Examples of cylindrical Fano fourfolds

Theorem 1.1

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 Open image in new window 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 Open image in new window , the complement Open image in new window 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 Open image in new window .

(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 Open image in new window , 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.

Notation

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

Open image in new window

Lemma 2.2

Lemma 2.3

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 Open image in new window 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 Open image in new window that are singular at some point of \({\Lambda }\) is Zariski closed in Open image in new window , of dimension

Open image in new window

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 Open image in new window 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:

Open image in new window

(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 Open image in new window , which contains some lines. Let l be a line on F, and consider the exact sequence

Open image in new window

We have

Open image in new window

where \(a+b+c=1\). Furthermore, \(a,b,c\le 1\) since Open image in new window 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 Open image in new window 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 Open image in new window , using deformation theory we obtain that \(\mathfrak L(W)\) is reduced, nonsingular and Open image in new window .

If \(d=3\), then W is a smooth cubic in Open image in new window . The tangent section T at a point P of W is a singular cubic threefold in the projective tangent space Open image in new window . In an affine chart in Open image in new window 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 Open image in new window cannot be a divisor in W. Indeed, otherwise this should be a hyperplane section generating the Picard group Open image in new window . However, for a generic point Open image in new window there is a hyperplane Open image in new window through \(P'\), which contains Open image in new window . This leads to a contradiction.

Any line through P in W is contained in the tangent space Open image in new window , 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. \(\square \)

Proposition 3.1

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

Open image in new window

(2)

where \(\rho \) is the blowup of l with exceptional divisor E, and \(\varphi \) is a birational morphism defined by the linear system Open image in new window . Furthermore, the following hold.

1. (i)

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

    
2. (ii)
   Letting L be the ample generator of Open image in new window we obtain

   Open image in new window
    
3. (iii)

   The divisor \(\rho (D)\) is cut out on W by a quadric in Open image in new window , and is swept out by lines meeting l. The image \(F=\varphi (D) \) is a surface in Open image in new window 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.

    
4. (iv)

   Open image in new window , where \(\varphi (E)\) is a quadric in Open image in new window .

    
5. (v)

   The quadric Open image in new window 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 Open image in new window is base points free. Hence the divisor Open image in new window is ample, i.e. \(\widetilde{W}\) is a Fano fourfold with Open image in new window . 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 Open image in new window . In particular, the divisor Open image in new window is not ample. So, Open image in new window yields a supporting linear function for the extremal ray generated by the curves contained in the fibers of \(\varphi \). Furthermore, we can write Open image in new window , where L is the ample generator of Open image in new window . We have Open image in new window . By the Riemann–Roch and Kodaira Vanishing Theorems we have Open image in new window . Therefore, Open image in new window defines a birational morphism Open image in new window , which coincides actually with the map \(\varphi \).

Since Open image in new window and Open image in new window , the linear system Open image in new window contains a unique divisor D contracted by \(\varphi \). Since Open image in new window , 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 Open image in new window , while the last equality in (ii) shows that \(\varphi (E)\) is a quadric in Open image in new window , 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 Open image in new window , cf. Lemma 2.3 (ii). Hence, the image \(F=\varphi (D)\) is a quintic surface in Open image in new window . 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 Open image in new window , 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 Open image in new window 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 Open image in new window is a line.

Lemma 3.3

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

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 Open image in new window , and

Open image in new window

(3)

If Open image in new window , then Open image in new window contains a cylinder. Assume that there is a plane \({\Pi }\subset W\) such that \(l\subset {\Pi }\subset H_0\). Then the condition Open image in new window holds.

Proof

Let \(\widetilde{H}_0\subset \widetilde{W}\) be the proper transform of \(H_0\). We may write Open image in new window for some \(k\ge 1\). Since Open image in new window , 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 Open image in new window , see Remark 3.2 (d). By Lemma 3.3, the projection of Open image in new window from the point Open image in new window produces a cylinder structure on Open image in new window . The existence of an isomorphism in (3) is straightforward by our construction. Now the remaining assertions follow.\(\square \)

Corollary 3.5

If \(H_0\subset W\) is a hyperplane section through a plane \({\Pi }\subset W\), then Open image in new window contains a cylinder.

Theorem 3.6

In the notation of Proposition 3.1, the Zariski open set Open image in new window contains a cylinder.

Proof

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

Theorem 4.1

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

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).

Proposition 4.3

Let Open image in new window be a Fano fourfold of index 3 and degree 5. Then the following hold.

1. (i)

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

    
2. (ii)

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

    
3. (iii)

   Any two \(\sigma _{3,1}\)-planes \({\Pi }_1\) and \({\Pi }_2\) meet at a point Open image in new window .

    
4. (iv)

   Let R be the union of all \(\sigma _{3,1}\)-planes on W. Then R is a hyperplane section of W and Open image in new window .

    

Lemma 4.4

For the Chern classes of Open image in new window we have

Open image in new window

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 Open image in new window , standard computations give the result, see e.g. [11, Example 14.5.2]. \(\square \)

Corollary 4.5

For \(W=W_5\) we have

Open image in new window

Proof

The Adjunction Formula for the total Chern classes yields

Open image in new window

Inversing the last factor gives the desired equalities.\(\square \)

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 Open image in new window and Open image in new window . Since Open image in new window , we obtain

Open image in new window

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, Open image in new window , and so, Open image in new window . 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 Open image in new window , and let R be as in Proposition 4.3 (iv).

1. (i)

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

    
2. (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 Open image in new window . 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. \(\square \)

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

Open image in new window

where

1. (i)

   \(\rho :\widetilde{W}\rightarrow W\) is the blowup of Open image in new window is the projection from Open image in new window is the blowup of a rational normal cubic curve Open image in new window ;

    
2. (ii)

   Open image in new window is defined by the linear system Open image in new window , where \(E =\rho ^{-1}({\Xi })\) is the exceptional divisor;

    
3. (iii)

   Open image in new window is the linear span of Y;

    
4. (iv)

   the exceptional divisor \(\widetilde{R}=\varphi ^{-1}(Y)\) of \(\varphi \) coincides with the proper transform of R in \(\widetilde{W}\), and Open image in new window on \(\widetilde{W}\).

    

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 Open image in new window , and Open image in new window . In particular, Open image in new window contains a cylinder.

Proposition 4.11

([9, Section 10], [22]) Let Open image in new window 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

Open image in new window

where

1. (i)

   \(\rho :\widetilde{W}\rightarrow W\) is the blowup of Open image in new window is the projection from Open image in new window is a smooth quadric, \(\varphi :\widetilde{W}\rightarrow Q\) is the blowup of a cubic scroll Open image in new window ;

    
2. (ii)

   Open image in new window is defined by the linear system Open image in new window , where \(E =\rho ^{-1}(l)\) is the exceptional divisor;

    
3. (iii)

   \(\varphi (E) = Q\cap \langle F\rangle \), where Open image in new window is the linear span of F, moreover, \(\varphi (E)\) is a quadratic cone;

    
4. (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). \(\square \)

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 Open image in new window and

Open image in new window

In particular, Open image in new window 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 Open image in new window contains a cylinder. \(\square \)

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 Open image in new window and Open image in new window . Then similarly as in Lemma 4.6, we obtain

Open image in new window

\(\square \)

Lemma 5.2

There exists a smooth section \(V=V_{14}\) of Open image in new window 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 Open image in new window 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 Open image in new window , and hence also in V. The latter yields a contradiction because Open image in new window . To show the claim, notice that Open image in new window consists of all lines in a plane Open image in new window in Open image in new window passing through a given point P. This plane N is the intersection of the two linear 3-subspaces, say, M and \(M'\) in Open image in new window that define our Schubert varieties \({\Pi }\) and \({\Pi }'\), respectively. Let Open image in new window be the linear span of \(M\cup M'\) in Open image in new window . Consider the Schubert variety Open image in new window in the Grassmannian Open image in new window , which consists of all lines through P contained in R. Its image under the Plücker embedding of Open image in new window 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 Open image in new window , so \(\dim G=32\). Consider further a \(\sigma _{3,3}\)-plane \({\Pi }'\) that meets \({\Pi }\) along a line. Then the plane in Open image in new window that corresponds to \({\Pi }'\) contains the point corresponding to \({\Pi }\) and is contained in the corresponding linear 3-subspace in Open image in new window . The set of all such planes in Open image in new window 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 Open image in new window . Then Open image in new window , 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 Open image in new window 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 Open image in new window , 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 Open image in new window is isomorphic to the variety of all the flags Open image in new window . 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 Open image in new window . 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 Open image in new window . 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 Open image in new window . Hence \(\fancyscript{V}\) is irreducible of dimension Open image in new window , and so, \(\fancyscript{V}_{4,2}\) has codimension 1 in \(\fancyscript{V}\). Now the assertion follows. \(\square \)

Lemma 5.4

We have Open image in new window , and Open image in new window .

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

Open image in new window

where

1. (i)

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

    
2. (ii)

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

    
3. (iii)

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

    
4. (iv)

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

    
5. (v)

   if \(D=\varphi ^{-1}(F)\) is the exceptional divisor of \(\varphi \), then Open image in new window and Open image in new window ;

    
6. (vi)

   Open image in new window , 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 Open image in new window is base point free. Hence the divisor Open image in new window is ample, i.e. \(\widetilde{V}\) is a Fano fourfold with Open image in new window . 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 Open image in new window . In particular, the divisor Open image in new window is not ample. So, Open image in new window defines a supporting function for the extremal ray generated by curves contained in the fibers of \(\varphi \). Moreover, we can write Open image in new window , where L is the ample generator of Open image in new window . We have Open image in new window . By the Riemann–Roch and Kodaira Vanishing Theorems we have Open image in new window . Therefore, Open image in new window defines a birational morphism Open image in new window . Further, Open image in new window and Open image in new window . Thus \(\varphi \) contracts a unique divisor Open image in new window . Since Open image in new window , the divisor D is irreducible. Furthermore, there is a commutative diagram

Open image in new window

where the map Open image in new window is given by the linear system Open image in new window and \(\upsilon :\widetilde{V}\xrightarrow {\varphi } W\rightarrow W'\) is the Stein factorization.

Since Open image in new window , the image \(F=\varphi (D)\) is a surface in W with Open image in new window . 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 Open image in new window passing through the point \(\upsilon (B)\). Moreover, \(\upsilon (E)\) is a cone over a surface which is an image of Open image in new window . Since Open image in new window , 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 Open image in new window is not movable, Open image in new window .

Finally, using Lemma 2.3, one can deduce that Open image in new window . 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 Open image in new window and Open image in new window , see Corollary 4.5 and Lemma 5.4. So by our construction, Open image in new window . 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 Open image in new window . 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 Open image in new window must be even, a contradiction.

Thus the divisor \(K_F+M\) is nef. This yields the inequality Open image in new window . Hence \(K_F^2\ge 3\), and so, F is a rational surface. By the Noether formula, \(K_F^2=3\) and Open image in new window . By Riemann–Roch and Kodaira Vanishing Theorems, Open image in new window . Since Open image in new window , the linear system Open image in new window is a base point free pencil. It defines a morphism Open image in new window such that \(-K_F\) is relatively ample. Hence \(\Phi _{|K_F+M|}\) is a conic bundle with Open image in new window degenerate fibers. Let \({\Sigma }\subset F\) be a section of this bundle with the minimal possible self-intersection number \({\Sigma }^2=-n\). Then Open image in new window , and so Open image in new window . 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 Open image in new window , as stated in (ii). \(\square \)

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 Open image in new window contains a cylinder.

Proof

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

Theorem 6.1

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

Proposition 6.2

([21]) Let Open image in new window 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 Open image in new window . 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.

1. (i)

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

    
2. (ii)
   There is a commutative diagram

   Open image in new window

   (4)

   where \(\rho \) is the blowup of F and \(\varphi \) is the blowup of a plane Open image in new window .
    
3. (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 Open image in new window . Then

   Open image in new window
    

Proof

Since F is a scheme-theoretic intersection of quadrics, the linear system Open image in new window is base point free. Hence the divisor Open image in new window is ample, i.e. \(\widetilde{W}\) is a Fano fourfold with Open image in new window . 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 Open image in new window for some \(k>0\). On the other hand, we have Open image in new window . Hence, \(k=1\) and Open image in new window . This means that the divisor class of Open image in new window 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 Open image in new window , where L is the ample generator of Open image in new window . We have Open image in new window . 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 Open image in new window . This yields the diagram

Open image in new window

where Open image in new window is given by the linear system Open image in new window , and \(\widetilde{W}\xrightarrow {\varphi } V\rightarrow V'\) is the Stein factorization.

Since Open image in new window , the image \(\varphi (D)\) is a surface with Open image in new window . 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 Open image in new window . 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. \(\square \)

Corollary 6.3

We have Open image in new window , 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 Open image in new window is the linear span of F in Open image in new window .

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 Open image in new window 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 Open image in new window is finite [24]. This yields a contradiction.

The group Open image in new window consists of the projective transformations of Open image in new window 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 Open image in new window being given by the linear system \(|- K_W/3|\), the latter group acts linearly on Open image in new window and preserves every degenerate member of the pencil of quadrics in Open image in new window passing through W. These degenerate members are seven quadric cones, whose vertices are points in Open image in new window 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 Open image in new window 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 Open image in new window is finite. \(\square \)

Construction 6.5

(cf. [23, 5.3.9]) Let Open image in new window 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 Open image in new window under its Plücker embedding in Open image in new window by a subspace of codimension 3. The projection from a general point \(P'\in X_5\) sends \(X_5\) to a singular Fano threefold Open image in new window 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 Open image in new window 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 Open image in new window 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

Open image in new window

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 Open image in new window is the exceptional divisor of the blowup \(\rho '\). Hence \({\Pi }_0\subset Y_4\).

We have Open image in new window , 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 Open image in new window . It follows that for any plane \({\Pi }'\) contained in \(Y_4\), the intersection numbers Open image in new window , 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 Open image in new window 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 Open image in new window . Consider the quadrics \(Q_i\) in Open image in new window 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 Open image in new window 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 Open image in new window 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.

Lemma 6.8

Let Open image in new window be a del Pezzo surface of degree 5 and \(Q_1,Q_2\) be general quadrics in Open image in new window 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 Open image in new window 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. \(\square \)

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 Open image in new window is an intersection of five linearly independent quadrics [7, Corollary 8.5.2]. Thus the space of all quadrics in Open image in new window passing through F has dimension \(5+7=12\). Pencils of quadrics passing through F are parametrized by the Grassmannian Open image in new window . Since the group Open image in new window is finite, and any automorphism of Open image in new window , which acts trivially on F, acts also trivially on Open image in new window , the algebraic group Open image in new window has dimension 6, while Open image in new window . Modulo the Open image in new window -action on Open image in new window , 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. \(\square \)