Examples of cylindrical Fano fourfolds

Abstract

We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form $Z\,{\times }\,{\mathbb {A}}^{1}$, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective ${\mathbb {G}}_{{\text {a}}}$-actions. Similar constructions of cylindrical Fano threefolds were done previously in the papers by Kishimoto et al. (Affine Algebraic Geometry. CRM Proceedings & Lecture Notes, vol 54, pp 123–163, 2011; Osaka J Math 51(4):1093–1112, 2014).

1 Introduction

A smooth projective variety V over ${\mathbb {C}}$ is called cylindrical if it contains a cylinder, i.e., a principal Zariski open subset U isomorphic to a product , where Z is a variety and stands for the affine line over ${\mathbb {C}}$ [16, 17].

Assuming , we let X be an affine cone over V. Due to the criterion of [17, Corollary 3.2], X admits an effective action of the additive group $\mathbb {G}_{{\text {a}}}$ if and only if V is cylindrical. This explains our interest in cylindrical projective varieties.

As a closely related topic, let us mention the well-known Hirzebruch Problem of classifying all possible smooth compactifications of the affine space such that $b_2(V)=1$; see [12] and references therein for studies on this problem. Similarly, it would be interesting to classify all cylindrical Fano varieties, or at least those with Picard number 1.

The answer to the latter question is known in dimension 2, even without the restriction on the Picard number. Namely, a smooth del Pezzo surface of degree d is cylindrical¹ if and only if $d\ge 4$ [4, 5, 16, 19].

In dimension 3, a cylindrical Fano threefold must be rational, see Remark 1.2 (a) below. However, even for cylindrical Fano threefolds with Picard number 1, the complete classification is lacking. Certain classes of such threefolds were described in [16] and [18]. Let us give the exhaustive list of known examples.

The projective space and the smooth quadric in are cylindrical, since they contain a Zariski open subset isomorphic to the affine space . By the same reason, the Fano threefold of index 2 and degree 5 is also cylindrical. A smooth intersection of two quadrics in is always cylindrical [16, Proposition 5.0.1]. The same is true for the Fano threefolds of index 1 and genus 12 [16, Proposition 5.0.2]. The moduli space of the latter family has dimension 6, while the subfamily of completions of is four-dimensional. There are two more families of cylindrical Fano threefolds with Picard number 1, index 1 and genera $g = 9$ and 10. These families fill in hypersurfaces in the corresponding moduli spaces [18].

In this paper we construct several examples of smooth, cylindrical Fano fourfolds with Picard rank 1. Let us recall first the standard terminology and notation. Given a smooth Fano fourfold V with Picard rank 1, the index of V is the integer r such that $-K_V=rH$, where H is the ample divisor generating the Picard group: . The degree $d=\deg V$ is the degree with respect to H. It is known that $1\le r\le 5$. Moreover, if $r=5$ then , and if $r=4$ then V is a quadric in . Smooth Fano fourfolds of index $r=3$ are called del Pezzo fourfolds; their degrees vary in the range $1\le d\le 5$ [8, 10]. Smooth Fano fourfolds of index $r=2$ are called Mukai fourfolds; their degrees are even and can be written as $d=2g-2$, where g is called the genus of V. The genera of Mukai fourfolds satisfy $2\le g\le 10$ [20]. The classification of Fano fourfolds of index $r=1$ is not known.

We let below $V_d$ be a Mukai fourfold of degree d, and $W_d$ be a del Pezzo fourfold of degree d. Our main results are summarized in the following theorem.

Theorem 1.1

There are the following families of cylindrical, smooth, rational Fano fourfolds:

• (i) the smooth intersections $V_{2\cdot 2}$ of two quadrics in ;

• (ii) the del Pezzo fourfold $W_5$ of degree 5;

• (iii) certain Mukai fourfolds $V_{14}$ of genus 8 varying in a family of codimension 1 in the moduli space;

• (iv) certain Mukai fourfolds $V_{12}$ of genus 7 varying in a family of codimension 2 (of dimension 13) in the moduli space.

The proof exploits explicit constructions of the Fano fourfolds as in (i)–(iv) via Sarkisov links. These constructions are borrowed from [21]. However, we recover some important details that are just sketched in [21]. The proof proceeds as follows. Starting with a (simpler) pair (V, D), where V is a smooth Fano fourfold and D an effective divisor on V such that contains a cylinder, we reconstruct it into a new (more complex) pair $(V',D')$ via a Sarkisov link that does not destroy the cylinder.

Remark 1.2

(a) Examples (i) and (ii) were alluded to in [16, 22], while (iii) and (iv) are new. The del Pezzo fourfold $W_5$ in (ii) is unique up to isomorphism, and, up to isomorphism of pairs, there are exactly four completions $(W_5, A)$ of by an irreducible divisor A, see [22]. However, there are much more cylinder structures on $W_5$. Namely, we show that for an ample divisor H generating the Picard group , the complement contains a cylinder provided the hyperplane section H is singular.

The analysis of examples (iii) and (iv) is based on the constructions that allowed us to establish rationality of these fourfolds [21].

(b) For certain popular families of rational Fano fourfolds, the existence of cylindrical members remains unknown. The latter concerns, for instance, the smooth, rational cubic fourfolds in .

(c) We do not know whether all the cylindrical Fano fourfolds are rational, while the varieties in Theorem 1.1 are. However, any cylindrical Fano threefold is rational. Indeed, for any smooth Fano variety V the plurigenera and irregularity vanish. If V contains a cylinder , then, for Z, the plurigenera and irregularity vanish too. If Z is a surface, then it must be rational due to the Castelnuovo Rationality Criterion. Of course, this argument fails in higher dimensions.

The paper is organized as follows. In Sect. 2 we give some technical preliminaries. We prove items (i), (ii), (iii), and (iv) of Theorem 1.1 in subsequent Sects. 3, 4, 5, and 6, in Theorems 3.6, 4.1, 5.6, and 6.1, respectively.

Note that in examples (i)–(iii) the construction goes from the resulting pair (claimed to have a desired cylinder) via a Sarkisov link to a simple birational model of a pair, where the cylinder is easy-to-see. It remains to observe that the inverse procedure does not affect the cylinder. In example (iv) we proceed the other way round, from an easy-to-see cylinder to the desired one.

2 Preliminaries

Notation

Let X be a smooth projective variety and $C\subset X$ be a smooth subvariety. We denote by

• $K_X$ the canonical divisor of X;

• ${\fancyscript{T}}_X$ the tangent bundle of X;

• ${\fancyscript{N}}_{C/X}$ the normal bundle of C in X;

• $c_i({\fancyscript{E}})$ the ith Chern class of a vector bundle ${\fancyscript{E}}$ on X, and $c_i(X)=c_i({\fancyscript{T}}_X)$;

• the Euler number of X.

As for the Schubert calculus, we follow the notation of [14].

In this section we gather some auxiliary facts that we need in the subsequent sections. The following lemma is a special case of the Riemann–Roch Theorem.

Lemma 2.1

For a smooth projective fourfold X and a divisor D on X we have

The next lemma can be deduced, for instance, from [11, Theorem 15.4].

Lemma 2.2

Let X be a smooth projective fourfold, and $\rho :\widetilde{X}\rightarrow X$ be a blowup with a smooth center $C \subset X$ and the exceptional divisor $E=\rho ^{-1}(C)$.

• (i) If C is a curve of genus g(C), then

  

  where A is the class of the fiber of $E\rightarrow C$.

• (ii) If C is a surface, then

  

Lemma 2.3

In the notation of Lemma 2.2, for a divisor H on X the following relations in the Chow ring $A (\widetilde{X})$ hold:

• (i) if C is a curve, then

  

• (ii) if C is a surface, then

  

Proof

The proof is straightforward. It uses the projection formula, the identification $E={\mathbb {P}}_C({\fancyscript{N}}_{C/X}^*)$, and the equality $E|_E= c_1\bigl ({\fancyscript{O}}_{{\mathbb {P}}_C({\fancyscript{N}}_{C/X}^*)}(-1)\bigr )$ in the Chow ring A(E), see e.g. [11]. $\square$

Lemma 2.4

Let be a smooth projective variety of dimension n. Assume that X contains a k-dimensional linear subspace ${\Lambda }$, and let H be a general hyperplane section of X containing ${\Lambda }$. If $n>2k$, then H is smooth.

Proof

By the Bertini Theorem, H is smooth outside ${\Lambda }$. We use a parameters count. The set $\fancyscript{H}$ of all hyperplane sections of that are singular at some point of ${\Lambda }$ is Zariski closed in , of dimension

On the other hand, the set of all hyperplane sections containing ${\Lambda }$ has dimension $N-(k+1)$. Now the result follows.$\square$

Lemma 2.5

Let be a del Pezzo fourfold of degree $d\ge 3$. Then W contains a line. Given a line $l\subset W$, one of the following holds:

(1)

Proof

Let $H_1, H_2$ be two generic hyperplane sections of W. Then $F= H_1\cap H_2$ is a smooth del Pezzo surface of degree d in W anticanonically embedded in , which contains some lines. Let l be a line on F, and consider the exact sequence

We have

where $a+b+c=1$. Furthermore, $a,b,c\le 1$ since for $x>y$. It follows also that $a,b,c\ge -1$. Assuming that $a\le b\le c$ we obtain that $(a,b,c)\in \{(0,0,1),(-1,1,1)\}$, as stated.$\square$

Corollary 2.6

Let be a del Pezzo fourfold of degree $d\ge 3$. Then the Hilbert scheme $\mathfrak L(W)$ of lines on W is reduced, nonsingular, and $\dim \mathfrak L(W)=4$. Through any point $P\in W$ there passes a family of lines on W of dimension greater than or equal to 1. For any $d\ge 4$ this family has dimension 1.

Proof

Since , using deformation theory we obtain that $\mathfrak L(W)$ is reduced, nonsingular and .

If $d=3$, then W is a smooth cubic in . The tangent section T at a point P of W is a singular cubic threefold in the projective tangent space . In an affine chart in centered at P, the equation of T has just the quadric and cubic homogeneous terms. The common zeros of these form a cone swept out by a family of lines through P of dimension greater than or equal to 1.

Let further $d\ge 4$, and so ${\text {codim}}_{{\mathbb {P}}^{d+2}} W\ge 2$. Then the tangent section cannot be a divisor in W. Indeed, otherwise this should be a hyperplane section generating the Picard group . However, for a generic point there is a hyperplane through $P'$, which contains . This leads to a contradiction.

Any line through P in W is contained in the tangent space , hence also in T, which is at most two-dimensional. Thus the family of lines in W through P is at most one-dimensional. At a generic point of W, it should be one-dimensional, because $\dim \mathfrak L(W)=4$. Therefore for any point $P\in W$, this family is of dimension 1. $\square$

3 Intersection of two quadrics in

It is known that any smooth intersection of two quadrics in contains a cylinder [16, Proposition 5.0.1]. Similarly, this holds for any smooth intersection $W_{2\cdot 2}$ of two quadrics in , see Theorem 3.6. The proof is based on a standard construction of $W_{2 \cdot 2}$ via a Sarkisov link, see (2). For the reader’s convenience, we recall this construction and some of its specific properties used in Sect. 6.

Proposition 3.1

Let as before $W=W_{2\cdot 2}$ be a smooth intersection of two quadrics in , and let H be an ample divisor on W whose class² generates . Given a line $l\subset W$, the projection with center l is a birational map , which fits in the diagram

(2)

where $\rho$ is the blowup of l with exceptional divisor E, and $\varphi$ is a birational morphism defined by the linear system . Furthermore, the following hold.

• (i) The $\varphi$-exceptional locus is an irreducible divisor $D\subset \widetilde{W}$.

• (ii) Letting L be the ample generator of we obtain

  

• (iii) The divisor $\rho (D)$ is cut out on W by a quadric in , and is swept out by lines meeting l. The image $F=\varphi (D)$ is a surface in of degree 5 with at worst isolated singularities. The singularities of F are the $\phi$-images of planes in W containing l. For a general line $l\subset W$ the surface F is smooth, and $\varphi$ is the blowup with center F.

• (iv) , where $\varphi (E)$ is a quadric in .

• (v) The quadric is singular, and its singular locus coincides with the locus of points $P\in \varphi (E)$, such that the restriction $\varphi |_E:E\rightarrow \varphi (E)$ is not an isomorphism over P.

Proof

Since l is a scheme-theoretic intersection of members of $|H-l|$, the linear system is base points free. Hence the divisor is ample, i.e. $\widetilde{W}$ is a Fano fourfold with . By the Cone Theorem there exists a Mori contraction $\varphi :\widetilde{W}\rightarrow U$ different from $\rho$. If $\widetilde{C}\subset \widetilde{W}$ is the proper transform of a line $C\subset W$ meeting l, then . In particular, the divisor is not ample. So, yields a supporting linear function for the extremal ray generated by the curves contained in the fibers of $\varphi$. Furthermore, we can write , where L is the ample generator of . We have . By the Riemann–Roch and Kodaira Vanishing Theorems we have . Therefore, defines a birational morphism , which coincides actually with the map $\varphi$.

Since and , the linear system contains a unique divisor D contracted by $\varphi$. Since , the divisor D is irreducible. This proves (i) and (ii).

The second equality in (ii) yields that $\rho (D)$ is cut out on W by a quadric in , while the last equality in (ii) shows that $\varphi (E)$ is a quadric in , as stated in (iii) and (iv), respectively.

Since $\phi$ is a projection and W is an intersection of quadrics, any positive dimensional fiber of the birational morphism $\varphi$ is the proper transform either of a line meeting l, or of a plane containing l. It follows that $\rho (D)$ is swept out by lines meeting l, as claimed in (iii). For a general line l in W, there is no plane in W containing l, and so, any fiber of $\varphi :D\rightarrow F$ has dimension at most 1. In this case F is smooth, and $\varphi$ is the blowup of F, see [2]. In general, there is at most a finite set of two-dimensional fibers of $\varphi$, mapped to the singular points of F. The local structure of $\varphi$ near “bad” fibers is described in [3]. Since $\varphi$ is the blowup of F outside of a finite number of points $P_i\in F$, we can use the standard formula , cf. Lemma 2.3 (ii). Hence, the image $F=\varphi (D)$ is a quintic surface in . The isomorphism in (iv) is straightforward from (2). This proves (iii) and (iv).

We finish with (v). The quadric $\varphi (E)$ is singular, because it contains the surface F of degree 5. Since , the restriction $\varphi |_E:E\rightarrow \varphi (E)$ is a crepant morphism. $\square$

Remark 3.2

(a) If the surface F is smooth, then F is a rational Castelnuovo quintic surface, see [1, Theorem 6 (XII)].

(b) Any smooth intersection of two quadrics contains exactly 64 planes, and the classes of these planes span the cohomology group $H^4(W, {\mathbb {Z}})$ [24]. Hence, the surface $F=\varphi (D)$ is smooth for a general line l on W, cf. Corollary 2.6, and singular for any l contained in a plane on W.

(c) The singular locus of the quadric $\varphi (E)$ is a point if and only if ${\fancyscript{N}}_{l/W}$ is of type (1), otherwise this is a line.

(d) Note that the divisor E is $\varphi$-ample. Hence E meets any nontrivial fiber of $\varphi$. Furthermore, it meets any two-dimensional fiber along a subvariety of positive dimension. Thus, if l is contained in a plane ${\Pi }\subset W$, then $\varphi (E)$ must be singular at $\phi ({\Pi })$. If l is contained in two planes ${\Pi }_1, {\Pi }_2\subset W$, then the quadric $\varphi (E)$ has two distinct singular points $\phi ({\Pi }_1)$ and $\phi ({\Pi }_2)$. Since the singular locus of a quadric is a linear subspace, in the latter case is a line.

The proof of the next simple lemma is left to the reader.

Lemma 3.3

Let be a quadric and be a hyperplane. Suppose that contains a point P. Then the projection with center P defines a cylinder structure in .

The next lemma and its corollary will be used in Sect. 6.

Lemma 3.4

In the notation of Proposition 3.1, let $H_0\subset W$ be a hyperplane section containing l. Then the image $L_0=\phi _*(H_0)$ is a hyperplane in , and

(3)

If , then contains a cylinder. Assume that there is a plane ${\Pi }\subset W$ such that $l\subset {\Pi }\subset H_0$. Then the condition holds.

Proof

Let $\widetilde{H}_0\subset \widetilde{W}$ be the proper transform of $H_0$. We may write for some $k\ge 1$. Since , we have $k=1$ and $\deg \varphi (\widetilde{H}_0)=1$. Hence $L_0$ is a hyperplane.

Furthermore, $P=\phi ({\Pi })$ is a point contained in both $L_0$ and , see Remark 3.2 (d). By Lemma 3.3, the projection of from the point produces a cylinder structure on . The existence of an isomorphism in (3) is straightforward by our construction. Now the remaining assertions follow.$\square$

The next corollary is immediate.

Corollary 3.5

If $H_0\subset W$ is a hyperplane section through a plane ${\Pi }\subset W$, then contains a cylinder.

Resuming, we obtain the main result of this section.

Theorem 3.6

In the notation of Proposition 3.1, the Zariski open set contains a cylinder.

Proof

Since $Q=\varphi (E)$ is a singular quadric, by Lemma 3.3, contains a cylinder . Now the result follows by Proposition 3.1 (iv). $\square$

4 The quintic del Pezzo fourfold

According to [9], a del Pezzo fourfold of degree 5 is unique up to isomorphism. It can be realized as a smooth section of the Grassmannian under its Plücker embedding in by a codimension 2 linear subspace. Clearly, the class of a hyperplane section H generates the group , and $-K_W\sim 3H$. The variety W is an intersection of quadrics, see [14, Chapter 1, Section 5].

The following theorem is the main result of this section.

Theorem 4.1

Let be a del Pezzo fourfold of degree 5, and let M be a hyperplane section of W. Then the Zariski open set contains a cylinder.

The proof is done at the end of the section. First we need some preliminaries. There are two types of planes in , namely, the Schubert varieties $\sigma _{3,1}$ and $\sigma _{2,2}$ [14, Chapter 1, Section 5], where

• with a fixed 3-dimensional subspace and a fixed point;

• with a fixed plane.

Remark 4.2

In the terminology of [9, Section 10], the $\sigma _{3,1}$-planes ($\sigma _{2,2}$-planes, respectively) are called planes of vertex type (non-vertex type, respectively).

The following proposition proven in [26] (see also [6, 3.3]) deals with the planes in the fourfold $W_5$.

Proposition 4.3

Let be a Fano fourfold of index 3 and degree 5. Then the following hold.

• (i) W contains exactly one $\sigma _{2,2}$-plane ${\Xi }$ and a one-parameter family of $\sigma _{3,1}$-planes.

• (ii) Any $\sigma _{3,1}$-plane ${\Pi }$ meets ${\Xi }$ along a tangent line to a fixed conic $C\subset {\Xi }$.

• (iii) Any two $\sigma _{3,1}$-planes ${\Pi }_1$ and ${\Pi }_2$ meet at a point .

• (iv) Let R be the union of all $\sigma _{3,1}$-planes on W. Then R is a hyperplane section of W and .

We need the following computational facts.

Lemma 4.4

For the Chern classes of we have

Proof

Let ${\fancyscript{I}}\rightarrow G$ be the universal subbundle and ${\fancyscript{Q}}\rightarrow G$ the universal factor-bundle, see [14, Chapter 3, Section 11]. Then $c_1({\fancyscript{I}})=\sigma _{1,0}$, $c_2({\fancyscript{I}})=\sigma _{1,1}$, and $c_r({\fancyscript{Q}})\sim \sigma _{r,0}$ (ibid). Since , standard computations give the result, see e.g. [11, Example 14.5.2]. $\square$

Corollary 4.5

For $W=W_5$ we have

Proof

The Adjunction Formula for the total Chern classes yields

Inversing the last factor gives the desired equalities.$\square$

The following lemma completes the picture.

Lemma 4.6

Let ${\Lambda }\subset W$ be a plane. Then $c_1({\fancyscript{N}}_{{\Lambda }/W})=0$ and $c_2({\fancyscript{N}}_{{\Lambda }/W})=2$ $(c_2({\fancyscript{N}}_{{\Lambda }/W})=1$, respectively) if ${\Lambda }$ is of type $\sigma _{2,2}$ $(\sigma _{3,1}$, respectively).

Proof

Let l be the class of a line on ${\Lambda }$. By Corollary 4.5, we have and . Since , we obtain

These lead to the desired equalities. $\square$

Corollary 4.7

The groups $H^q(W,{\mathbb {Z}})$ vanish if q is odd, $H^2(W,{\mathbb {Z}})\simeq H^6(W,{\mathbb {Z}})\simeq {\mathbb {Z}}$, and ${\text {rk}}H^4(W,{\mathbb {Z}})=2$. Moreover, $H^4(W,{\mathbb {Z}})/{\text {Tors}}$ is generated by the classes of $\sigma _{3,1}$-plane ${\Pi }$ and $\sigma _{2,2}$-plane ${\Xi }$.

Proof

The first two statements follow by the Lefschetz Hyperplane Section Theorem. By Corollary 4.5, we have ${\text {rk}}H^4(W,{\mathbb {Z}})=2$. By Lemma 4.6, ${\Pi }^2=1$ and ${\Xi }^2=2$. Furthermore, , and so, . Hence the intersection matrix of ${\Xi }$ and ${\Pi }$ is unimodular. Now the last assertion follows by the Poincaré duality. $\square$

Lemma 4.8

Let M be a hyperplane section of , and let R be as in Proposition 4.3 (iv).

• (i) If $M=R$, then M contains a $\sigma _{2,2}$-plane.

• (ii) If $M\ne R$, then there exists a line $l\subset M$ such that $l\not \subset R$.

Proof

In case (i), the assertion follows by Proposition 4.3. In case (ii), we pick a point . By Corollary 2.6, there exists a one-parameter family of lines in W passing through P. Let ${\Delta }\subset W$ be the cone with vertex P swept out by these lines. The intersection $M\cap {\Delta }$ is of positive dimension, so there exists a line $l\subset M\cap {\Delta }$ through P. $\square$

Theorem 4.1 claims that the complement contains a cylinder. We construct such a cylinder in Proposition 4.9 and Corollary 4.10 in the case $M=R$, and in Proposition 4.11 and Corollary 4.12 in the case $M\ne R$.

Proposition 4.9

([6, 3.6], [9, (7.8)], [22]) Let ${\Xi }\subset W$ be the $\sigma _{2,2}$-plane. Then there is a commutative diagram

where

• (i) $\rho :\widetilde{W}\rightarrow W$ is the blowup of is the projection from is the blowup of a rational normal cubic curve ;

• (ii) is defined by the linear system , where $E =\rho ^{-1}({\Xi })$ is the exceptional divisor;

• (iii) is the linear span of Y;

• (iv) the exceptional divisor $\widetilde{R}=\varphi ^{-1}(Y)$ of $\varphi$ coincides with the proper transform of R in $\widetilde{W}$, and on $\widetilde{W}$.

Sketch of the proof Using Lemmas 2.3 and 4.6, it is easy to deduce that and . Hence $\varphi$ is a birational morphism, , and . Since for some $k \ge 2$, we have . Thus $k = 2$ and $\dim \varphi (R) \le 2$. $\square$

The next corollary is straightforward.

Corollary 4.10

In the notation as before, let $M\subset W$ be a hyperplane section containing the $\sigma _{2,2}$-plane ${\Xi }$ in W (the case $M=R$ is not excluded), and let $\widetilde{M}$ be the proper transform of M in $\widetilde{W}$. Then $\varphi (\widetilde{M})$ is a hyperplane in , and . In particular, contains a cylinder.

Now we consider the case where a hyperplane section $M\subset W$ does not contain the $\sigma _{2,2}$-plane ${\Xi }$.

Proposition 4.11

([9, Section 10], [22]) Let be a del Pezzo fourfold of degree 5 and let $l\subset W$ be a line such that $l\not \subset R$. Then there is a commutative diagram

where

• (i) $\rho :\widetilde{W}\rightarrow W$ is the blowup of is the projection from is a smooth quadric, $\varphi :\widetilde{W}\rightarrow Q$ is the blowup of a cubic scroll ;

• (ii) is defined by the linear system , where $E =\rho ^{-1}(l)$ is the exceptional divisor;

• (iii) $\varphi (E) = Q\cap \langle F\rangle$, where is the linear span of F, moreover, $\varphi (E)$ is a quadratic cone;

• (iv) the image $D=\rho (\widetilde{D})\subset W$ of the exceptional divisor $\widetilde{D}=\varphi ^{-1}(F)\subset \widetilde{W}$ of $\varphi$ is a hyperplane section of W singular along l and swept out by lines meeting l.

Proof

It is similar to the proof of Propositions 3.1 and 4.9; we leave the details to the reader. The important thing here is that l is not contained in a plane in W by our assumption $l\not \subset R$, and so, $\varphi$ has no two-dimensional fiber (otherwise Q must be singular). $\square$

Now we can deduce the following corollary.

Corollary 4.12

In the notation as before, let M be a hyperplane section of W containing l. Then $\phi (M)$ is a hyperplane section of and

In particular, contains a cylinder.

Proof of Theorem 4.1

By Lemma 4.8, one of conditions (i) and (ii) of this lemma is fulfilled. In any case, by Corollaries 4.10 and 4.12 the complement contains a cylinder. $\square$

5 Cylindrical Mukai fourfolds of genus 8

In this section we construct a family of cylindrical Mukai fourfolds of genus 8 in ${\mathbb {P}}^{10}$, see Theorem 5.6 below. According to [20] any Fano fourfold $V=V_{14}\subset {\mathbb {P}}^{10}$ of index 2 and genus 8 with is a section of the Grassmannian under its Plücker embedding in ${\mathbb {P}}^{14}$ by a linear subspace of dimension 10. The Grassmannian contains planes of two kinds, namely, the two-dimensional Schubert varieties $\sigma _{3,3}$ and $\sigma _{4,2}$, where

• with a fixed plane, and

• with a fixed linear 3-subspace, and $p\in {\Lambda }$ a fixed point.

Lemma 5.1

Let $V=V_{14}$ be a Fano–Mukai fourfold of index 2 and genus 8. Suppose that V contains a $\sigma _{4,2}$-plane ${\Pi }$. Then $c_2({\fancyscript{N}}_{{\Pi }/V})=2$.

Proof

Let l be the class of a line on ${\Pi }$. Likewise as in Corollary 4.5, we have and . Then similarly as in Lemma 4.6, we obtain

$\square$

Lemma 5.2

There exists a smooth section $V=V_{14}$ of by a linear subspace $L\simeq {\mathbb {P}}^{10}$ containing a $\sigma _{4,2}$-plane ${\Pi }$. Furthermore, such a section V can be chosen so that ${\Pi }$ does not meet along a line any other plane contained in V.

Proof

The first assertion follows immediately from Lemma 2.4. To show the second one, assume that ${\Pi }$ meets another plane ${\Pi }'\subset V$ along a line, and let be the linear span of ${\Pi }\cup {\Pi }'$ in $L\simeq {\mathbb {P}}^{10}$.

We claim that if ${\Pi }'$ is a $\sigma _{4,2}$-plane, then K is contained in the Grassmannian , and hence also in V. The latter yields a contradiction because . To show the claim, notice that consists of all lines in a plane in passing through a given point P. This plane N is the intersection of the two linear 3-subspaces, say, M and $M'$ in that define our Schubert varieties ${\Pi }$ and ${\Pi }'$, respectively. Let be the linear span of $M\cup M'$ in . Consider the Schubert variety in the Grassmannian , which consists of all lines through P contained in R. Its image under the Plücker embedding of in ${\mathbb {P}}^{14}$ is a linear 3-subspace containing ${\Pi }\cup {\Pi }'$. Hence this image coincides with K. This proves the claim.

The latter argument does not work in the case, where ${\Pi }'$ is a $\sigma _{3,3}$-plane. However, this possibility can be ruled out as well by choosing carefully a section L through ${\Pi }$.

Indeed, let G be the set of all linear subspaces of dimension 10 in ${\mathbb {P}}^{14}$ through the given $\sigma _{4,2}$-plane ${\Pi }$. Then , so $\dim G=32$. Consider further a $\sigma _{3,3}$-plane ${\Pi }'$ that meets ${\Pi }$ along a line. Then the plane in that corresponds to ${\Pi }'$ contains the point corresponding to ${\Pi }$ and is contained in the corresponding linear 3-subspace in . The set of all such planes in is two-dimensional, hence also the set of all such possible $\sigma _{3,3}$-planes ${\Pi }'$ in ${\mathbb {P}}^{14}$ is.

Fixing ${\Pi }'$ we consider the set $G'$ of all linear subspaces of dimension 10 in ${\mathbb {P}}^{14}$ through the linear 3-space . Then , and so $\dim G'=28$. Finally, let $\fancyscript{E}$ be the variety of all possible configurations $({\Pi }', L)$ as before. Due to our observations we have $\dim \fancyscript{E}\le 28+2=30<32=\dim G$. Hence a general section through ${\Pi }$ does not contain a $\sigma _{3,3}$-plane ${\Pi }'$ that meets ${\Pi }$ along a line. $\square$

Lemma 5.3

Let $\fancyscript{V}$ be the family of all smooth fourfold linear sections $V_{14}$ of the Grassmannian , and $\fancyscript{V}_{4,2}$ be the subfamily of those sections that contain a $\sigma _{4,2}$-plane. Then $\fancyscript{V}_{4,2}$ has codimension 1 in $\fancyscript{V}$.

Proof

We keep the notation from the proof of Lemma 5.2. The variety $\fancyscript{P}$ of all the $\sigma _{4,2}$-planes in the Grassmannian is isomorphic to the variety of all the flags . The latter variety has dimension 11. It follows that $\dim \fancyscript{V}_{4,2}\le \dim G+\dim \fancyscript{P}=32+11=43$. Let us show that actually $\dim \fancyscript{V}_{4,2}=43$.

Indeed, consider the incidence variety . We claim that the natural surjection ${\text {pr}}_2:\fancyscript{I}\rightarrow \fancyscript{V}_{4,2}$ is generically finite, or, which is equivalent, that a generic member $V\in \fancyscript{V}_{4,2}$ contains at most finite number of $\sigma _{4,2}$-planes ${\Pi }$. Assume that ${\Pi }$ belongs to a family of $\sigma _{4,2}$-planes ${\Pi }_t\subset V$. By Lemma 5.1, we have . Since ${\Pi }$ and ${\Pi }_t$ are planes, they cannot meet each other at two points. Hence ${\Pi }\cap {\Pi }_t$ is a line. On the other hand, it was shown in the proof of Lemma 5.2 that ${\Pi }_t$ and ${\Pi }$ cannot meet along a line, a contradiction. Hence a generic Mukai fourfold $V\in \fancyscript{V}_{4,2}$ contains a finite number of $\sigma _{4,2}$-planes, as claimed.

Since the projection ${\text {pr}}_1:\fancyscript{I}\rightarrow \fancyscript{P}$ is surjective, and its fiber G over a given $\sigma _{4,2}$-plane ${\Pi }\in \fancyscript{P}$ has dimension 11, we have $\dim \fancyscript{I}=\dim G +\dim \fancyscript{P}=43$. Furthermore, since the second projection ${\text {pr}}_2:\fancyscript{I}\rightarrow \fancyscript{V}_{4,2}$ is surjective and generically one-to-one, we get $\dim \fancyscript{V}_{4,2}=\fancyscript{I}=43$, as desired. On the other hand, the variety $\fancyscript{V}$ of all the Mukai fourfolds $V_{14}$ in ${\mathbb {P}}^{14}$ can be naturally identified with an open set in the Grassmannian . Hence $\fancyscript{V}$ is irreducible of dimension , and so, $\fancyscript{V}_{4,2}$ has codimension 1 in $\fancyscript{V}$. Now the assertion follows. $\square$

Similarly to Corollary 4.5 we can prove

Lemma 5.4

We have , and .

Proposition 5.5

Let $V=V_{14}\subset {\mathbb {P}}^{10}$ be a Mukai fourfold of genus 8 containing a $\sigma _{4,2}$-plane ${\Pi }$, see Lemma 5.2. Then there is a commutative diagram

where

• (i) $\rho :\widetilde{V}\rightarrow V$ is the blowup of ${\Pi }$ and is the projection with center ${\Pi }$, which sends V birationally to a quintic Fano fourfold $W=W_5$ in with ;

• (ii) $\varphi :\widetilde{V}\rightarrow W$ is the blowup of a smooth rational surface $F\subset W$ of degree 7 with , contained in a singular hyperplane section L of W. This surface F can be obtained by blowing up six points in ;

• (iii) is defined by the linear system on $\widetilde{V}$, where $E =\rho ^{-1}({\Pi })\subset \widetilde{V}$ is the exceptional divisor of $\rho$ and H is a hyperplane in ${\mathbb {P}}^{10}$;

• (iv) $\varphi (E) = L = \langle F\rangle$ is the linear span of F;

• (v) if $D=\varphi ^{-1}(F)$ is the exceptional divisor of $\varphi$, then and ;

• (vi) , where $\rho (D)$ is cut out in V by a hyperplane section ${\mathbb {P}}^{10}$ which is singular along ${\Pi }$.

Proof

Since ${\Pi }$ is a scheme-theoretic intersection of members of the linear system $|H-{\Pi }|$, the linear system is base point free. Hence the divisor is ample, i.e. $\widetilde{V}$ is a Fano fourfold with . By the Cone Theorem there exists a Mori contraction $\varphi :\widetilde{V}\rightarrow W$ different from $\rho$. If $\widetilde{C}\subset \widetilde{W}$ is the proper transform of a line $C\subset V$ meeting ${\Pi }$, then . In particular, the divisor is not ample. So, defines a supporting function for the extremal ray generated by curves contained in the fibers of $\varphi$. Moreover, we can write , where L is the ample generator of . We have . By the Riemann–Roch and Kodaira Vanishing Theorems we have . Therefore, defines a birational morphism . Further, and . Thus $\varphi$ contracts a unique divisor . Since , the divisor D is irreducible. Furthermore, there is a commutative diagram

where the map is given by the linear system and $\upsilon :\widetilde{V}\xrightarrow {\varphi } W\rightarrow W'$ is the Stein factorization.

Since , the image $F=\varphi (D)$ is a surface in W with . Note that $E\simeq {\mathbb {P}}_{{\mathbb {P}}^2}\bigl ({\fancyscript{N}}_{{\Pi }/V}^*\bigr )$. Suppose first that there is a two-dimensional fiber $B\subset \widetilde{V}$ of $\varphi$ not contained in E. Then by our construction $\rho (B)$ is a plane meeting ${\Pi }$ along a conic. This contradicts our assumption.

Assume further that there is a two-dimensional fiber $B\subset \widetilde{V}$ of $\varphi$ contained in E. Then $\upsilon (E)$ is a cone. Indeed, the images of the fibers of $E\rightarrow {\Pi }$ are lines in passing through the point $\upsilon (B)$. Moreover, $\upsilon (E)$ is a cone over a surface which is an image of . Since , we get a contradiction.

Therefore, all the fibers of $\varphi$ have dimension less than or equal to 1, the contraction $E\rightarrow \varphi (E)$ is small, and the variety $\varphi (E)$ is a del Pezzo threefold with isolated singularities. (A description of such threefolds can be found in [23, 5.3.5].) By [2], both V and F are smooth, and $\varphi$ is the blowup of F. Since $-K_W=\varphi _*(-K_{\widetilde{V}})=3L$, W is a Fano fourfold of index 3 and degree $L^4=5$. By the classification $L=- K_V/3$ is very ample, so $V'\rightarrow V$ is an isomorphism. Since the divisor is not movable, .

Finally, using Lemma 2.3, one can deduce that . So, a general hyperplane section of F is a smooth curve of genus 2, and F is a surface of negative Kodaira dimension, i.e. F is birationally ruled. For the Euler numbers we obtain and , see Corollary 4.5 and Lemma 5.4. So by our construction, . Let $M\subset F$ be a general hyperplane section. If the divisor $K_F+M$ is not nef, then there exists an extremal ray R such that . Since M is ample, R cannot be generated by a $(-1)$-curve. Hence F is a geometrically ruled surface, and R is generated by its rulings. In the latter case the Euler number must be even, a contradiction.

Thus the divisor $K_F+M$ is nef. This yields the inequality . Hence $K_F^2\ge 3$, and so, F is a rational surface. By the Noether formula, $K_F^2=3$ and . By Riemann–Roch and Kodaira Vanishing Theorems, . Since , the linear system is a base point free pencil. It defines a morphism such that $-K_F$ is relatively ample. Hence $\Phi _{|K_F+M|}$ is a conic bundle with degenerate fibers. Let ${\Sigma }\subset F$ be a section of this bundle with the minimal possible self-intersection number ${\Sigma }^2=-n$. Then , and so . On the other hand, $n\ge 1$. It is possible to contract extra components of the five degenerate fibers of F in order to get a relatively minimal rational ruled surface $F'$ with a section ${\Sigma }'\subset F'$ such that ${{\Sigma }'}^2=-1$, i.e. $F'\simeq {\mathbb {F}}_1$. This means that F can be obtained by blowing up six points on , as stated in (ii). $\square$

We can deduce now the main result of this section.

Theorem 5.6

Let $V=V_{14}$ be the Mukai fourfold of genus 8 constructed in Lemma 5.2, and let $\rho (D)$ be the divisor on V constructed in Proposition 5.5. Then the Zariski open set contains a cylinder.

Proof

Indeed, by (iv) and (vi) of Proposition 5.5, we have . Using Theorem 4.1 with $M=L$, the result follows. $\square$

6 Cylindrical Mukai fourfolds of genus 7

In this section we prove the following theorem.

Theorem 6.1

There exists a family of smooth cylindrical Mukai fourfolds of genus 7 with . Its image in the corresponding moduli space has codimension 2.

The proof exploits several auxiliary results. Actually, our Mukai fourfold V is obtained starting with a del Pezzo fourfold $W_{2\cdot 2}$ via a Sarkisov link, as described in Proposition 6.2. We choose this link in such a way that the cylinder structure is preserved. This is the main point of our construction.

Proposition 6.2

([21]) Let be a smooth intersection of two quadrics, and H be a hyperplane section of W, so that the class of H is the ample generator of . Suppose that W contains an anticanonically embedded del Pezzo surface $F=F_5$ of degree 5 and does not contain any plane that meets F along a conic. Then the following hold.

• (i) The linear system $|2H-F|$ of quadrics passing through F defines a birational map , where $V=\phi (W)$ is a Mukai fourfold of genus 7 with .

• (ii) There is a commutative diagram

  

  (4)

  where $\rho$ is the blowup of F and $\varphi$ is the blowup of a plane .

• (iii) Let $E\subset \widetilde{W}$ $(D\subset \widetilde{W}$, respectively) be the $\rho$-exceptional $(\varphi$-exceptional, respectively) divisor, and let L be the ample generator of . Then

  

Proof

Since F is a scheme-theoretic intersection of quadrics, the linear system is base point free. Hence the divisor is ample, i.e. $\widetilde{W}$ is a Fano fourfold with . By the Cone Theorem, there exists a Mori contraction $\varphi :\widetilde{W}\rightarrow U$ different from $\rho$. Let $H_F$ be the hyperplane section of W that passes through F, and let D be its proper transform in $\widetilde{W}$. We can write for some $k>0$. On the other hand, we have . Hence, $k=1$ and . This means that the divisor class of is not ample, and so it yields a supporting linear function of the extremal ray generated by the curves in the fibers of $\varphi$. Moreover, we can write , where L is the ample generator of . We have . It follows that $\dim V=4$, i.e. $\varphi$ is birational, and its exceptional locus coincides with D. In particular, it is an irreducible divisor. Using the Riemann–Roch and Kodaira Vanishing Theorems we obtain the equality . This yields the diagram

where is given by the linear system , and $\widetilde{W}\xrightarrow {\varphi } V\rightarrow V'$ is the Stein factorization.

Since , the image $\varphi (D)$ is a surface with . Let $B\subset \widetilde{W}$ be a two-dimensional component of a fiber of $\varphi$ which is not contained in E. According to our geometric construction, $\rho (B)\subset \rho (B)\subset W\cap \langle F\rangle$. Thus $\rho (B)$ is a plane meeting F along a conic. However, the latter contradicts our assumption.

If $B\subset \widetilde{W}$ is a two-dimensional fiber of $\varphi$ contained in E, then the restriction $\rho |_B:B \rightarrow F$ is a finite morphism. Hence . This contradicts the classification of fourfold contractions [3].

Finally, all the fibers of $\varphi$ have dimension less than or equal to 1. By [2], both V and $\varphi (D)$ are smooth, and $\varphi$ is the blowup of $\varphi (D)$. Since $-K_V=\varphi _*(-K_W)=2L$, the variety V is a Fano fourfold of index 2 and of genus $g= L^4/2+1=7$. By virtue of the classification, $L=- K_V/2$ is very ample, so $V'\rightarrow V$ is an isomorphism. $\square$

The following corollary is immediate.

Corollary 6.3

We have , where $\varphi (E)$ and $\rho (D)$ are hyperplane sections of V and W, respectively. Moreover, $\varphi (E)$ is singular along ${\Xi }$, and $\rho (D) =W\cap \langle F\rangle$, where is the linear span of F in .

Corollary 6.4

Let V be a variety as in Proposition 6.2. Then the number of planes contained in V is finite, and the group is finite as well.

Proof

Assume that V contains a family of planes ${\Xi }_t$. The map $\phi ^{-1}$ is a projection from ${\Xi }$. Hence the image of a general plane ${\Xi }_t$ is again a plane. On the other hand, the set of planes contained in is finite [24]. This yields a contradiction.

The group consists of the projective transformations of preserving V, so this is a linear algebraic group. Since the number of planes contained in V is finite, the identity component ${\text {Aut}}^0(V)$ preserves each of these planes. In particular, it preserves the center ${\Xi }$ of the blowup $\varphi$. Hence diagram (4) is ${\text {Aut}}^0(V)$-equivariant with respect to a faithful ${\text {Aut}}^0(V)$-action on W.

However, the group ${\text {Aut}}^0(W)$ is trivial. Indeed, the embedding being given by the linear system $|- K_W/3|$, the latter group acts linearly on and preserves every degenerate member of the pencil of quadrics in passing through W. These degenerate members are seven quadric cones, whose vertices are points in in general position fixed under the ${\text {Aut}}^0(W)$-action. There is a unique (up to permutations) homogeneous coordinate system $(x_0\,{:}\,\ldots \,{:}\, x_6)$ in with our seven vertices as nodes, such that the pencil of quadrics is generated by $\sum x_i^2=0$ and $\sum \lambda _i x_i^2=0$ with $\lambda _i\ne \lambda _j$ for $i\ne j$, see e.g. [24, Proposition 2.1]. It follows that the group ${\text {Aut}}^0 (W )$ is trivial. Hence ${\text {Aut}}^0(V)$ is trivial too, and so is finite. $\square$

Due to Corollary 6.3, to prove Theorem 6.1 it suffices to show the existence of a cylinder in , where $H_0=\rho (D)$ is a hyperplane section of which contains a quintic del Pezzo surface F. To this end, we apply Corollary 3.5. The assumptions of Corollary 3.5 are satisfied once there is a plane ${\Pi }_0\subset H_0$ which does not meet F along a conic, see Proposition 6.2.

Using the following construction we produce examples, where these geometric restrictions are fulfilled. This gives the first part of Theorem 6.1.

Construction 6.5

(cf. [23, 5.3.9]) Let be a Fano threefold of index 2 and degree 5. It is well known, see e.g. [15, Theorem 3.3.1], that X can be realized as a section of the Grassmannian under its Plücker embedding in by a subspace of codimension 3. The projection from a general point $P'\in X_5$ sends $X_5$ to a singular Fano threefold of degree 4. The latter threefold is a complete intersection of two quadrics, say, $Q_1'$ and $Q_2'$, see [25, Corollary 0.8].

Lemma 6.6

The variety Y constructed above contains an anticanonically embedded del Pezzo surface of degree 5 and a unique plane ${\Pi }_0$. This plane meets F at three points, say, $A_j$, $j=1,2,3$, that are the only singular points of Y, and these singularities are ordinary double points.

Proof

By [13], through a general point $P'\in X_5$ there pass exactly three lines, say, $l_j$, $j=1,2,3$, on $X_5$. These lines do not belong to the same plane. Under our projection, they are contracted to three distinct non-collinear points, say, $A_j$, $j=1,2,3$, of $Y_4$. By the van der Waerden’s Purity Theorem, the points $A_j$ are the only singularities (actually, nodes) of $Y_4$. Let ${\Pi }_0$ be the plane in through the points $A_j$, $j=1,2,3$. We claim that ${\Pi }_0$ is contained in $Y_4$ and is a unique plane contained in $Y_4$. Indeed, consider the diagram

where $\rho '$ is the blowup of $P'$ and $\varphi '$ is the (small) contraction of the proper transforms $\widetilde{l_j}\subset \widetilde{X_5}$ of the lines $l_j$, $j=1,2,3$. Clearly, ${\Pi }_0=\varphi '(E')$, where is the exceptional divisor of the blowup $\rho '$. Hence ${\Pi }_0\subset Y_4$.

We have , and by duality ${\text {rk}}{{\text {N}}_1(\widetilde{X_5}})_{{\mathbb {R}}}=2$. For a general hyperplane section H of $Y_4$ we have . It follows that for any plane ${\Pi }'$ contained in $Y_4$, the intersection numbers , are simultaneously all zero or not, where $\widetilde{{\Pi }'}$ is the proper transform of ${\Pi }'$ in $\widetilde{X_5}$. If $\widetilde{{\Pi }'}$ does not meet the curves $\widetilde{l_j}$, then ${\Pi }'$ does not pass through the singular points of $Y_5$ and so is a Cartier divisor on $Y_4$. Then $1=\deg {\Pi }'\equiv 0 \,\mathrm{mod}\, 4$, a contradiction. Thus $A_j\in {\Pi }'$, $j=1,2,3$, hence ${\Pi }'={\Pi }_0$, as claimed.

A general hyperplane section $F'$ of $X_5$ in is a smooth del Pezzo surface of degree 5. Since $F'$ meets transversally the lines $l_j$ and does not contain $P'$, it maps under $\phi '$ isomorphically onto its image, say, F in $Y_4$, and $F\cap {\Pi }_0=\{A_1,A_2,A_3\}$. $\square$

Remark 6.7

Let $q_i(x_0,\ldots ,x_5)=0$ be the equation of $Q_i'$ in . Consider the quadrics $Q_i$ in with equations $q_i(x_0,\ldots ,x_5)+x_6f_i(x_0,\ldots ,x_5)=0$, $i=1,2$, where $f_1$ and $f_2$ are generic linear forms. We claim that the fourfold $W=W_{2 \cdot 2}=Q_1\cap Q_2$ in is smooth and satisfies all the assumptions of Proposition 6.2. To show the claim, let us notice that the hyperplane $x_6=0$ in cuts the quadric $Q_i$ along $Q_i'$ and cuts W along $Y_4=Q_1'\cap Q_2'$. Hence W contains the smooth del Pezzo surface $F\subset Y_4$ of degree 5, and does not contain any plane which meets F along a conic. Indeed, otherwise such a plane would be contained in the hyperplane $x_6=0$, and so would coincide with ${\Pi }_0$ by virtue of Lemma 6.6. Since ${\Pi }_0$ meets $Y_4$ just in the points $A_j$, $j=1,2,3$, we get a contradiction. This proves the claim.

In the following lemma we provide an alternative construction of the family of pairs (Y, F) as in Construction 6.5 and Lemma 6.6, which will be used in Lemma 6.9.

Lemma 6.8

Let be a del Pezzo surface of degree 5 and $Q_1,Q_2$ be general quadrics in containing F. Then $Y=Q_1\cap Q_2$ is a threefold as in Construction 6.5. In particular, Y contains a plane meeting F in three points.

Proof

Let be another general quadric containing F. Then $Y\cap Q= F\cup F'$, where $F'$ is a cubic surface scroll. The linear span ${\Lambda }=\langle F'\rangle$ is a subspace of dimension 4. Hence $Y\cap {\Lambda }= F'\cup {\Pi }_0$, where ${\Pi }_0$ is a plane contained in Y. By Construction 6.5, for a general choice of $Q_1,Q_2$ and Q, the variety Y has only isolated singularities, and these singularities are nodes. Moreover, Y contains no plane different from ${\Pi }_0$. Such varieties Y are described in [23, 5.3.9], and their construction coincides with that of Construction 6.5. $\square$

Using [20], one can deduce that the moduli space of the Mukai fourfolds of genus 7 has dimension 15. The second assertion of Theorem 6.1 follows now from the next lemma.

Lemma 6.9

The image in the moduli space of the family of all Fano fourfolds of genus 7 obtained by our Construction 6.5 has dimension 13.

Proof

Recall that is an intersection of five linearly independent quadrics [7, Corollary 8.5.2]. Thus the space of all quadrics in passing through F has dimension $5+7=12$. Pencils of quadrics passing through F are parametrized by the Grassmannian . Since the group is finite, and any automorphism of , which acts trivially on F, acts also trivially on , the algebraic group has dimension 6, while . Modulo the -action on , we have $20- 7=13$-dimensional family of such pencils of quadrics. Hence the dimension of the family of all the Fano fourfolds that can be obtained by our construction equals 13. Its image in the moduli space has the same dimension due to Corollary 6.4. $\square$