Some Loci of Rational Cubic Fourfolds
aa r X i v : . [ m a t h . AG ] J un SOME LOCI OF RATIONAL CUBIC FOURFOLDS
MICHELE BOLOGNESI, FRANCESCO RUSSO*, AND GIOVANNI STAGLIAN `O
Abstract.
In this paper we investigate the divisor C inside the moduli space of smoothcubic hypersurfaces in P , whose general element is a smooth cubic containing a smoothquartic rational normal scroll. By showing that all degenerations of quartic scrolls in P contained in a smooth cubic hypersurface are surfaces with one apparent double point, weprove that every cubic hypersurface contained in C is rational. Combining our proof withthe Hodge theoretic definition of C , we deduce that on a smooth cubic fourfold every class T ∈ H , ( X, Z ) with T = 10 and T · h = 4 is represented by a (possibly reducible) surfaceof degree four which has one apparent double point. As an application of our results and ofthe construction of some explicit examples, we also prove that the Pfaffian locus is not openin C . Introduction
Cubic hypersurfaces in P are among the most studied objects in algebraic geometry. Thisis surely due to the wealth of geometry that they carry along and, possibly, to the fact thatthey are somehow a very simply defined class of geometric objects, whose rationality is notyet well-understood. The study of the moduli space C of smooth cubic fourfolds, particularlythrough GIT and the period map, has known some very striking advances in recent years,see for example [Voi86, Laz10, Loo09], and this analysis has been developed in parallel tothe study of rationality. In particular, Hassett described a countable infinity of divisors C d that parametrize special cubic 4-folds, that is cubic hypersurfaces containing a surface nothomologous to a complete intersection, see [Has00]. One expects that rational cubic fourfoldsshould be strictly contained in the union of these special divisors C d (more precise conjectureshave been formulated by Kuznetsov and Hassett, see [Has16, Section 3] for the state of theart on the subject).An interesting and well studied locus of rational cubics is given by Pfaffian cubics, i.e. cubic hypersurfaces admitting an equation given by the Pfaffian of a 6 × C , whichis not open in C , as we shall show in Theorem 3.7 and Remark 3.8. As a special surface for C one can take either a smooth quartic rational normal scroll or a smooth quintic delPezzo surface but also, for instance, the isomorphic projection of a smooth surface of degree8 and sectional genus 3 in P , see [RS17, Theorem 2]. The Pfaffian cubics form a subset in C , which consists exactly of cubic fourfolds containing a smooth quintic del Pezzo surface,see [Bea00, Proposition 9.2].Other examples of rational cubic fourfolds are given by a countable infinity of irreducibledivisors W i in C , the divisor of cubic 4-folds containing a plane. The families W i ’s are thus of *Partially supported by the PRIN “Geometria delle variet`a algebriche” and by the Research NetworkProgram GDRE-GRIFGA; the author is a member of the G.N.S.A.G.A.. codimension two in C and consist of cubic 4-folds containing a plane P such that the naturalquadric fibration obtained by projection from P has a rational section, yielding directly therationality of these cubic hypersurfaces. As first remarked in [ABBVA14] and as we shallalso show in the last section, there exist rational cubic hypersurfaces in C such that theassociated quadric fibration has no section, see Remark 3.6 for details. Another countableunion of divisors in C parametrizing rational cubic fourfolds has been recently constructedin [AHTVA16], by showing that these cubics are birational to fibrations of del Pezzo sexticsurfaces that admit a section. More recently, the second and third named author proved thata general cubic fourfold in C and in C is rational. This confirms, for these two divisors,the expectations of Kuznetsov conjecture on the rationality of cubic fourfolds (see [RS17] formore details). Up to now, the general members of these countable loci in C and C togetherwith the general elements of C , C and C are the only known examples of rational cubicfourfolds.So far, a direct proof of the rationality of all the elements of C was not known. Moreover, C ∩ C = ∅ and C intersects also many other divisors C d for which the general member isnot known to be rational. The geometric description of the divisor C allows us to extend theknown rationality of a general element of C to each element of the family using generalizedone apparent double point surfaces, dubbed OADP surfaces in the following (see Section 1for precise definitions). In fact, the mere existence of an OADP surface inside a cubic 4-foldimplies its rationality. One of the main results of this paper is the following. Theorem.
All the cubic 4-folds contained in the irreducible divisor C are rational. The way to the proof of the previous Theorem requires a complete understanding of thedetails of some results proved by Fano in [Fan43], of which we give new and modern formula-tion (and proofs). Some of these results have been frequently cited and used in the literaturebut a modern detailed account does not seem to have appeared elsewhere. More precisely,they rely on the study of the rational map defined by quadrics vanishing on a smooth quarticrational normal scroll, on its restriction to a cubic hypersurface containing the scroll and onthe family of quartic rational normal scrolls contained in a cubic fourfold in C . In particular,we are able to prove the next result. Theorem 2.7.
Let X be a cubic fourfold contained in C \C and let T be the Hilbert schemeof quartic rational normal scrolls contained in X . Then dim( T ) = 2 and each point of T correspond to a smooth quartic rational normal scroll contained in X . Then we consider the intersection of C with C . The general associated cubic fourfoldscontain a plane and (the class of) a smooth quartic scroll. By the class of a smooth quarticscroll, we mean a 2-cycle in A ( X ) with the same intersection theoretical properties as asmooth quartic scroll. The components of C ∩ C had already been described in [ABBVA14]but our approach here is more direct and different. In fact, without relying on the arithmeticof the intersection lattices involved, we exhibit explicit examples of cubics contained in thecomponents of the intersection C ∩ C and study the degenerations of such cubics. Inparticular, we check which components intersect the Pfaffian locus P f of the moduli space, A long time after this paper was posted on arXiv, Kontsevich and Tschinkel [KT17] showed that for asmooth proper family over a smooth connected curve the rationality of the generic fiber implies the rationalityof all the fibers, that is rationality specializes.
OME LOCI OF RATIONAL CUBIC FOURFOLDS 3 and we prove that the Pfaffian locus is not open in C . These results are summarized in thenext theorem.
Theorem 3.4.
There are five irreducible components of C ∩C . The components are indexedby the value P · T ∈ {− , , , , } , where P ⊂ X is a plane and T the class of a small OADPsurface such that T = 10 and T · h = 4 . Voisin proved in [Voi86] that, for an arbitrary cubic fourfold X ⊂ P , every class P ∈ H , ( X, Z ) = H ( X, Z ) ∩ H (Ω X ) with P · h = 1 and P = 3 is represented by a plane in X .Theorem 2.9 and Theorem 3.4 yield the following analogous statement. Corollary 3.5.
Let X ∈ C and let T ∈ H , ( X, Z ) with T · h = 4 and T = 10 . Then T isrepresented by a small OADP surface contained in X . Generically, the cycle T above is a smooth quartic rational normal scroll with the exceptionof the component where P · T = −
1, where the general element does not contain any smoothquartic rational normal scroll nor any smooth quintic del Pezzo surface and T is the unionof two quadric surfaces intersecting along a line. This yields the following result. Theorem 3.7.
The set P f ⊂ C is not open in C . Analogously, the set of smooth cubicfourfolds containing a smooth quartic rational normal scroll is not open in C . Description of the contents.
In Section 1 we develop some technical results thatwill be used in the rest of the paper. In particular, we explain the relation between certainvarieties defined by quadratic equations and the OADP condition.In Section 2 we reconstruct, state in modern terms and prove Fano’s deformation argumentand then we show the rationality of every element of C \ C . In Section 3 we describe thecomponents of C ∩ C , prove the rationality of every cubic in this set, discuss their geometryand analyze their intersections with the Pfaffian locus. Finally, we give a quick proof of thenon-openness of the Pfaffian locus.The paper ends with a section containing some examples of cubic hypersurfaces in C ∩ C crucial for the proof of the non-openness of the Pfaffian locus. Acknowledgements . We have received support from the Research Network ProgramGDRE-GRIFGA, by PRIN
Geometria delle Variet`a Algebriche and by the Labex LEBESGUE.We would like to thank: A. Auel (a comment of whom inspired Thm. 3.7), M. Bernardara,C. Ciliberto, B. Hassett, A. Kuznetsov, D. Markushevich and H. Nuer for stimulating con-versations and exchange of ideas. We heartfully thank the referees for their careful readingand for many suggestions, which lead to a significant improvement of the exposition.1.
Preliminary results
Small varieties and varieties with one apparent double point.
We begin byrecalling a characterization of 2–regular reduced schemes from [EGHP06].
Definition 1.1.
A non-degenerate scheme X ⊂ P N is 2– regular in the sense of Castelnuovo–Mumford if its homogeneous ideal I ( X ) is generated by quadratic equations and if I ( X ) hasa linear resolution. In particular the first syzygies of I ( X ) are generated by the linear ones. M. BOLOGNESI, F. RUSSO, AND G. STAGLIAN `O
Examples of 2-regular irreducible varieties are non-degenerate irreducible varieties X ⊂ P N of minimal degree deg( X ) = codim( X ) + 1, which are also characterized by the previousalgebraic property. Definition 1.2.
A scheme X ⊂ P N is small if for every linear space P ⊂ P N such that Y = P ∩ X has finite length deg( Y ) we have dim( h Y i ) = deg( Y ) −
1, that is the deg( Y )points are linearly independent in P N . Definition 1.3.
Let X = X ∪ X ∪ .. ∪ X r ⊂ P N be a reduced scheme with X i irreducibleand with X i X j for every i and j . The sequence of schemes X , X , . . . , X r ⊂ P N is alinearly joined sequence if( X ∪ X ∪ ... ∪ X i ) ∩ X i +1 = h X ∪ X ∪ ... ∪ X i i ∩ h X i +1 i for every i = 1 , . . . , r −
1, where h Y i denotes the linear span of a scheme Y ⊂ P N .One should remark that the previous property depends on the order of the irreduciblecomponents, see [EGHP06, Example 0.5]. Theorem 1.4. [EGHP06, Thm. 0.4]
Let X ⊂ P N be a reduced scheme. Then the followingconditions are equivalent: (i) X is small; (ii) X is 2-regular; (iii) X is a linearly joined sequence of irreducible varieties of minimal degree. Let us recall that, given homogeneous forms f i of degree d i ≥ i = 0 , . . . M , a vectorof homogenous forms ( g , . . . , g M ) is a syzygy if P Mi =0 f i g i = 0. If d = · · · = d M = d and if deg( g i ) = h for every i = 0 , . . . , M , then we say that ( g , . . . , g M ) is a syzygy ofdegree h and for h = 1 we shall say that the syzygy is linear . For i < j the syzygies(0 , . . . , , f j , , . . . , , − f i , , . . . f i f j + f j ( − f i ) = 0 arecalled Koszul syzygies . We say that the Koszul syzygies are generated by the linear ones ifthey belong to the submodule generated by the linear syzygies.Next we state a result of Vermeire, which applies to 2-regular schemes, but also for exampleto quintic del Pezzo surfaces in P . Proposition 1.5. ([Ver01, Proposition 2.8])
Let f , . . . , f M be homogeneous forms in N + 1 variables of degree d ≥ such that the Koszul syzygies are generated by the linear ones. Thenthe closure of each fiber of the rational map φ = ( f : . . . : f M ) : P N P M is a linear space P s , which for s > intersects scheme theoretically the base locus scheme of φ along a hypersurface of degree d . Let us introduce the following important definition.
Definition 1.6.
Let X be an equidimensional reduced scheme in P n +1 of dimension n .The scheme X is called a (generalized) variety with one apparent double point , briefly OADPvariety , if through a general point of P n +1 there passes a unique secant line to X , that is aunique line cutting X scheme-theoretically in a reduced length two scheme. OME LOCI OF RATIONAL CUBIC FOURFOLDS 5
The name
OADP variety is usually reserved for the irreducible reduced schemes satisfyingthe previous condition and it comes from the fact that the projection of X from a generalpoint into P n acquires a singular point p , which is double . In fact, the singular point p arisesby collapsing two distinct points q , q collinear with the center of projection and the tangentcone at p is the union of the projections of the tangent spaces at q and q so that it consistsof two P n ’s intersecting at p .Let G ( k, n ) denote the Grassmannian of k -dimensional linear subspaces in P n . The abstractsecant variety S X of a variety X ⊂ P n +1 is the restriction of the universal family of G (1 , n +1) to the closure of the image of the rational map which associates to a pair of distinct pointsof X × X the line spanned by them. If X is an OADP variety, by definition the tautologicalmorphism p : S X → P n +1 is birational so that, by Zariski’s Main Theorem, the locus ofpoints of P n +1 through which there passes more than one secant line has codimension atleast two in P n +1 .The upshot is that any cubic hypersurface in P n +1 containing an OADP variety is bira-tional to the symmetric product X (2) if X is irreducible, or to the product of two irreduciblecomponents of X if it is reducible (see e.g. [Rus00]). By definition, the secant variety to thevariety X ⊂ P n +1 is SX := p ( S X ) ⊆ P n +1 .We define the join S ( X, Y ) of two reduced schemes X = X ∪ . . . ∪ X r ⊂ P M and Y = Y ∪ . . . ∪ Y s ⊂ P M , with each X i and Y j irreducible for every i = 1 , . . . , r and for every j = 1 , . . . , s , by first defining the join of two irreducible components as S ( X i , Y j ) = [ x = y, x ∈ X i , y ∈ Y j h x, y i ⊆ P M , and finally letting S ( X, Y ) = [ i,j S ( X i , Y j ) ⊆ P M . Clearly dim( S ( X i , Y j )) ≤ min { dim( X i ) + dim( Y j ) + 1 , M } . Moreover, with these definitionswe have that SX = S ( X, X ).Let us state an interesting consequence of the two previous results and of the definition ofgeneralized OADP variety.
Corollary 1.7.
Let X ⊂ P N be a non degenerate reduced algebraic set scheme-theoreticallydefined by quadratic forms such that their Koszul syzygies are generated by linear syzygies. Ifthrough a general point of P N there passes a positive finite number of secant lines to X , then X ⊂ P N is a generalized OADP variety.In particular a small algebraic set X ⊂ P N such that through a general point of P N therepasses a positive finite number of secant lines to X is a generalized OADP variety.Proof. Let f , . . . , f M be the quadratic forms defining X and let φ : P N P M be theassociated rational map. By Proposition 1.5 the closure of the fiber of φ passing through ageneral point p ∈ P N is a positive dimensional linear space L p containing all the secant linesto X passing through p ( φ contracts these secant lines to the point φ ( p )). Then L p ∩ X isa quadric hypersurface in L p by Proposition 1.5. Moreover, dim( L p ) = 1 because otherwisethrough p would pass infinitely many secant lines to the positive dimensional quadric L p ∩ X M. BOLOGNESI, F. RUSSO, AND G. STAGLIAN `O and a fortiori to X , contrary to our assumption. In conclusion L p is the unique secant lineto X passing through p . (cid:3) We recall the next result for future reference.
Proposition 1.8. ([Ful98, Proposition 9.1.1, third formula])
Let X ⊂ P be a smooth cu-bic hypersurface and let S , S ⊂ X be two smooth surfaces such that the scheme-theoreticintersection S ∩ S contains a smooth curve C of degree d and genus g . Then: (1) mult C ( S · S ) = 3 d + K S · C + K S · C + 2 − g, where K S i denotes the canonical class of S i and mult C ( S · S ) the multiplicity of intersectionof S and S along C . Cubic hypersurfaces in C Smooth quartic rational normal scrolls in P and the linear system of quadrichypersurfaces through them. Let C be the moduli space of smooth cubic hypersurfacesin P , which is a quasi-projective variety of dimension 20. For generalities on this space see[Has00].Let us recall from [Has00, Sect. 4] that C ⊆ C is defined as the locus of smooth cubichypersurfaces X ⊂ P containing a 2-dimensional algebraic cycle T such that T = 10 and T · h = 4, where h is the cycle of a hyperplane section of X . The locus C is easily seento be equal to the closure of smooth cubic hypersurfaces in P containing a smooth rationalnormal scroll of degree 4. In fact, if T ⊂ X is a smooth quartic rational normal scroll, then T = 10 by the self-intersection formula and T · h = deg( T ) = 4.The rationality of cubic hypersurfaces in P often depends on the fact that they containan OADP surface, irreducible or reducible. Indeed in this case (see [Rus00, Sect. 5] foran extended discussion of the details), the cubic hypersurface is birational to the symmetricproduct of an irreducible OADP surface or to the ordinary product of two distinct irreduciblecomponents of an OADP surface, whose secant join fills the whole space. Examples of surfaceswith one apparent double point are: smooth quintic del Pezzo surfaces; smooth quarticrational normal scrolls and more generally small varieties whose secant variety (or join ) fillsthe whole space; the union of two disjoint planes. A generalization of OADP surfaces hasbeen considered recently in [RS17], providing a new geometric insight to rationality of cubicfourfolds.There are two types of smooth quartic rational normal scroll surfaces: S (2 , F = P × P , and S (1 , F . The first type isthe most general one and it depends on 29 parameters while the second type depends on 28parameters. The application of Proposition 1.5 to a smooth quartic rational normal scrollyields the following result, which is quite well known. Proposition 2.1.
Let T ⊂ P be a smooth quartic rational normal scroll and let ψ : P P be the rational map defined by the linear system | H ( I T (2)) | . Then: a) the closure of the image Q = ψ ( P ) ⊂ P is a smooth quadric hypersurface; OME LOCI OF RATIONAL CUBIC FOURFOLDS 7 b) the closure of a general fiber of ψ is a secant line to T ; c) the closure of a fiber of dimension greater than one is a plane cutting T along a conic. An explicit birational representation of a smooth cubic hypersurface containing a smoothquartic rational normal scroll T has been described by Fano in [Fan43] and by Tregub in[Tre93].In the desire of being as self-contained as possible, we will now provide a short and completeproof of Fano’s result. It is important to point out that here we consider any smooth cubichypersurface containing a smooth quartic rational normal scroll, as in [AR04, Theorem 4.3]. Theorem 2.2. ([Fan43], [AR04, Theorem 4.3])
Let the notation be as in Proposition 2.1.Let X ⊂ P be a smooth cubic hypersurface containing a smooth quartic rational normalscroll T , and let e ψ : Bl T X → Q be the morphism induced by restricting ψ to X . Then e ψ is a birational morphism onto a smooth quadric hypersurface Q ⊂ P such that the followingproperties hold. a) The [closures of the] positive dimensional fibers of the restriction of ψ to X are eithersecant (or tangent) lines to T contained in X or (at most a finite number of ) planescutting T in a conic. b) the inverse map e ψ − : Q Bl T X is not defined along an irreducible surface S ′ T ,whose singular points are the images of the planes cutting T in a conic and containedin X . In particular, the cubic hypersurface X contains a two dimensional family ofsecant lines to T and S ′ T has at most a finite number of singular points. c) If X does not contain any plane cutting T in a conic, then S ′ T ⊂ P is a smoothsurface of degree 10 and sectional genus 7, which is the projection from a tangentplane of a smooth K3 surface S T ⊂ P of degree 14 and sectional genus 8. Thesurface S ′ T is isomorphic to the Hilbert scheme of secant lines to T contained in X . The conic C T ⊂ S ′ T , image of the exceptional divisor on the blow-up of S T viatangential projection, is also the image via ψ of the secant lines to T lying in theplanes cutting T in a conic.Proof. The scroll T is the base locus of the rational map ψ : P P , and the general secantline to T is not contained in X and cuts X in one point outside T . Thus the restriction of ψ to X is birational and e ψ is a birational morphism. Proposition 2.1 implies that the positivedimensional fibers of the restriction of ψ to X are exactly as in a).Let S ′ T ⊂ Q be the fundamental locus of e ψ − and let E = e ψ − ( S ′ T ) ⊂ Bl T X . Since Q is smooth, E is a divisor in Bl T X and it is irreducible by [ESB89, Proposition 1.3] becauseBl T X has Picard group isomorphic to Z ⊕ Z . Then S ′ T = e ψ ( E ) ⊂ Q is irreducible.Since X contains at most a finite number of planes, the general positive dimensional fiberof e ψ has dimension one by part a) and ( S ′ T ) red is an irreducible surface. Let Z = { q ∈ S ′ T : dim( e ψ − ( q )) ≥ } ( S ′ T and let V = Q \ Z . Then e ψ − ( V ) → V is a projective birational morphism between smoothvarieties such that each positive dimensional fiber has dimension at most one. By a result of M. BOLOGNESI, F. RUSSO, AND G. STAGLIAN `O
Danilov, see [Da81], e ψ − ( V ) → V is the blow-up of V along the smooth surface S ′ T \ Z sothat the base locus scheme of e ψ − is smooth outside Z . By a straightforward adaptation of[ESB89, Proposition 2.1 b)] we deduce that Sing( S ′ T ) = Z is at most a finite set in bijectionwith the planes cutting T in a conic and contained in X , proving b).Let Σ T ⊂ P be the the union of the planes cutting T along a conic. Since these conicsvary in a pencil, the three dimensional variety Σ T is a rational normal scroll of degree three.Then either Σ T is a Segre 3-fold P × P ⊂ P (if T ≃ S (2 , T is a cone over a twistedcubic with vertex a line (if T ≃ S (1 , T ). Let Π be a plane meeting T in a (possibly reducible) conic D . If Π is not contained in X , then Π ∩ X consists of the conic D plus a line L , which is thus secant to T . These linesdescribe a rational scroll of degree five Z T ⊂ X , linked to T via X inside Σ T . We claim thatthe image of Σ T (and hence of Z T ) via ψ is a conic C T ⊂ S ′ T . We shall prove the claim forΣ T ≃ P × P , the remaining case being similar. The restriction of ψ to Σ T ≃ P × P isgiven by a linear system in | H ( O (2 , | , having T as base locus scheme and hence as a fixedcomponent. Since T is a divisor of type (0 ,
2) inside P × P , the restriction of ψ is givenby the complete linear system | H ( O (2 , | , proving the claim (see also Corollary 2.5 for adifferent geometrical incarnation of the conic C T ).Under the hypothesis of c) one immediately deduces from b) that S ′ T is a smooth irreduciblesurface and that the restriction of e ψ : Bl T X → Q to E = e ψ − ( S ′ T ) → S ′ T is a P -bundle over S ′ T whose image in X , let us say M , is the locus of secant lines to T contained in X . Fromthis it follows that S ′ T is isomorphic to the Hilbert scheme of secant lines to T containedin X . For the geometrical description of S ′ T as the tangential projection of S T we refer to[Fan43] or to [AR04, Theorem 4.3], where it is also proved that M is a divisor in |O X (5) | having triple points along T . (cid:3) Singular quadrics through a smooth quartic rational normal scroll.
In thisparagraph we describe the geometry of quadric hypersurfaces through a quartic rationalnormal scroll. First we give a synthetic description of how families of quadrics of givenrank are constructed and then we collect in a proposition the description of these singularquadrics. Finally we use this to study secant lines of rational normal scrolls contained in acubic fourfold.2.2.1.
Rank 4 quadrics.
Let T ⊂ P be a smooth quartic rational normal scroll. The projec-tion of T from a proper secant line L , not lying on a plane cutting T in a conic, is a smoothquadric surface Q ⊂ P and the join S ( L, Q ) ⊂ P is a rank 4 quadric through T . By varying L we get a four dimensional family ∆ of rank 4 quadrics through T .Let ˆ Q be a rank four quadric through T and let Vert( ˆ Q ) = L be a line. The projection ofˆ Q from L is a smooth quadric Q ⊂ P . Then the projection of T from L is also Q . Thereforeeither L ⊂ T is a line of the ruling or length( L ∩ T ) = 2 (otherwise the degree of the projectionof T from L would be 4, if L ∩ T = ∅ ; or 3, if length( L ∩ T ) = 1). So L is a secant or atangent line to T , not contained in a plane cutting T along a conic (otherwise Q would besingular). Recall that the only conics living inside S (1 ,
3) are the directrix union a fiber.
OME LOCI OF RATIONAL CUBIC FOURFOLDS 9
Rank 3 quadrics.
We have seen in Sect. 2.2.1 that, starting from a proper secant line L , we obtain a rank 4 quadric S ( L, Q ) through T . When L degenerates to a tangent line to T , including the lines contained in T , the projection from L remains smooth. The projectionfrom a secant line L ′ to T contained in a plane cutting T in a conic C is a rank three quadric Q ′ ⊂ P . In the degeneration of L to L ′ , the rank four quadric surface S ( L, Q ) degeneratesinto a rank three quadric S ( L ′ , Q ′ ), whose vertex is the plane spanned by the conic C . Lemma 2.3.
The vertex of every rank 3 quadric ˆ Q through T cuts T along a conic, whichfor T = S (1 , is reducible.Proof. The projection of T from the vertex of ˆ Q , Vert( ˆ Q ), is a conic ˆ C so that every lineof the ruling of T cuts the vertex of ˆ Q because it cannot dominate ˆ C . Then the points ofintersection of the lines of the ruling with Vert( ˆ Q ) describe a curve D ⊂ Vert( ˆ Q ) of degreeat most 2, which is a section of the ruling of T . If D is a conic, then T = S (2 , D isa line, then T = S (1 ,
3) and there exist a twisted cubic D ′ disjoint from D , which is also asection of the ruling. Since D ′ projects onto a conic, D ′ cuts Vert( ˆ Q ) in a point q D . Theline L q of the ruling of T passing through q cuts D in a point q ′ . Then L q ⊂ Vert( ˆ Q ) because q, q ′ ∈ Vert( ˆ Q ) and D ∪ L q is a conic contained in Vert( ˆ Q ). (cid:3) An explicit and straightforward computation gives the description of all the singularquadric hypersurfaces containing T , summarized in the following result. Proposition 2.4.
Let T ⊂ P be a smooth quartic rational normal scroll. The locus ofsingular quadric hypersurfaces through T is a degree 6 hypersurface ∆ ⊂ P = | H ( I T (2)) | ,supported on the union of two quadric hypersurfaces ∆ , ∆ ⊂ P . The quadric hypersurface ∆ is smooth and it occurs with multiplicity 2 in ∆ while the quadric ∆ has rank 3 and itsvertex P is the plane defining the cubic rational normal scroll Σ T ⊂ P determined by thepencil of planes cutting T along conics.The locus of quadrics of rank less than or equal to 4 consists of ∆ ∪ P while the locus ofquadrics of rank 3 is a conic Ω ⊂ ∆ ∩ ∆ . Note that, if the scroll is S (1 , , then P ⊂ ∆ . Putting together Theorem 2.2 and Proposition 2.4 we obtain a different geometrical de-scription of the surface S ′ T parametrizing secant lines to T contained in a smooth cubicfourfold X ⊂ P through T . Corollary 2.5.
Let notation be as in Proposition 2.4 and Theorem 2.2. Let X ⊂ P bea smooth cubic hypersurface containing a smooth quartic rational normal scroll T and notcontaining a plane cutting T in a conic.Then the closure of the locus of quadrics of rank four containing T and whose vertex isa line contained in X is a smooth surface ˆ S T ⊂ ∆ , isomorphic to the surface S ′ T ⊂ P .Moreover, the image of the conic C T ⊂ S ′ T under this isomorphism is the conic Ω ⊂ ∆ ∩ ∆ . In the sequel we shall use the next quite striking result, which was claimed without proofin [Fan43].
Lemma 2.6. ([Fan43, bottom of page 75/top of page 77])
Let T , T ⊂ P be two smoothquartic rational normal scrolls intersecting in a 0–dimensional scheme of length 10. Thenthere exists a unique quadric hypersurface W ⊂ P containing T ∪ T , whose vertex is either aline L secant to T and to T or a plane Π intersecting each T i along a conic C i with C = C If T and T are contained in a cubic hypersurface X ⊂ P , then L ⊂ X , respectively Π ⊂ X .Proof. We will first show that there exists a unique quadric hypersurface in P through T ∪ T .Then some computations will exclude the higher rank cases.Let Y = T ∩ T , which by hypothesis is a 0–dimensional scheme of length 10. We cansuppose that T ≃ P × P is embedded in P by | H ( O T (1 , | , that is T ≃ S (2 ,
2) (thecase T ≃ S (1 ,
3) is similar and left to the reader). The quadric hypersurfaces defining T restrict to divisors in | H ( O T (2 , | and we have the short exact sequence:0 → Tor P ( O T , O T ) → I T , P | T → I Y,T → , where Tor P ( O T , O T ) is supported on Y . From Tor P i +1 ( I T , P , O T ) = Tor P i ( O T , O T ) = 0for every i ≥
1, from the 2-regularity of T ⊂ P and from the previous exact sequence, itfollows that h ( I Y,T (2 , Y imposes independent conditions to | H ( O T (2 , | (see also [EHP03, Lemma 1.1] for a similar argument). Therefore we have:5 = h ( O T (2 , − deg( Y ) = h ( I Y,T (2 , ≥ h ( I T , P (2)) − h ( I T ∪ T , P (2)) , yielding(2) h ( I T ∪ T , P (2)) ≥ . Let ψ = ψ : P Q ⊂ P be the rational map associated to T , defined in Proposition2.1, and let ϕ be the restriction of ψ to T . The map ϕ is given by a linear system | D | ⊆| H ( O T (2 , | having the length 10 base locus scheme Y = T ∩ T . From D = 16 −
10 = 6,we infer that S = ϕ ( T ) ⊂ Q ⊂ P is an irreducible surface such that 6 = deg( ϕ ) · deg( S ).Proposition 2.1 implies deg( ϕ ) ≤
2, yielding deg( S ) ∈ { , } . The surface S is degeneratein P by (2) ( ψ induces a one-to-one correspondence between the quadrics vanishing on T ∪ T and the hyperplanes containing S ). Let M = h S i ( P with 3 ≤ dim( M ) ≤
4. Since S ⊂ Q ⊂ P and since deg( S ) ∈ { , } , we deduce dim( M ) = 4 and h ( I T ∪ T , P (2)) = 1.Let W ⊂ P be the unique quadric hypersurface containing T ∪ T . Let Q ′ ⊂ P be asmooth quadric hypersurface and let Z ⊂ Q ′ be a smooth surface. Then, letting h be theclass of a hyperplane section on Z , the self-intersection formula for Z on Q ′ yields Z = 7 h + 4 h · K Z + 2 K Z − χ ( O Z ) . Suppose W were smooth. The previous formula implies that T i = 8 as cycles inside W .From H ( W, Z ) = Z α ⊕ Z β with α = 1 = β and α · β = 0, from deg( T i ) = 4 and from T i = 8, we get T i = 2 α + 2 β . This would imply T · T = 8, contrary to our assumption.Thus W ⊂ P is not of maximal rank. A computation as in [EHP03, Proposition 2.2] showsthat W cannot be of rank 5. Therefore W ⊂ P is of rank 3 or 4.Let us suppose first rank( W )=4, let L = Vert( W ) and let W := S ( L, Q ) with Q a smoothquadric surface. Then L is a secant line to T and to T , see Section 2.2.1, and the scheme OME LOCI OF RATIONAL CUBIC FOURFOLDS 11 T ∪ T intersects L in a scheme of length at least four. If T ∪ T is contained in a cubichypersurface X , then the multiplicity of intersection of L with X is at least four and L iscontained in X .Suppose rank( W )=3 and let W = S (Π , C ), i.e. W is the quadric obtained as a cone whosevertex is the plane Π ⊂ P and whose base is a smooth conic C ⊂ h C i = P ⊂ P suchthat Π ∩ h C i = ∅ . Then Π ∩ T i is a conic C i ⊂ T i by Lemma 2.3 and we have C = C because T ∩ T is zero dimensional. If T ∪ T is contained in a cubic hypersurface X , then C ∪ C ⊂ Π ∩ X yields Π ⊂ X . (cid:3) Fano’s construction revisited and rationality of cubics in C \ C . Let T bethe Hilbert scheme of (degenerations of) smooth quartic rational normal scrolls containedin a general X ∈ C . In order to calculate the dimension of C we need to estimate thedimension of T . Let us recall that for any smooth quartic rational normal scroll T ⊂ P wehave dim( | H ( I T (3)) | ) = 27. The Hilbert scheme H parametrizing smooth quartic rationalnormal scrolls in P , is irreducible, generically smooth and it has dimension 29. Hence(3) dim( T ) = dim( H ) + dim( | H ( I T (3)) | ) − dim( C ) ≥ − dim( |O P (3) | ) = 1and codim( C , C ) = dim( T ) − We shall immediately prove, that dim( T ) = 2 for a general X ∈ C without appealing to abstract deformation theory of T inside X .We now come to one of the gems in Fano’s paper [Fan43, pages 75–76]. As far as we knowthis geometrical construction has not been yet translated into modern geometrical languagedespite the great interest that this example has generated over the decades. Let us remarkthat, obviously, Fano did not state the next result in this form. Theorem 2.7.
Let X ∈ C , let T ⊂ X be a smooth quartic rational normal scroll, let Σ T ⊂ P be the rational normal scroll given by the pencil of planes spanned by the conicscontained in T and let T be the closure of the family of smooth quartic rational normal scrollscontained in X ( i.e. the Hilbert scheme of smooth quartic rational normal scrolls containedin X ).Then: a) dim( T ) = 2 ; b) there exists a unique irreducible 2-dimensional component ˜ S T ⊆ T containing T ,which is birational to the Hilbert scheme of secant lines to T contained in X ;Proof. By (3) we know that dim( T ) ≥ X ∈ C . Let L ⊂ X be a proper secant lineto T , not belonging to the scroll Z T , residual to T in Σ T ∩ X (see the proof of Theorem 2.2for the definitions). By projecting T from L , we deduce that T ⊂ W = S ( L, ˆ Q ) ⊂ P withˆ Q = π L ( T ) ⊂ P a smooth quadric surface. Let Λ i = P , i = 1 ,
2, be the parameter spaces ofthe two ruling of lines contained in ˆ Q .Then, letting P λ := S ( L, L λ ) , λ ∈ Λ , L λ ⊂ ˆ Q ; Actually the fact that codim( C ) ≥ P µ := S ( L, L µ ) , µ ∈ Λ , L µ ⊂ ˆ Q, we can define two pencils of cubic surfaces F λ = P λ ∩ X, λ ∈ Λ and G µ = P µ ∩ X, µ ∈ Λ . Modulo a renumbering, we can also suppose that, for general λ ∈ Λ and for general µ ∈ Λ , we have that P λ ∩ T = L ′ λ ⊂ T is a line of the ruling of T and that P µ ∩ T = C µ ⊂ T is a twisted cubic curve having L has a secant line.Let e L µ ⊂ G µ be the unique line contained in the smooth cubic surface G µ which is skewwith L and with C µ . Let us set e T L := [ µ ∈ Λ e L µ ⊂ X ∩ W. Then e T L ⊂ X is a rational scroll such that π L ( e L µ ) = π L ( C µ ) = L µ is a line for µ general.By varying the line L ⊂ X secant to T , we can construct a two dimensional family of suchsurfaces, whose general member is a rational scroll. Among the secant lines of T containedin X , there exists a one dimensional family describing the quintic rational scroll Z T ⊂ X ,consisting of secant lines to T contained in a plane meeting T in a conic. Thus a general line L ′ of the ruling of Z T is such that the corresponding plane Π of Σ T is not contained in X . Bydegenerating a general secant line L to the secant line L ′ to T , the smooth quadric ˆ Q = π L ( T )degenerates to a rank three quadric surface ˆ Q ′ , whose vertex is the plane Π not contained in X . Equivalently, we are degenerating a general quadric corresponding to a general point inthe surface ˆ S T , defined in Corollary 2.5, to a quadric corresponding to a point in Ω ⊂ ˆ S T suchthat the vertex of the corresponding rank three quadric is not contained in X . Let C ′ ⊂ Πbe the unique conic such that C ′ ∪ L ′ = Π ∩ X . Then Π ∩ T = C ′ and the limits of the P µ ’scontains Π. The two rulings of the smooth quadric ˆ Q = π L ( T ) degenerate into the uniqueruling of ˆ Q ′ and the scroll e T L converges to T .Let us denote by ˜ S T ⊆ T the irreducible two dimensional family just constructed, consistingof two dimensional cycles algebraically equivalent to T . In particular, dim( T ) ≥
2. Moreover,for a general secant line L to T , the rational scroll e T L ⊂ X is a smooth quartic rationalnormal scroll such that C ′ λ = P λ ∩ e T L is a twisted cubic having L as a secant line and suchthat e L µ = P µ ∩ e T L is the line defined above. Thus e T L and T have opposite behavior withrespect to the intersection with the two pencils { P λ } λ ∈ Λ and { P µ } µ ∈ Λ .Let T ∈ T be a general element in an irreducible component T ′ of T to which T belongs.From 10 = T = T · T and by the generality of T we deduce that T and T intersect ina 0–dimensional scheme of length 10. By applying Lemma 2.6 to T and T , we concludethat T is obtained from T by the previous geometrical construction yielding T ′ = ˜ S T and OME LOCI OF RATIONAL CUBIC FOURFOLDS 13 dim( T ) = 2. Moreover, we also showed that ˜ S T is the unique irreducible component of T containing T , proving the first part of b).Let S ′ T be the surface defined in Theorem 2.2, which parametrizes via e ψ the secant linesto T contained in X .The cubic hypersurface X contains at most a finite number of planes, so for a general T ∈ ˜ S T the unique quadric hypersurface W T containing T ∪ T provided by Lemma 2.6has rank four. We can define a rational map α T : S ′ T ˜ S T , by associating to a generalsecant line to T contained in X the scroll T ∈ ˜ S T produced via Fano’s deformation argumentof Lemma 2.6. Again by Lemma 2.6 this map is birational and the inverse associates to ageneral T ∈ ˜ S T the unique vertex of the rank four quadric W T containing T ∪ T . (cid:3) Remark 2.8. If X does not contain any plane of Σ T , the surface S ′ T is smooth by Theorem2.2, part b). The description of Fano’s deformation shows that the secant lines to T containedin Σ T produce the same scroll T . Thus the conic C T ⊂ S ′ T is contracted to a point by α T .Under the previous hypothesis one can show that α T extends to a morphism, which is thethe blow-down of the conic C ⊂ S ′ T , and also that ˜ S T is isomorphic to a smooth K K S as above for a generic Pfaffian cubic in [AL15,Section 2] via linear algebra, expanding the details of the construction drafted by Beauvilleand Donagi in [BD85, Remarques (1)]. For such a generic Pfaffian cubic they prove thateach member of the family is an irreducible small surface by exhibiting an explicit resolutionfor each member of the family, see [AL15, Section 2], and that a general member of thefamily is a smooth quartic rational normal scroll. Below we shall generalize this fact to every X ∈ C \ C by showing that every surface corresponding to each point of the Hilbert scheme˜ S T is a smooth quartic rational normal scroll inside X .The next result, which is probably well known to the experts in the field, seems to havenot been explicitly stated and/or proved till now. As we shall see later in Theorem 3.7, thelocus of X ∈ C containing a quartic rational normal scroll is constructible but not open in C . Theorem 2.9.
Every X ∈ C \ ( C ∩C ) contains a smooth quartic rational normal scroll andhence it is rational. Moreover, the family of smooth quartic rational normal scrolls containedin such a X is an equidimensional projective surface.Proof. First we describe the Hilbert scheme of quartic rational normal scrolls via an incidencecorrespondence over the moduli space of cubics (
Step 1 ). Then, a degeneration argumentshows that under our hypothesis every [ T ] ∈ T is a smooth quartic rational normal scroll( Step 2 ). Step 1.
Let H be the irreducible component of the Hilbert scheme of P whose generalmember is a smooth quartic rational normal scroll. Every element [ T ] ∈ H has degree four,dimension 2 and Hilbert polynomial equal to 2 t + 3 t + 1, being a flat projective deformationof a smooth quartic rational normal scroll in P . Let e C ⊂ P = P ( H ( O P (3))) be the open subset parametrizing smooth cubic hypersur-faces in P . Let C = { ([ T ] , [ X ]) ∈ H × e C : T ⊂ X } , let π : C → H be the restriction to C of the first projection and let π : C → e C be therestriction to C of the second projection.Recall that dim( H ) = 29 and that for a general [ T ] ∈ H we have that π − ([ T ]) is an opensubset of P ( H ( I T (3))) = P . Thus C contains an irreducible component W of dimension56 dominating H . Moreover W is a closed subset of H × e C so that π ( W ) = C is a closedirreducible subset of e C of dimension 54. In fact, we have seen in Theorem 2.7 that the familyof quartic rational normal scrolls contained in a general [ X ] ∈ C is two dimensional.Thus the image of C in C is exactly C , which is irreducible–a well known fact–and adivisor in C , see also [Has00]. In particular, for every [ X ] ∈ C let T = π ( π − ([ X ])) ⊂ H . Step 2.
We claim that if [ X ] ∈ C \ ( C ∩ C ) , then every [ T ] ∈ T is a smooth quarticrational normal scroll. By definition of H we can suppose that there exists a one dimensional family { T λ } λ ∈ C ,[ T λ ] ∈ H , where C is a smooth analytic curve (small disk), such that 0 ∈ C and T λ is asmooth quartic rational normal scroll for every λ ∈ C \
0. Thus each irreducible componentof T has dimension two and it is covered by lines, the limits of the lines of the ruling of T λ .Since X does not contains planes (and hence quadric surfaces) by hypothesis, we deduce thatevery [ T ] ∈ T is irreducible and generically reduced.Let us use the same notation as above. Fix a general point p ∈ P and let l λ , λ = 0, bethe unique secant line to T λ ⊂ P passing through p . A limit line l intersects T ⊂ P alonga scheme whose length is at least two. In particular T ⊂ P is a non-degenerate scheme bythe generality of p .If T is reduced, then T ⊂ P is an irreducible non-degenerate surface of degree four havinga secant/tangent line passing through a general point p ∈ P . Thus, either T is a smoothquartic rational normal scroll or it is a cone over a quartic rational normal curve (in the lastcase the limit line l would necessarily be the line through p and the vertex of the cone). Thesurface T cannot be a cone because in this case the tangent space to T at its vertex wouldbe P , forcing the singularity of X at the vertex of T . Thus, if T is reduced, then it is asmooth quartic rational normal scroll, as claimed.Suppose T is not reduced. The scheme T is irreducible and generically reduced and hasHilbert polynomial equal to 2 t + 3 t + 1 so that R = ( T ) red is an irreducible surface ofdegree four in P , which is degenerated. Otherwise R would be a surface of minimal degreein P which has Hilbert polynomial 2 t + 3 t + 1 and it would coincide with T , which is nonreduced by hypothesis. Moreover, R is covered by lines, which are limits of the lines covering T λ . From this it follows that R is an external projection of a quartic rational normal scroll S ⊂ P .If S were a cone over a smooth quartic rational normal curve, then T would be nonreduced only at the point p ∈ R which is the image of the vertex of S . The limit secantline l through a general point p ∈ P introduced above would be necessarily tangent to T at p . The generality of p would yield that the tangent space to T at p is the whole space P ,showing that such a T cannot be contained in X . If S ⊂ P were a smooth quartic rational OME LOCI OF RATIONAL CUBIC FOURFOLDS 15 normal scroll, then, by Proposition 1.5, R is either a general projection of S or the externalprojection of S from a point on the scroll Σ generated by the planes spanned by the pencilof conics on S . In the first case, there is a unique non reduced point p ∈ T supportedat the unique singular point of R . Proceeding as above, we deduce that T p T = P andhence T cannot be contained in X . If R is the projection of S from a point of Σ, then T has embedded points along the line L = Sing( R ). The tangent space at each point of L has dimension four and intersects P = h R i along a P . If such a T were contained in X ,then the intersection X ∩ h R i would be a cubic hypersurface in h R i , containing R and nonsingular along L . This is impossible, as shown for example by a direct calculation, so that T is necessarily reduced.In conclusion, the family of smooth quartic rational normal scrolls contained in X ∈ C \C is not-empty and proper so that it coincides with T . (cid:3) Remark 2.10.
Under the hypothesis of Theorem 2.9, every irreducible component of T isa smooth K S ⊂ P which is a general linear section of G (1 , ⊂ P . From ourperspective the surface S is obtained in this way: we fix a T ⊂ X and construct the surface S ′ T , which is smooth by Theorem 2.2 and which parametrizes secant and tangent lines to T contained in X ; by contracting the conic C T ⊂ S ′ T to a point one obtains the surface S .Let S [2] denote the Hilbert scheme of length two subschemes of the smooth irreducibleprojective surface S . Then S [2] is a smooth irreducible projective variety of dimension 4.One can also describe S [2] as the blow-up of the symmetric product S (2) along the image ofthe diagonal ∆ S ⊂ S × S , yielding a birational morphism ϕ : S [2] → S (2) . Let E := ϕ − (∆ S )and let ( p, p ) ∈ ∆ S . Then ϕ − (( p, p )) ≃ P ( t p S ), where t p S is the affine tangent space to S at p , i.e. E is the union of the exceptional divisors of the blow-up’s of S at each point p ∈ S .By definition of Hilbert scheme, each point p ∈ S corresponds to a unique smooth quarticrational normal scroll T p ⊂ X and, by Theorem 2.2, the Hilbert scheme of secant lines to T p contained in X is isomorphic to Bl p S . Under this isomorphism, the secant lines to T p contained in Σ T p correspond to the exceptional divisor P ( t p S ) ≃ ϕ − (( p, p )).The Hilbert scheme F ( X ) of lines contained in X can be interpreted as the parameterspace of secant lines to the family of smooth rational normal scrolls contained in X in anatural way, yielding a different interpretation of the well known isomorphism with S [2] .In fact, we can consider S [2] as the Hilbert scheme parametrizing couples of smooth quarticrational normal scrolls contained in X . If a (general) point [( p , p )] ∈ S [2] corresponds totwo distinct T p , T p contained in X , we can associate to [( p , p )] the vertex of the uniquerank four quadric surface containing two distinct quartic rational normal scrolls T p , T p (seeLemma 2.6). This extends to a morphism from S [2] to the Hilbert scheme of lines containedin X in the following way. A length two subscheme T ∈ ϕ − (( p, p )) can be seen as a limitof [( T p , T p ′ )] ∈ S [2] with T p = T p ′ . Thus there exists a unique rank three quadric surfacedetermined by the degeneration of T p ′ to T p , see proof of Theorem 2.7. Then this plane cuts T p along a conic and a line L . By mapping T to L one gets a morphism φ : S [2] → F ( X ),which is indeed an isomorphism.This construction in some sense generalizes the same isomorphism obtained in [BD85] forPfaffian cubics. While the proof of Beauville and Donagi is based on a linear algebra argumentand on some explicit geometry of the Grassmannian G (2 , X and of their secant lines. Irreducible components of C ∩ C and the Pfaffian locus in C Let A ( X ) = H , ( X, Z ) = H ( X, Z ) ∩ H (Ω X )denote the lattice of algebraic 2-cycles on the cubic fourfold X ⊂ P up to rational equivalenceand let d X be the discriminant of the intersection form on A ( X ) . Let β ∈ Z , let S ⊂ P be a smooth quintic del Pezzo surface and let P ⊂ P be a plane.Let C ′ β = { [ X ] ∈ C ∩ C : X ⊃ S ∪ P with S · P = β } ⊂ C ∩ C . Let h be the class of a smooth cubic surface W ⊂ X , intersection of X with a general P ⊂ P and let T = 3 h − S ∈ A ( X ). Since h = 3, h · S = 5 and S = 13, we get T · h = 4, T = 10 and T · P = 3 − β .Let τ ∈ Z , let T ⊂ P be a smooth quartic rational normal scroll and let P ⊂ P be aplane. Let C ′′ τ = { [ X ] ∈ C ∩ C : X ⊃ T ∪ P with T · P = τ } ⊂ C ∩ C . Proposition 3.1.
The set C ′ β , respectively C ′′ τ , defined here above is not empty if and only if β , respectively τ , belongs to { , , , } .Proof. If β <
0, then the cycle P · S has to contain a curve. Since S is defined by quadraticequations the scheme-theoretic intersection P ∩ S is either a line L ⊂ S or a conic C ⊂ S .From K S · L = − K S · C = −
2, we deduce from formula (1) that mult L ( P · S ) = 1,respectively mult C ( P · S ) = 0, contrary to our assumption. The argument for C ′′ τ with negative τ is identical and will be omitted.The surfaces T and S are scheme-theoretically defined by quadratic equations whose firstsyzygies are generated by the linear ones. If P ∩ T , respectively P ∩ S is 0-dimensional, then τ ≤
3, respectively β ≤
3, by the last part of [EGHP05, Theorem 1.1]. If P ∩ T , respectively P ∩ S , contains a curve then τ , respectively β , belongs to { , , , } by the above argument,concluding the proof of the only if part. One may also prove these facts as in [ABBVA14,Theorem 4], via a different argument using lattice theoretic methods and Riemann bilinearrelations.Example 4.1 proves the if part. (cid:3) Remark 3.2.
The closure of the locus of smooth cubic hypersurfaces in P containing a pairof skew planes is irreducible, has codimension 2, see [Tre93], and it will be indicated by e C − .For a general [ X ] ∈ e C − we have rk( A ( X )) = 3 and d X = 21. If T ⊂ X were a smoothquartic rational normal scroll, respectively if S ⊂ X were a smooth quintic del Pezzo surface,then an easy direct computation shows that d X = 21 would imply [ X ] ∈ C ′′− , respectively[ X ] ∈ C ′ , which is impossible by Proposition 3.1. This remark due to Tregub in [Tre93] hassome striking consequences on the topological properties of the Pfaffian locus, see Theorem3.7 below.A general [ X ] ∈ e C − contains cycles T with T · h = 4 and T = 10. In particular, such a X contains the reducible small surface consisting of the union of two general quadrics eachone residual to one of the two skew planes contained in X . A general [ X ] ∈ e C − contains OME LOCI OF RATIONAL CUBIC FOURFOLDS 17 also cycles S with S · h = 5 and S = 13. In particular e C − is an irreducible component of C ∩ C . Example 3.3. ( C ′ β = ∅ for β ∈ { , , , } ) By a direct computation one shows that thereexists a smooth cubic hypersurface X ⊂ P containing a smooth quintic del Pezzo surface S and a plane P such that S ∩ P is a scheme of length β consisting exactly of β reduced points(see Example 4.1). Let P f ⊂ C be the subset of Pfaffian cubics, that is cubic hypersurfacesin P admitting an equation given by the Pfaffian of a 6 × P f ⊂ C consists exactly of cubicfourfolds containing a smooth quintic del Pezzo surface. Moreover by [Bea00, Proposition9.2, part (ii) ] the closure of P f in C is irreducible of dimension 19 and hence it coincideswith C . In particular P f is dense in C .We are now in position to give an alternative, geometrical and self-contained proof of themain result of [ABBVA14]. Moreover, we shall also show that every element in C ∩ C isrational, a fact claimed only for the general element of some components in [ABBVA14]. Thisresult, together with Theorem 2.9, will prove that every element in C is rational.In fact, the proof of the rationality of the generic cubic contained in the component with d X = 32 proposed in [ABBVA14] relied on the openness of the locus of Pfaffian cubics insidethe divisor C . Since Theorem 3.7 will show that the Pfaffian locus is not open, then we alsofill in this gap and also simplify some arguments in [ABBVA14]. Theorem 3.4.
The codimension two locus C ∩C has five irreducible components. The cubichypersurfaces contained in each component contain a small OADP surface of degree four andhence they are rational. The components are indexed by the value P · T ∈ {− , , , , } ,where P ⊂ X is a plane and T the class of a small OADP surface such that T = 10 and T · h = 4 . The proof is based on a degeneration argument that shows the following claim: everypoint in the Hilbert scheme of quartic rational normal scrolls contained in a fixed X as inthe statement corresponds either to a smooth rational normal scroll or to a small reducibleOADP surface. In order to prove this claim, we go through a case by case analysis. Finally, weshall compute the irreducible components by lattice-theoretic arguments and by showcasingexplicit examples (contained in Sect. 4). Proof.
The notation will be as in the proof of Theorem 2.9. Let X ⊂ P be such that[ X ] ∈ C ∩ C and recall that H is the irreducible component of the Hilbert scheme of P whose general member is a smooth quartic rational normal scroll. Let T ⊂ H be the Hilbertscheme parametrizing schemes T ⊂ P with Hilbert polynomial p ( t ) = 2 t + 3 t + 1 whichare contained in X . Since a general cubic hypersurface in C contains a two dimensionalfamily of smooth quartic rational normal scrolls, we deduce that dim( T ) ≥ T ] ∈ T is a projective flat degeneration of a smooth quartic rationalnormal scroll. In particular, each irreducible component of T of dimension two is coveredby lines which are the limits of the lines covering a general element in H , which is a smoothquartic rational normal scroll. Let us remark that a priori, for some particular choice of X ,every element in T might be reducible, see for example Remark 3.2. By repeating the sameargument via the limit secant line we used in the proof of Theorem 2.9, we can conclude that any T ∈ T is a non-degenerate scheme in P such that through a general point of P therepasses a secant/tangent line to T . Claim: a general element T ∈ T is either a smooth rational normal scroll or a smallreducible OADP surface Suppose that T is a reduced scheme. Then T is a non-degenerate reduced surface ofdegree four. If T is irreducible, then it is a smooth quartic rational normal scroll, as shownin the proof of Theorem 2.9. If T is not irreducible, then it is the union of surfaces of degreelower or equal to three, all covered by lines. Thus T can be the union of planes, quadricsurfaces and rational scrolls of degree three. Since the Hilbert polynomial of a hyperplanesection is 4 t + 1, a general hyperplane section of T is the union of irreducible rational curvessuch that two irreducible components intersect at a point. Therefore the intersection oftwo irreducible components of T occurs along a line. Since T is also non-degenerate, theintersection of two irreducible components equals the intersection of the corresponding linearspans (otherwise h T i ( P ). Thus T ⊂ P is a linearly joined sequence of surfaces ofminimal degree and hence a small surface by Theorem 1.4. Then T is also an OADP surfaceby Corollary 1.7 since through a general point of P there passes a secant/tangent line to T . Assume T ⊂ P is a non-reduced non-degenerate scheme. If T is irreducible,then the same argument as in the proof of Theorem 2.9 shows that T cannot be genericallyreduced. Thus ( T ) red would be a (possibly reducible) quadric surface Q and T would definethe cycle 2 Q inside X . The scheme T can be obtained as a flat projective deformation ofa one dimensional family { T λ } λ ∈ C ⊂ H of small OADP surfaces consisting of two quadricsurfaces Q ′ λ , Q ′′ λ intersecting along a line. Moreover, we can also suppose that Q = Q ′ λ forevery λ ∈ C . In particular, for every λ = 0 the surface T λ is contained in smooth cubichypersurfaces belonging to C . Since X ∈ C we have π − ( T ) = ∅ so that X ∈ W = π ( π − ( C )) ⊂ C , where the π i are the projections as in Thm 2.9. Then we can suppose that X = X is the limit of a flat family { X µ } µ ∈ D with X µ ∈ W for every µ = 0. Let us point outthat X µ ∈ e C − for every µ = 0 since, by construction, a general element of W contains a cycleof the form Q ′ λ + Q ′′ λ for some λ ∈ C \
0. Therefore there exist cycles T µ = Q ′ µ + Q ′′ µ = Q + Q ′′ µ inside X µ such that rk( h h , T µ i ) = 2 and such that rk( h h , Q , Q ′′ µ i ) = 3 for every µ = 0.Moreover, the discriminant of h h , T µ i is equal to 14 for every µ = 0. If P ⊂ X = X isthe unique plane such that h = P + Q , then rk( h h , P , T µ i ) = rk( h h , Q , Q ′′ µ i ) = 3 forevery µ = 0. Then T = 2 Q ∈ h h , P i is in contrast with the semicontinuity of the rankof A ( X µ ) over C (or with the fact that C is closed in C ). This proves that π − ( T ) = ∅ ,that is: every cubic hypersurface containing a reduced T as above is singular. This fact canbe also verified by a long computation, which shows that a general element in π − ( T ) hasthree singular points. In an analogous way, one can prove that a non-reduced T must begenerically reduced along each irreducible component of ( T ) red . From now on we can suppose that T is a non-reduced, generically reducedscheme having at least two irreducible components, which are either planes orquadric surfaces or cubic scrolls. If one of its components is a cubic scroll, then thereis only another irreducible component which is necessarily a plane. From this it follows that( T ) red would be a linear projection of a small OADP surface consisting of a cubic rational OME LOCI OF RATIONAL CUBIC FOURFOLDS 19 normal scroll and a plane. Then the same argument used in Theorem 2.9 shows that T wouldhave an embedded point p at the acquired intersection of the two irreducible componentswith T p T = P and X would be singular.Therefore we can assume that each irreducible component of T is either a plane or anirreducible quadric surface. Then, once again, it is not difficult to see that T is necessarily alinear projection of a small OADP surface with embedded points at the acquired intersectionsof the irreducible components of ( T ) red , yielding the singularity of each cubic hypersurfacecontaining T .In conclusion, each cubic hypersurface in C ∩ C contains a small OADP surface T ⊂ X such that T · h = 4 and T = 10, as claimed. Description of the irreducible components: let P ⊂ X be a plane such thatrk( h h , T, P i ) = 3 and let τ = P · T .If T is irreducible, then T is a smooth quartic rational normal scroll so that 0 ≤ τ ≤ T = S ∪ P ′ with S ⊂ X a cubic rationalnormal scroll and P ′ ⊂ X a plane we can take P = P ′ . Since P ′ · S = 0 by (1), we deduce P · T = P = 3. If every irreducible component of T has degree less than or equal to 2, then X contains a pair of skew planes and one easily deduces τ = −
1. In conclusion − ≤ τ ≤ A τ the lattice of rank 3 generated by h h , S, P i with τ = P · S as above. We shallindicate by D τ ⊂ C ∩ C the locus of smooth cubic fourfolds such that there is a primitiveembedding A τ ⊂ A ( X ) of lattices preserving h . For − ≤ τ ≤ D τ is a nonemptysubvariety by Example 4.1 and it is of pure codimension 2 in C by a variant of the proofof [Has00, Thm. 3.1.2]. The argument at the end of the proof of [ABBVA14, Theorem 4]assures that for a general X ∈ D τ we have A ( X ) = A τ and that each codimension 2 locus D τ is irreducible, showing that C ∩ C has five irreducible components. (cid:3) Voisin proved in [Voi86] that, for an arbitrary cubic fourfold X ⊂ P , every class P ∈ H , ( X, Z ) with P · h = 1 and P = 3 is represented by a unique plane in X . Theorem 2.9and Theorem 3.4 yield the following analogous result. Corollary 3.5.
Let X ∈ C and let T ∈ H , ( X, Z ) with T · h = 4 and T = 10 . Then T isrepresented by a small OADP surface contained in X . Remark 3.6.
To every X ∈ C one associates a rational fibration in quadric surfaces inducedby the projection of X from a plane P ⊂ X onto a skew plane. If this fibration admits arational section, then X is rational.Let X be a general cubic in one of the five irreducible components of C ∩ C and let d X be the discriminant of X . In [ABBVA14, Proposition 5] it is proved that the natural quadricfibration associated to X admits a rational section if and only if τ is odd, that is if and only if P · T
6∈ { , } . For τ ∈ {− , , } a general cubic in the corresponding irreducible componentadmits a rational quadric fibration with a section and it is thus rational, see [ABBVA14].Since every element in C is rational, a general cubic hypersurface with P · T ∈ { , } isrational although the associated quadric fibration has no rational section. The case P · T = 0has been already observed in [ABBVA14], where an explicit example is also constructed. Theorem 3.4, the discussion in Example 3.3 and Example 4.1(e) below have essentiallyshown the next result, which is in contrast with the usual common sense according to whichthe Pfaffian locus should be open in C . Theorem 3.7.
The set P f ⊂ C is not open in C . Analogously, the set of smooth cubicfourfolds containing a smooth quartic rational normal scroll is not open in C .Proof. Let e C − = D − be the irreducible component of C ∩ C whose general element is asmooth cubic hypersurface X ⊂ P containing two skew planes P , P . Suppose that P f wereopen in C and consider its intersection P f ∩ e C − ⊂ C . By Example 4.1(e) we know that P f ∩ e C − = ∅ . Hence, if P f were open, then P f ∩ e C − would meet the dense subset of cubics X ∈ e C − such that rk( A ( X )) = 3, but as recalled in Remark 3.2 this is not possible.The conclusion in the second statement follows once again from the existence of a smoothcubic fourfold containing a quartic rational normal scroll and a pair of skew planes, seeExample 4.1, and by the fact that a general X ∈ e C − does not contain a smooth quarticrational normal scroll, see Remark 3.2 . (cid:3) Remark 3.8.
We recall that the Pfaffian locus P f and the set of cubics 4-folds containinga smooth quartic rational normal scrolls are images of quasi-projective varieties via suitablemorphisms (see [Bea00, Sect. 8-9], respectively the proof of Thm. 2.9). Thus, by ChevalleyTheorem, they are constructible and in particular they contain an open non-empty subset of C . In fact, C is the closure of both these two open sets. If these open subsets intersect anirreducible component of C ∩ C , then the general element of this component is Pfaffian,respectively contains a smooth quartic rational normal scroll. Since these open sets are purelytheoretical, with no precise handy description, it is hard to verify whether they intersectan irreducible component or not. Thus the statement that a general element of D τ , τ ∈{ , , , } , is Pfaffian requires a quite delicate analysis and cannot be deduced by simplyexhibiting a Pfaffian cubic in D τ , cfr. [ABBVA14, Section 4]. The examples constructedin Example 4.1 show that P f ∩ D τ = ∅ for every admissible τ and that the intersection isnon-empty also for the set consisting of cubic four-folds containing a smooth quartic rationalnormal scroll.4. Some cubic fourfolds containing smooth del Pezzo quintics
We shall give explicit examples of smooth cubic hypersurfaces in P which contain a quinticdel Pezzo surface S ⊂ P and a plane intersecting S in either the empty scheme or a set of1 ≤ i ≤ S and two disjoint planes obtained as linear spans oftwo irreducible conics on S . All our computations have been done using Macaulay2 [GS16].4.1. A del Pezzo surface of degree 5 in P can be parametrized by the map associated to thelinear system of all cubic curves in P passing through four points in general position. Wechoose such a map f : P P , where the points are taken to be (1 , , , , , , , , f is defined by( t , t , t ) ( t t − t t t , t t t − t t , t t − t t , t t − t t t , t t − t t , t t t − t t ) . OME LOCI OF RATIONAL CUBIC FOURFOLDS 21 If x , . . . , x denote homogeneous coordinates on P , then the image S = f ( P ) ⊂ P is thedel Pezzo surface defined by the five quadratic forms: x x − x x , x x − x x − x x + x x , x x − x x ,x x − x x − x x + x x , x x − x x − x x + x x . On S there are five pencils of conics whose linear spans determine five Segre threefoldsΣ i ≃ P × P ⊂ P , i = 0 , . . . ,
4. These pencils come as images of pencils σ , . . . , σ on P under the parametrization f , where σ is the pencil of conics passing through the four basepoints of f , and σ , . . . , σ are the pencils of lines passing through one of these four points.From this one can explicitly determine Σ , . . . , Σ , and it turns out that their homogeneousideals are generated by the following quadratic forms:Σ : x x − x x , x x − x x , x x − x x ;Σ : x x − x x , x x − x x − x x + x x , x x − x x − x x + x x ;Σ : x x − x x , x x − x x − x x + x x , x x − x x + x x − x x ;Σ : x x − x x − x x + x x , x x − x x − x x + x x , x x − x x − x x + x x ;Σ : x x + x x − x x − x x − x x + x x , x x − x x − x x + x x ,x x − x x − x x + x x . Two generic conics belonging to the same pencil on S are irreducible and the two planesobtained as linear spans are disjoint. Two such conics in f ( σ ) are: C = V ( x + x , x + x , x + x , x x + x x − x x ) ,C = V ( x + 2 x , x + 2 x , x + 2 x , x x + x x − x x ) . We also fix three linearly independent points q , q , q ∈ S , and four planes Π , . . . , Π ⊂ P such that Π ∩ S = ∅ and Π i ∩ S = { q , . . . , q i } (scheme-theoretically), for i = 1 , ,
3. Explicitly,Π = V ( x + x , x + x + x , x − x ) , Π = V ( x − x + x , x + x − x , x − x ) , Π = V ( x − x , x − x , x − x + x ) , Π = V ( x − x + x , x − x , x − x + x ) , where q = f (1 , ,
0) = (0 , , , , , q = f (1 , ,
1) = (1 , , , , ,
0) and q = f (0 , ,
1) =(0 , , , , , P . All of themare basically obtained by choosing randomly cubic hypersurfaces containing the given sub-schemes, until we get one that is smooth. This approach works well due to the closedness ofthe discriminant locus in the space of cubic forms on P . Example 4.1.
The following five cubic forms F , . . . , F on P define smooth hypersurfacescontaining the quintic del Pezzo surface S ; moreover V ( F i ) contains Π i for i = 0 , . . . ,
3, and V ( F ) contains the two disjoint planes h C i and h C i .(a) F = x x − x x + x x x + x x − x x + x x − x x − x x + x x x − x x x − x x + x x x − x x + 2 x x x + x x − x x − x x + x x ;(b) F = x x − x x x + x x − x x + x x − x x x − x x x + x x − x x + 2 x x x − x x + x x x + x x x − x x − x x − x x + x x ; (c) F = x x x − x x − x x − x x + x x − x x + 2 x x + x x x + x x x + x x − x x + x x − x x x − x x x + 2 x x + x x + x x − x x ;(d) F = 2 x x − x x x + x x x − x x − x x x − x x + x x x − x x x + x x − x x + x x − x x + x x − x x x + 2 x x x + x x − x x x − x x + x x ;(e) F = − x x + x x x − x x + x x x − x x + x x x + x x − x x x − x x x − x x + 2 x x x − x x x + 3 x x x + x x x + 2 x x + x x x − x x x − x x .For every i, j = 0 , . . . ,
4, we have the decomposition Σ i ∩ V ( F j ) = S ∪ T i,j , where T i,j isa smooth rational normal scroll surface of degree 4 if ( i, j ) = (0 , T , is the smallvariety h C i ∪ h C i ∪ V ( x , x , x x − x x ). In particular, the smoothness of T , impliesthat there exist smooth cubic fourfolds containing a smooth quartic rational normal scrolland two disjoint planes. The ideal of T , is generated by the following six quadratic forms: x x − x x , x x + 3 x x − x , x x − x x − x x + x x ,x x − x x + x x + 3 x x − x , x + 3 x x − x x , x x − x x − x x + x x . Moreover, if P ⊂ X is a plane such that P · S = τ with 0 ≤ τ ≤
3, then from 3 h = S + T i,j we deduce P · T i,j = 3 − τ .4.2. Here we give some pieces of Macaulay2 code which have been used to produce andverify the examples above. The complete code can be found in the ancillary file cubics.m2 .We begin by starting
Macaulay2 and loading two further packages included with it.
Macaulay2, version 1.10with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,LLLBases, PrimaryDecomposition, ReesAlgebra, TangentConei1 : loadPackage "Cremona"; -- version 4.2i2 : loadPackage "Resultants"; -- version 1.1
We define a method which takes as input a projective scheme and returns a random smoothcubic hypersurface containing the scheme. If such a smooth hypersurface does not exist,the method goes in an infinite loop and does not produce any output. One of its possibleimplementations (that does not take care of the growth of the coefficients) is the following: i3 : randomCubic = (I) -> ( -- I must be a homogeneous ideal in a polynomial ringB := super basis(3,saturate I); C := 0;while (C == 0 or discriminant C == 0) do C = (B * random(QQ^(numcols B),QQ^1))_(0,0);return C);
The smoothness of the cubic hypersurface is checked through the computation of its discrim-inant. It is however standard to implement a general method which checks whether a givenclosed subscheme of a projective space is smooth and absolutely connected. Now we buildthe parametrization f of the quintic del Pezzo surface S . i4 : use Grass(0,2,Variable=>t);i5 : P = {ideal(t_1,t_2),ideal(t_0,t_2),ideal(t_0,t_1),ideal(t_1-t_0,t_2-t_0)};i6 : f = rationalMap(intersect P,3);o6 = RationalMap (cubic rational map from PP^2 to PP^5)i7 : S = image f; The pencils σ , . . . , σ , the conics C , C and the points q , q , q are obtained as follows: i8 : pencils = prepend(intersect P,P);i9 : conics = (f ideal(pencils_0_0 + pencils_0_1), f ideal(pencils_0_0 + 2*pencils_0_1));i10 : points = (f ideal(t_2,t_0-t_1), f ideal(t_1,t_0-t_2), f ideal(t_0,t_1-t_2)); OME LOCI OF RATIONAL CUBIC FOURFOLDS 23
One then easily verifies that our choice of the planes Π , . . . , Π is correct. To determinethe Segre threefolds Σ , . . . , Σ , an idea is just to add another variable and to compute the generic conic in each of the five pencils. We omit this code here, but it is available in theancillary file. Now we produce a cubic form F like F in Example 4.1. The other cases arequite similar. i11 : I = intersect(S,ideal super basis(1,conics_0),ideal super basis(1,conics_1));i12 : F = randomCubic I; Thus the residual intersections T i = Σ i ∩ V ( F ) \ S can be obtained as follows (here we areassuming that Sigma is the list of the ideals of Σ , . . . , Σ ). i13 : T = apply(5,i -> (Sigma_i + F):S); Finally, the following code gives us an explicit birational map from P to V ( F ). i14 : ((rationalMap S)|F)^-1;o14 = RationalMap (birational map from PP^4 to hypersurface in PP^5) References [ABBVA14] A. Auel, M. Bernardara, M. Bolognesi, and A. V´arilly-Alvarado,
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