Fano weighted complete intersections of large codimension

TL;DR

This paper classifies smooth Fano weighted complete intersections with large codimension, advancing understanding of their structure and properties in algebraic geometry.

Contribution

It provides a classification of smooth Fano weighted complete intersections specifically in the case of large codimension, a previously less understood area.

Findings

Complete classification achieved for large codimension cases.

Identification of key properties characterizing these intersections.

New insights into the structure of Fano varieties.

Abstract

We classify smooth Fano weighted complete intersections of large codimension.

Equations

$$\mathbb{P}=\mathbb{P}(a_0,\dots ,a_N)=\mathrm{Proj}\mathbb{k}[x_0,\dots ,x_N].$$

$${\mathrm{codim}}_X\left(X\cap \mathrm{Sing}\mathbb{P}\right)⩾2,$$

$$\mathrm{Sing}X=X\cap \mathrm{Sing}\mathbb{P}.$$

$$\begin{array}{c}\mathbb{P}(a_0,\dots ,a_{N-1},a_N)\text{dasharrow}{\mathbb{P}}^{\prime }=\mathbb{P}(a_0,\dots ,a_{N-1}),\\ (x_0:\dots :x_{N-1}:x_N)\mapsto (x_0:\dots :x_{N-1}),\end{array}$$

$${\omega }_X={\mathcal{O}}_X\left(-i_X\right).$$

$$1⩽i_X=\sum _{i=0}^N{a}_i-\sum _{j=1}^k{d}_j=a_0+\dots +a_n+(a_{n+1}-d_1)+\dots +(a_N-d_k)⩽n+1-k⩽1.$$

$$1=d_k-a_N⩾a_N,$$

$$\begin{array}{c}1⩽i_X=\sum _{i=0}^N{a}_i-\sum _{j=1}^k{d}_j=a_0+\dots +a_{n-1}+a_n+(a_{n+1}-d_1)+\dots +(a_N-d_k)=\\ =a_0+a_1+(a_{n+1}+a_2-d_1)+\dots +(a_{N-1}+a_{n-1}-d_{k-1})+(a_N+a_n-d_k)=\\ =2+(a_{n+1}+1-d_1)+\dots +(a_{N-1}+1-d_{k-1})+(a_N+a_n-d_k)⩽\\ ⩽2+(a_N+a_n-d_k).\end{array}$$

$$1⩽2+(a_n+a_N-d_k)<2+(2a_N-d_k)⩽2$$

$$d_1=a_{n+1}+1.$$

$$2a_N⩽d_1=a_{n+1}+1=a_N+1,$$

Full text

Victor Przyjalkowski was supported by grant MD–30.2020.1. Constantin Shramov was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”. Both authors are Young Russian Mathematics award winners and would like to thank its sponsors and jury.

1. Introduction

Given a smooth Fano variety $X$ over an algebraically closed field $\Bbbk$ of characteristic zero, we denote by $i_{X}$ the Fano index of $X$, that is, the maximal positive integer $i$ such that the canonical class $K_{X}$ is divisible by $i$ in the Picard group of $X$. It is well known that $i_{X}\leqslant\dim X+1$, see [IP99, Corollary 3.1.15]. The goal of this note is to prove the following (we refer the reader to [Do82] and [IF00], or to §2 below, for the relevant definitions).

Theorem 1.1**.**

Let $X\subset\mathbb{P}=\mathbb{P}(a_{0},\ldots,a_{N})$ be a smooth well formed Fano weighted complete intersection of dimension $n\geqslant 2$ and codimension $k=N-n$ that is not an intersection with a linear cone. The following assertions hold.

• (i)

One has $k\leqslant n-i_{X}+1$.

• (ii)

If $k=n-i_{X}+1$, then $X$ is a complete intersection of $n-i_{X}+1$ quadrics in $\mathbb{P}=\mathbb{P}^{N}$.

• (iii)

Suppose that $k=n-i_{X}\geqslant 2$ (so that in particular $n\geqslant 3$). Then $X$ is a complete intersection of $n-i_{X}-1$ quadrics and a cubic in $\mathbb{P}=\mathbb{P}^{N}$.

Note that the assumption that $X$ in Theorem 1.1 is well formed, as well as the assumption that $X$ is not an intersection with a linear cone, can be omitted provided that $X$ is sufficiently general and $n\geqslant 3$, see Proposition 2.9 below.

We are grateful to M. Ovcharenko who found a mistake in the first version of our paper.

2. Preliminaries

Let $a_{0},\ldots,a_{N}$ be positive integers. Consider the graded algebra $\Bbbk[x_{0},\ldots,x_{N}]$, where the grading is defined by assigning the weights $a_{i}$ to the variables $x_{i}$. Put

[TABLE]

The weighted projective space $\mathbb{P}$ is said to be well formed if the greatest common divisor of any $N$ of the weights $a_{i}$ is $1$. Every weighted projective space is isomorphic to a well formed one, see [Do82, 1.3.1]. A subvariety $X\subset\mathbb{P}$ is said to be well formed if $\mathbb{P}$ is well formed and

[TABLE]

where the dimension of the empty set is defined to be $-1$.

We say that a subvariety $X\subset\mathbb{P}$ of codimension $k$ is a weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ if its weighted homogeneous ideal in $\Bbbk[x_{0},\ldots,x_{N}]$ is generated by a regular sequence of $k$ homogeneous elements of degrees $d_{1},\ldots,d_{k}$. Note that $\mathbb{P}$ can be thought of as a weighted complete intersection of codimension [math] in itself.

Theorem 2.1** (see [Di86, Proposition 8]).**

Let $X\subset\mathbb{P}$ be a well formed weighted complete intersection. Then

[TABLE]

The following property is an analog of smoothness for subvarieties in a weighted projective space.

Definition 2.2** (see [IF00, Definition 6.3]).**

Let $p\colon\mathbb{A}^{N+1}\setminus\{0\}\to\mathbb{P}$ be the natural projection to the weighted projective space. A subvariety $X\subset\mathbb{P}$ is called quasi-smooth if the preimage $p^{-1}(X)$ is smooth.

Lemma 2.3**.**

Let $X\subset\mathbb{P}$ be a weighted complete intersection. Then $X$ is isomorphic to a weighted complete intersection $X^{\prime}$ of the same codimension in a well formed weighted projective space $\mathbb{P}^{\prime}$. Moreover, if $X$ is quasi-smooth, $X^{\prime}$ is also quasi-smooth, and if $X$ is a general weighted complete intersection of the corresponding multidegree in $\mathbb{P}$, then $X^{\prime}$ is also a general weighted complete intersection of the corresponding multidegree in $\mathbb{P}^{\prime}$.

Proof.

We may assume that the greatest common divisor of the weights $a_{0},\ldots,a_{N}$ equals $1$. Suppose that $\mathbb{P}$ is not well formed. This means that, up to renumbering, one has $a_{i}=aa_{i}^{\prime}$, where $a_{i}^{\prime}\in\mathbb{Z}_{\geqslant 1}$ for $i=1,\ldots,N$, and $a\in\mathbb{Z}_{\geqslant 2}$. Then, if we put $a_{0}^{\prime}=a_{0}$, we get an isomorphism $\mathbb{P}\cong\mathbb{P}^{\prime}=\mathbb{P}(a_{0}^{\prime},\ldots,a_{N}^{\prime})$ given by the $a$-th Veronese map, see, for instance, [IF00, Lemma 5.7]. The image of $X$ under this isomorphism is a weighted complete intersection $X^{\prime}\subset\mathbb{P}^{\prime}$. Indeed, by homogenicity $a$ divides all degrees of the variable $x_{0}$ in the equations defining $X$, so equations that define $X^{\prime}$ are given from ones that define $X$ by replacing $x_{0}^{a}$ by $x_{0}^{\prime}$ and $x_{i}$ by $x_{i}^{\prime}$, $i=1,\ldots,N$, where $x_{i}$ and $x_{i}^{\prime}$ are weighted homogeneous coordinates on $\mathbb{P}$ and $\mathbb{P}^{\prime}$ of weights $a_{i}$ and $a_{i}^{\prime}$, respectively. Repeating this procedure sufficiently many times, we may assume that $\mathbb{P}^{\prime}$ is well formed.

The affine cone over $X^{\prime}$ is a quotient of the affine cone over $X$ by a group generated by quasi-reflections, so $X^{\prime}$ is quasi-smooth if and only if $X$ is quasi-smooth. The assertion about the generality of $X^{\prime}$ is obvious. ∎

Although quasi-smoothness is a natural condition to consider, we will be mostly interested in weighted complete intersections that are smooth in the usual sense.

Remark 2.4* ([PSh16, Corollary 2.14]).*

Let $X$ be a smooth well formed weighted complete intersection. Then $X$ is quasi-smooth.

The following lemma provides a smoothness criterion for weighted complete intersections.

Lemma 2.5** ([PSh16, Lemma 2.5]).**

Let $X\subset\mathbb{P}$ be a smooth well formed weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$. Then for every $r$ and every choice of $r$ weights $a_{i_{1}},\ldots,a_{i_{r}}$, $i_{1}<\ldots<i_{r}$, such that their greatest common divisor $\delta$ is greater than $1$, there exist $r$ degrees $d_{s_{1}},\ldots,d_{s_{r}}$, $s_{1}<\ldots<s_{r}$, such that their greatest common divisor is divisible by $\delta$.

The following definition describes weighted complete intersections that are to a certain extent analogous to complete intersections in a usual projective space that are contained in a hyperplane.

Definition 2.6** (cf. [IF00, Definition 6.5]).**

A weighted complete intersection $X\subset\mathbb{P}$ is said to be an intersection with a linear cone if one has $d_{j}=a_{i}$ for some $i$ and $j$.

Theorem 2.7** (see [IF00, Theorem 6.17]).**

Let $X\subset\mathbb{P}$ be a quasi-smooth weighted complete intersection of dimension at least $3$. Suppose that $\mathbb{P}$ is well formed and $X$ is not an intersection with a linear cone, and that $X$ is general in the family of weighted complete intersections of the same multidegree in $\mathbb{P}$. Then $X$ is well formed.

Remark 2.8*.*

Theorem 2.7 does not hold in dimensions less than $3$. A counterexample is a hypersurface of degree $9$ in $\mathbb{P}(1,2,2,3)$, which is smooth and quasi-smooth, but not well formed, see [IF00, 6.15(ii)].

The following result shows that a general enough quasi-smooth weighted complete intersection is isomorphic to a weighted complete intersection with nice properties.

Proposition 2.9**.**

Let $X\subset\mathbb{P}$ be a quasi-smooth weighted complete intersection of dimension at least $3$. Assume that $X$ is general in the family of weighted complete intersections of the same multidegree in $\mathbb{P}$. Then there exists a quasi-smooth well formed weighted complete intersection $X^{\prime}$ isomorphic to $X$ which is not an intersection with a linear cone.

Proof.

Suppose that $X$ is an intersection with a linear cone. Let $(d_{1},\ldots,d_{k})$ be the multidegree of $X$. We may assume that $d_{k}=a_{N}$. Let $x_{i}$, $i=0,\ldots,N$, be a weighted homogeneous coordinate of weight $a_{i}$ on $\mathbb{P}$. Since $X$ is general in the family of weighted complete intersections of the same multidegree in $\mathbb{P}$, there exists a homogeneous polynomial of degree $d_{k}$ in the weighted homogeneous ideal of $X$ of the form $f_{k}=f-x_{N}$, where the polynomial $f$ depends only on variables $x_{0},\ldots,x_{N-1}$. Moreover, substituting $f$ instead of $x_{N}$ into the remaining polynomials $f_{1},\ldots,f_{k-1}$ that generate the ideal of $X$, one can assume that these polynomials do not depend on $x_{N}$. Thus the natural projection

[TABLE]

induces the isomorphism of $X$ with a weighted complete intersection $X^{\prime}$ of multidegree $(d_{1},\ldots,d_{k-1})$ in $\mathbb{P}^{\prime}$. Quasi-smoothness of $X$ means that the Jacobian matrix of the polynomials defining $X$ is non-degenerate. Since the variable $x_{N}$ appears linearly and only in $f_{k}$, quasi-smoothness of $X$ implies that $X^{\prime}$ is quasi-smooth as well. By the assumption on the generality of $X$, we conclude that $X^{\prime}$ is a general weighted complete intersection of multidegree $(d_{1},\ldots,d_{k-1})$ in $\mathbb{P}^{\prime}$.

Repeating the above procedure sufficiently many times, we may assume that $X^{\prime}$ is not an intersection with a linear cone. Applying Lemma 2.3, we may assume that $X^{\prime}$ is a quasi-smooth weighted complete intersection in a well formed weighted projective space; however, as a result of the application of Lemma 2.3, it may become an intersection with a linear cone again. Alternating these two steps, we will finally arrive to a quasi-smooth weighted complete intersection $X^{\prime}$ in a well formed weighted projective space that is not an intersection with a linear cone, because projection (2.1) decreases the dimension of the ambient weighted projective space, while the application of Lemma 2.3 leaves it the same. It remains to apply Theorem 2.7 to conclude that $X^{\prime}$ is well formed. ∎

If a weighted complete intersection $X$ is smooth and well formed, then $X$ is quasi-smooth by Remark 2.4. However a priori we cannot assume smoothness instead of quasi-smoothness in Proposition 2.9 since $X$ is not necessarily well formed.

Question 2.10**.**

Does Proposition 2.9 nevertheless hold if we replace quasi-smoothness by smoothness?

3. Proof of the main result

In this section we prove Theorem 1.1. By $\mathbb{P}$ we will always denote the weighted projective space $\mathbb{P}(a_{0},\ldots,a_{N})$. We will say that a weighted complete intersection $X\subset\mathbb{P}$ of multidegree $(d_{1},\ldots,d_{k})$ is normalized if the inequalities $a_{0}\leqslant\ldots\leqslant a_{N}$ and $d_{1}\leqslant\ldots\leqslant d_{k}$ hold. Recall that if $X\subset\mathbb{P}$ is a smooth well formed Fano weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ and dimension at least $2$, then $i_{X}=\sum a_{i}-\sum d_{j}$, see [PSh19, Corollary 2.8]. Note that in a more general situation the following version of adjunction formula holds.

Theorem 3.1** (see [Do82, Theorem 3.3.4], [IF00, 6.14]).**

Let $X$ be a quasi-smooth well formed weighted complete intersection. Let $\omega_{X}$ be the dualizing sheaf on $X$. Then

[TABLE]

In particular, Theorem 3.1 shows that if $X$ is a quasi-smooth well formed weighted complete intersection of arbitrary dimension, then $X$ is Fano if and only if $i_{X}>0$.

In the proof of Theorem 1.1 we will use the following auxiliary results.

Theorem 3.2** (see [PST17, Theorem 1.2]).**

Suppose that $X\subset\mathbb{P}$ is a normalized smooth well formed Fano or Calabi–Yau weighted complete intersection of codimension $k$ that is not an intersection with a linear cone. Then $a_{k-1}=1$. Moreover, a general element of the linear system $|\mathcal{O}_{X}(1)|$ is smooth.

The following result is implicitly contained in [PST17]. We prove it for the reader’s convenience.

Lemma 3.3**.**

Suppose that $X\subset\mathbb{P}$ is a normalized smooth well formed Fano weighted complete intersection of dimension $n$ multidegree $(d_{1},\ldots,d_{k})$ that is not an intersection with a linear cone. Let $X^{\prime}$ be a general element of the linear system $|\mathcal{O}_{X}(1)|$. Then $X^{\prime}$ is a well formed weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ in $\mathbb{P}^{\prime}=\mathbb{P}(a_{1},\ldots,a_{N})$.

Proof.

By the classification of one-dimensional smooth well formed Fano weighted complete intersections (see for instance [PSh19, Lemma 2.6]), we may assume that $\dim X\geqslant 2$.

The linear system $|\mathcal{O}_{X}(1)|$ is non-empty by Theorem 3.2, and every element of $|\mathcal{O}_{X}(1)|$ is cut out on $X$ by a weighted hypersurface of degree $1$ in $\mathbb{P}$ (see, for instance, [PSh19, Corollary 3.3]). Therefore, one can consider $X^{\prime}$ as a weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ in $\mathbb{P}^{\prime}$.

Suppose that the weighted projective space $\mathbb{P}^{\prime}$ is not well formed. Then there are $N-1$ numbers among $a_{1},\ldots,a_{N}$ such that their greatest common divisor is at least $2$. We may assume that these are $a_{2},\ldots,a_{N}$. Let $x_{i}$ be the weighted homogeneous coordinate of weight $a_{i}$ on $\mathbb{P}$, and let $\Lambda\subset\mathbb{P}$ be the subset given by $x_{0}=x_{1}=0$. Then $\dim\Lambda=N-2$. Since $\dim X\geqslant 2$, the intersection $X\cap\Lambda$ is not empty. On the other hand, by [IF00, 5.15] the weighted projective space $\mathbb{P}$ is singular along $\Lambda$, and by Theorem 2.1 the weighted complete intersection $X$ is singular along $X\cap\Lambda$. The obtained contradiction shows that $\mathbb{P}^{\prime}$ is well formed.

By [IF00, 5.15] one has $\mathrm{Sing}\,\mathbb{P}^{\prime}\subset\mathrm{Sing}\,\mathbb{P}$. On the other hand, since $X$ is smooth and well formed, it is disjoint from $\mathrm{Sing}\,\mathbb{P}$ by Theorem 2.1. Hence $X^{\prime}$ is disjoint from $\mathrm{Sing}\,\mathbb{P}$, and thus also from $\mathrm{Sing}\,\mathbb{P}^{\prime}$, which implies that $X^{\prime}$ is well formed. ∎

Corollary 3.4**.**

Suppose that $X\subset\mathbb{P}$ is a normalized smooth well formed Fano or Calabi–Yau weighted complete intersection of codimension $k$ that is not an intersection with a linear cone. Then $a_{k+i_{X}-1}=1$.

Proof.

If $X$ is Calabi–Yau, the assertion follows from Theorem 3.2. Thus, we assume that $X$ is Fano. Let $n=\dim X$. By [PSh19, Lemma 2.6], we may assume that $\dim X\geqslant 2$. Furthermore, by [PSh19, Corollary 3.8(i),(ii)], we may assume that $i_{X}\leqslant n-1$.

Applying Theorem 3.2 consecutively $i_{X}$ times and keeping in mind that a general element of $|\mathcal{O}_{X}(1)|$ is well formed by Lemma 3.3, we see that $a_{0}=\ldots=a_{i_{X}-1}=1$, and there exists a normalized smooth well formed Calabi–Yau weighted complete intersection of codimension $k$ in $\mathbb{P}(a_{i_{X}},\ldots,a_{N})$, cf. Theorem 3.1. Also, the obtained weighted complete intersection is not an intersection with a linear cone. It remains to apply Theorem 3.2 once again to conclude that $a_{i_{X}}=\ldots=a_{k+i_{X}-1}=1$. ∎

Theorem 3.5** ([IF00, Lemma 18.4]).**

Suppose that $X\subset\mathbb{P}$ is a normalized quasi-smooth well formed weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ that is not an intersection with a linear cone. Then $d_{k-j}>a_{N-j}$ for all $1\leqslant j\leqslant k$.

Lemma 3.6** (see [CCC11, Proposition 3.1(1)]).**

Suppose that $X\subset\mathbb{P}$ is a normalized quasi-smooth well formed weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ that is not an intersection with a linear cone. Then $d_{k}\geqslant 2a_{N}$.

The following is a particular case of Theorem 1.1.

Lemma 3.7** (cf. [CCC11, Proof of Theorem 1.3]).**

Suppose that $X\subset\mathbb{P}$ is a smooth well formed Fano weighted complete intersection of dimension $n$ and codimension $k$ that is not an intersection with a linear cone. Then $k\leqslant n$. Moreover, if $k=n$, then $X$ is a complete intersection of $n$ quadrics in $\mathbb{P}=\mathbb{P}^{2n}$.

Proof.

We may assume that $X$ is normalized. Suppose that $k\geqslant n$. Then $a_{0}=\ldots=a_{n}=1$ by Corollary 3.4. Let $(d_{1},\ldots,d_{k})$ be the multidegree of $X$. Using Theorem 3.5, we obtain

[TABLE]

This implies that $k=n$, and $d_{j}-a_{n+j}=1$ for all $1\leqslant j\leqslant k$. In particular, by Lemma 3.6 we have

[TABLE]

so that $a_{N}=1$ and $d_{k}=2$. This means that $a_{i}=1$ for all $0\leqslant i\leqslant N$ and $d_{j}=2$ for all $1\leqslant j\leqslant k$. ∎

Remark 3.8*.*

In fact, the first assertion of Lemma 3.7 holds under a weaker assumption that $X$ is a quasi-smooth well formed Fano weighted complete intersection, see [CCC11, Theorem 1.3]. However, quasi-smoothness is not enough for the second assertion to hold. For instance, a general hypersurface of degree $30$ in $\mathbb{P}(6,10,15)$ is a quasi-smooth well formed Fano curve for which such an assertion fails.

Similarly to Lemma 3.7, we prove the following result (which is also a particular case of Theorem 1.1).

Lemma 3.9**.**

Suppose that $X\subset\mathbb{P}$ is a smooth well formed Fano weighted complete intersection of dimension $n$ and codimension $k$ that is not an intersection with a linear cone. Suppose that $k=n-1\geqslant 2$. Then $X$ is either a complete intersection of $n-1$ quadrics in $\mathbb{P}=\mathbb{P}^{2n-1}$, or a complete intersection of $n-2$ quadrics and a cubic in $\mathbb{P}=\mathbb{P}^{2n-1}$.

Proof.

We may assume that $X$ is normalized. Then $a_{0}=\ldots=a_{n-1}=1$ by Corollary 3.4. Let $(d_{1},\ldots,d_{k})$ be the multidegree of $X$. Using Theorem 3.5, we obtain

[TABLE]

Suppose that $a_{N}=1$. This means that $a_{i}=1$ for all $0\leqslant i\leqslant N$, so that $X$ is a complete intersection in the usual projective space. From (3.1) we conclude that $d_{k}<4$. If $d_{k}=2$, then $X$ is a complete intersection of $k=n-1$ quadrics. If $d_{k}=3$, then we look at (3.1) once again and see that for all $1\leqslant j\leqslant k-1$ one has $d_{j}=a_{n+j}+1=2$. This means that $X$ is a complete intersection of $k=n-2$ quadrics and a cubic.

Now suppose that $a_{N}>1$. If $a_{n}=a_{N}$, then by Lemma 2.5 at least $k+1$ degrees among $d_{1},\ldots,d_{k}$ are divisible by $a_{n}$, which is absurd. Thus $a_{n}<a_{N}$. From (3.1) we have

[TABLE]

by Lemma 3.6. This means that $d_{k}=2a_{N}$ and $a_{N}=a_{n}+1$. Looking at (3.1) once again and keeping in mind that $k\geqslant 2$ by assumption, we see that

[TABLE]

Since $a_{N}=a_{n}+1$, one has either $a_{n+1}=a_{n}$ or $a_{n+1}=a_{N}$. In the former case (3.2) gives $d_{1}=a_{n+1}+1=a_{n}+1=a_{N}$. This contradicts the assumption that $X$ is not an intersection with a linear cone. In the latter case by Lemma 2.5 the number $a_{N}$ divides all $k$ degrees $d_{j}$. Since $X$ is not an intersection with a linear cone, we see that $d_{j}\geqslant 2a_{N}$ for all $1\leqslant j\leqslant k$. Using (3.2), we get

[TABLE]

which is again a contradiction. ∎

Now we are ready to prove Theorem 1.1 in full generality.

Proof of Theorem 1.1.

Assume that $i_{X}\geqslant 2$. By Theorem 3.2, the linear system $|\mathcal{O}_{X}(1)|$ is not empty. Let $X^{\prime}$ be a general divisor from $|\mathcal{O}_{X}(1)|$. Then $X^{\prime}$ is smooth by Theorem 3.2. Moreover, $X^{\prime}$ is a well formed weighted complete intersection of multidegree $(d_{1},\ldots,d_{k})$ in $\mathbb{P}^{\prime}=\mathbb{P}(a_{1},\ldots,a_{N})$ by Lemma 3.3, and $X^{\prime}$ is not an intersection with a linear cone. Furthermore, $X^{\prime}$ is Fano by Theorem 3.1. Observe that Theorem 1.1 holds for $X$ if and only if it holds for $X^{\prime}$. Note also that if the dimension of $X^{\prime}$ is at most $3$, then the assertion of Theorem 1.1 holds for $X^{\prime}$ by the classification of smooth Fano weighted complete intersections in low dimensions (see, for instance, [PSh18, Tables 1 and 2]).

Therefore, repeating the above procedure several times and keeping in mind Theorem 3.1, we may assume that $i_{X}=1$, since both $k$ and $n-i_{X}+1$ do not change when we take the section. In this case we know that $k\leqslant n=n-i_{X}+1$ by Lemma 3.7, which proves assertion (i). Furthermore, if $k=n=n-i_{X}+1$, then $X$ is a complete intersection of $n$ quadrics in $\mathbb{P}^{2n}$ by Lemma 3.9, which proves assertion (ii). Finally, if $k=n-1=n-i_{X}$ and $k\geqslant 2$, then $X$ is a complete intersection of $n-2$ quadrics and a cubic in $\mathbb{P}^{2n-1}$ by Lemma 3.9, which proves assertion (iii). ∎

Remark 3.10*.*

Assertion (iii) of Theorem 1.1 does not hold without the assumption that $k\geqslant 2$. Indeed, except for a cubic hypersurface in $\mathbb{P}=\mathbb{P}^{n+1}$, there exist two other types of smooth well formed Fano weighted hypersurfaces of dimension $n$ and Fano index $n-1$: a hypersurface of degree $6$ in $\mathbb{P}=\mathbb{P}(1^{n},2,3)$, and a hypersurface of degree $4$ in $\mathbb{P}=\mathbb{P}(1^{n+1},2)$, cf. [PSh19, Corollary 3.8(iii)].

It seems that the analog of Theorem 1.1 for $k=n-i_{X}-1$ does not look that nice. For instance, there are $5$ families of five-dimensional weighted complete intersections of such codimension, as well as $5$ families of six-dimensional weighted complete intersections. One of the latter varieties is a six-dimensional weighted complete intersection of multidegree $(2,2,2,6)$ in $\mathbb{P}(1^{10},3)$.

Question 3.11**.**

Is there a finite number of series of smooth well formed weighted complete intersections of codimension $n-i_{X}-1$?