Examples of cylindrical Fano fourfolds
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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).
Keywords
Affine cone Fano variety Automorphism Group action of the additive groupMathematics Subject Classification
14R20 14J45 14J50 14R051 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 Open image in new window , where Z is a variety and Open image in new window stands for the affine line over \({\mathbb {C}}\) [16, 17].
Assuming Open image in new window , 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 wellknown Hirzebruch Problem of classifying all possible smooth compactifications Open image in new window 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 cylindrical^{1} 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 Open image in new window and the smooth quadric in Open image in new window are cylindrical, since they contain a Zariski open subset isomorphic to the affine space Open image in new window . By the same reason, the Fano threefold of index 2 and degree 5 is also cylindrical. A smooth intersection of two quadrics in Open image in new window 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 Open image in new window is fourdimensional. 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: Open image in new window . 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 Open image in new window , and if \(r=4\) then V is a quadric in Open image in new window . 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 \(d=2g2\), 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
 (i)
the smooth intersections \(V_{2\cdot 2}\) of two quadrics in Open image in new window ;
 (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 Open image in new window 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 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.
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 easytosee. It remains to observe that the inverse procedure does not affect the cylinder. In example (iv) we proceed the other way round, from an easytosee cylinder to the desired one.
2 Preliminaries
Notation

\(K_X\) 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 \(c_i(X)=c_i({\fancyscript{T}}_X)\);

Open image in new window the Euler number of X.
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
The next lemma can be deduced, for instance, from [11, Theorem 15.4].
Lemma 2.2
 (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
 (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 Open image in new window be a smooth projective variety of dimension n. Assume that X contains a kdimensional linear subspace \({\Lambda }\), and let H be a general hyperplane section of X containing \({\Lambda }\). If \(n>2k\), then H is smooth.
Proof
Lemma 2.5
Proof
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 twodimensional. Thus the family of lines in W through P is at most onedimensional. At a generic point of W, it should be onedimensional, 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 Open image in new window
It is known that any smooth intersection of two quadrics in Open image in new window contains a cylinder [16, Proposition 5.0.1]. Similarly, this holds for any smooth intersection \(W_{2\cdot 2}\) of two quadrics in Open image in new window , 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
 (i)
The \(\varphi \)exceptional locus is an irreducible divisor \(D\subset \widetilde{W}\).
 (ii)Letting L be the ample generator of Open image in new window we obtain
 (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.
 (iv)
Open image in new window , where \(\varphi (E)\) is a quadric in Open image in new window .
 (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 schemetheoretic intersection of members of \(Hl\), 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 twodimensional 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 twodimensional 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.
The proof of the next simple lemma is left to the reader.
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 .
The next lemma and its corollary will be used in Sect. 6.
Lemma 3.4
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.\(\square \)
The next corollary is immediate.
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.
Resuming, we obtain the main result of this section.
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). \(\square \)
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 Open image in new window of the Grassmannian Open image in new window under its Plücker embedding in Open image in new window by a codimension 2 linear subspace. Clearly, the class of a hyperplane section H generates the group Open image in new window , 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 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.

Open image in new window with Open image in new window a fixed 3dimensional subspace and Open image in new window a fixed point;

Open image in new window with Open image in new window 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 (nonvertex 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
 (i)
W contains exactly one \(\sigma _{2,2}\)plane \({\Xi }\) and a oneparameter 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 Open image in new window .
 (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 .
We need the following computational facts.
Lemma 4.4
Proof
Let \({\fancyscript{I}}\rightarrow G\) be the universal subbundle and \({\fancyscript{Q}}\rightarrow G\) the universal factorbundle, 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
Proof
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
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
 (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 Open image in new window . By Corollary 2.6, there exists a oneparameter 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 Open image in new window 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
 (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 ;
 (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;
 (iii)
Open image in new window 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 Open image in new window on \(\widetilde{W}\).
Sketch of the proof Using Lemmas 2.3 and 4.6, it is easy to deduce that Open image in new window and Open image in new window . Hence \(\varphi \) is a birational morphism, Open image in new window , and Open image in new window . Since Open image in new window for some \(k \ge 2\), we have Open image in new window . 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 Open image in new window , and Open image in new window . In particular, Open image in new window 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
 (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 ;
 (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;
 (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;
 (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 twodimensional fiber (otherwise Q must be singular). \(\square \)
Now we can deduce the following corollary.
Corollary 4.12
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. \(\square \)
5 Cylindrical Mukai fourfolds of genus 8

Open image in new window with Open image in new window a fixed plane, and

Open image in new window with Open image in new window a fixed linear 3subspace, 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
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 3subspaces, 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 3subspace 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 3subspace in Open image in new window . The set of all such planes in Open image in new window is twodimensional, 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 3space 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 onetoone, 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. \(\square \)
Similarly to Corollary 4.5 we can prove
Lemma 5.4
We have Open image in new window , and Open image in new window .
Proposition 5.5
 (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 ;
 (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 ;
 (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 \({\mathbb {P}}^{10}\);
 (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 Open image in new window and Open image in new window ;
 (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 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 twodimensional 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 twodimensional 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 selfintersection 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). \(\square \)
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 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. \(\square \)
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 Open image in new window of genus 7 with Open image in new window . 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
 (i)
The linear system \(2HF\) 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 .
 (ii)There is a commutative diagram where \(\rho \) is the blowup of F and \(\varphi \) is the blowup of a plane Open image in new window .
 (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
Proof
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 twodimensional 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 twodimensional 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. \(\square \)
The following corollary is immediate.
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. \(\square \)
Due to Corollary 6.3, to prove Theorem 6.1 it suffices to show the existence of a cylinder in Open image in new window , where \(H_0=\rho (D)\) is a hyperplane section of Open image in new window 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 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
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.
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 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. \(\square \)
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 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. \(\square \)
Footnotes
 1.
If Open image in new window , by cylindricity we mean here the polar cylindricity, see [19].
 2.
By abuse of notation, we denote the class in Open image in new window of the hyperplane section H by the same letter.
Notes
Acknowledgments
This work was done during a stay of the first author at the Institute Fourier, Grenoble, and of the second author in the Max Planck Institute for Mathematics, Bonn. The authors thank these institutions for their hospitality and a generous support. Our thanks are due also to the referee for useful comments.
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