}=\dim\operatorname{Cl}_{G}(\sigma)+\dim G_{r}(\sigma)

for all diagonal matrices $\sigma\neq\pm\operatorname{Id}$ of order $2$ .

The action of $\sigma$ on $V$ induces a decomposition $V\cong V_{+}\oplus V_{-}$ with eigenvalues $\pm 1$ . Then the symmetric tensor $\operatorname{\mathrm{Sym}}^{d}V^{*}$ , under the action of $\sigma$ , decomposes into eigenspaces

S_{+}=\bigoplus_{j\text{~{}even}}S^{d-j}V_{+}\otimes S^{j}V_{-},~{}S_{-}=\bigoplus_{j\text{~{}odd}}S^{d-j}V_{+}\otimes S^{j}V_{-}.

Let us set

	$\displaystyle\dim V_{+}=n_{1},\dim V_{-}=n_{2},~{}n_{1}+n_{2}=n+1,n_{1},n_{2}>0,$
	$\displaystyle\dim S_{+}=e_{1},\dim S_{-}=e_{2},~{}e_{1}+e_{2}=\binom{n+d}{d}.$

Suppose that $W\subset\operatorname{\mathrm{Sym}}^{d}V^{*}$ is a subspace stabilized by $\sigma$ . Then $W$ admits a decomposition $W_{+}\oplus W_{-}$ where $W_{+}\subset S_{+},W_{-}\subset S_{-}$ . Hence the set $G_{r}(\sigma$ consists of pairs of subspaces in $S_{+}$ and $S_{-}$ . Let $f_{1}:=\dim W_{+}$ , $f_{2}:=\dim W_{-}$ . The dimension of $G_{r}(\sigma)$ is bounded by

\operatorname{max}\{f_{1}(e_{1}-f_{1})+f_{2}(e_{2}-f_{2})~{}|~{}0\leq f_{i}\leq e_{i},f_{1}+f_{2}=r\}.

Let $Z_{G}(\sigma)<G$ denote the center of $\sigma$ . As a diagonal matrix $\sigma$ with $\mathrm{sign}(\sigma)=(n_{1},n_{2})$ , it is direct to see $\dim Z_{G}(\sigma)=n_{1}^{2}+n_{2}^{2}$ . By the formula

\dim\operatorname{Cl}_{G}(\sigma)+\dim Z_{G}(\sigma)=\dim G,

we have $\dim\operatorname{Cl}_{G}(\sigma)=2n_{1}n_{2}$ . Then $\dim\mathbf{G}(r,\operatorname{\mathrm{Sym}}^{d}V^{*})-\dim\mathcal{I}_{\sigma}$ is equal to

\operatorname{min}\{f_{2}(e_{1}-f_{1})+f_{1}(e_{2}-f_{2})-2n_{1}n_{2}~{}|~{}0\leq f_{i}\leq e_{i},f_{1}+f_{2}=r\}.

Let us show that $f_{2}(e_{1}-f_{1})+f_{1}(e_{2}-f_{2})-2n_{1}n_{2}$ are positive numbers within the cases $d\geq 3$ and $d=2,r\geq 3$ .

•

For $d\geq 3$ , we consider the two subcases:

(a) Assuming $f_{1}\geq f_{2}>0$ , we have the inequality

f_{2}(e_{1}-f_{1})+f_{1}(e_{2}-f_{2})-2n_{1}n_{2}\geq f_{2}(e_{1}+e_{2}-2f_{1})-2n_{1}n_{2}.

Observe that $n_{1}n_{2}\leq(\frac{n+1}{2})^{2}$ and $f_{1}<r<n$ . Therefore,

	$\displaystyle f_{2}(e_{1}+e_{2}-2f_{1})-2n_{1}n_{2}$	$\displaystyle>f_{2}(e_{1}+e_{2}-2r)-\frac{(n+1)^{2}}{2}$
		$\displaystyle>f_{2}(\binom{n+d}{d}-2n)-\frac{(n+1)^{2}}{2}.$

The term $\binom{n+d}{d}$ increases with $d$ . For $d=3$ , it is easy to verify that

\binom{n+3}{3}-2n-\frac{(n+1)^{2}}{2}>0,\forall n\geq 3.

(b) Assuming $f_{2}=0,~{}f_{1}=r$ , we have

f_{2}(e_{1}-f_{1})+f_{1}(e_{2}-f_{2})-2n_{1}n_{2}=re_{2}-2n_{1}n_{2}.

The dimension $e_{2}=\dim S_{-}$ , which depends on the integers $n_{1},n_{2}$ and $d$ , also increases with $d$ . For $d=3$ , we get $e_{2}=\binom{n_{2}+2}{3}+n_{2}\cdot\binom{n_{1}+1}{2}$ . This leads to

	$\displaystyle re_{2}-2n_{1}n_{2}$	$\displaystyle=r\cdot\binom{n_{2}+2}{3}+rn_{2}\cdot\binom{n_{1}+1}{2}-2n_{1}n_{2}$
		$\displaystyle\geq 2\cdot\binom{n_{2}+2}{3}+n_{2}n_{1}(n_{1}+1)-2n_{1}n_{2}.$

Hence $re_{2}-2n_{1}n_{2}>0$ unless $n_{1}=1,n_{2}=0$ . However, thess values violates the condition $n_{1}+n_{2}=n+1>2$ . Swapping $f_{1}$ and $f_{2}$ does not affect our argument. Thus the assertion follows.

•

For $d=2,r\geq 3$ , we get

e_{1}=\binom{n_{1}+1}{2}+\binom{n_{2}+1}{2},~{}e_{2}=n_{1}n_{2}.

Therefore the experssion $f_{2}(e_{1}-f_{1})+f_{1}(e_{2}-f_{2})-2n_{1}n_{2}$ equals

f_{2}(e_{1}-2f_{1})+(f_{1}-2)n_{1}n_{2}.

We claim that $e_{1}-2f_{1}>0$ . Since $n_{1}+n_{2}-2=n-1\geq r\geq f_{1}$ , it suffices to prove that $e_{1}>2(n_{1}+n_{2}-2)$ . We have

		$\displaystyle\binom{n_{1}+1}{2}+\binom{n_{2}+1}{2}-2(n_{1}+n_{2}-2)$
	$\displaystyle=$	$\displaystyle\frac{1}{2}(n_{1}^{2}+n_{2}^{2}-3(n_{1}+n_{2})+8)$
	$\displaystyle=$	$\displaystyle\frac{1}{2}((n_{1}-\frac{3}{2})^{2}+(n_{2}-\frac{3}{2})^{2}+\frac{7}{2})>0.$

Furthermore, $e_{1}>2(n_{1}+n_{2}-2)$ ensures $e_{1}>4$ since $n_{1}+n_{2}=n+1>r+1\geq 4$ . When $f_{1}\geq 2$ , it follows that $f_{2}(e_{1}-2f_{1})+(f_{1}-2)n_{1}n_{2}$ is positive.

Now consider the specific cases $(f_{1},f_{2})=(0,r)$ and $(1,r-1)$ , then the expression $f_{2}(e_{1}-2f_{1})+(f_{1}-2)n_{1}n_{2}$ becomes $re_{1}-2n_{1}n_{2}$ and $(r-1)(e_{1}-2)-n_{1}n_{2}$ repsectively. We claim that $e_{1}-2>n_{1}n_{2}$ . In fact, the condition $n_{1}+n_{2}>4$ implies that

\binom{n_{1}+1}{2}+\binom{n_{2}+1}{2}-n_{1}n_{2}-2=\frac{1}{2}((n_{1}-n_{2})^{2}+n_{1}+n_{2}-4)>0.

Using $r\geq 3$ , it is easy to see that both $re_{1}-2n_{1}n_{2}$ and $(r-1)(e_{1}-2)-n_{1}n_{2}$ are positive numbers.

∎

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