Remarks on the geometry of the variety of planes of a cubic fivefold

This note presents some properties of the variety of planes $F_2(X)\subset G(3,7)$ of a cubic $5$-fold $X\subset \mathbb P^6$. A cotangent bundle exact sequence is first derived from the remark made by Iliev and Manivel that $F_2(X)$ sits as a Lagrangian subvariety of the variety of lines of a cubic $4$-fold, which is a hyperplane section of $X$. Using the sequence, the Gauss map of $F_2(X)$ is then proven to be an embedding. The last section is devoted to the relation between the variety of osculating planes of a cubic $4$-fold and the variety of planes of the associated cyclic cubic $5$-fold.


Introduction
To understand the topology and the geometry of smooth complex hypersurfaces X ⊂ P(V * ) ≃ P n+1 , various auxiliary manifolds have been introduced in the past century, of which the intermediate Jacobian when n = 2k + 1 is odd is one of the most widely known since the seminal work of ) on the cubic 3-fold.
Cubic 5-folds are classically (cf.[Gri69]) known to be the only hypersurfaces of dimension greater than 3 for which the intermediate Jacobian, which is in general just a (polarised) complex torus, is a (non-trivial) principally polarised abelian variety.
Another interesting series of varieties classically associated to X are the varieties F m (X) ⊂ G(m + 1, V ) of m-planes contained in X.
Starting from Collino ([Col86]), some properties of the variety of planes F 2 (X) ⊂ G(3, V ) of a cubic 5-fold X have been studied in connection with the 21-dimensional intermediate Jacobian J 5 (X).In loc.cit., the following is proven.
Theorem 1.1.For a general cubic X ⊂ P(V * ) ≃ P 6 , F 2 (X) is a smooth irreducible surface, and the Abel-Jacobi map of the family of planes Φ P : F 2 (X) → J 5 (X) is an immersion; i.e., the associate tangent map is injective and induces an isomorphism of abelian varieties φ P : Alb(F 2 (X)) ∼ − − → J 5 (X), where P ∈ CH 5 (F 2 (X) × X) is the universal plane over F 2 (X).Equivalently, q * p * : H 3 (F 2 (X), Z) / torsion → H 5 (X, Z) is an isomorphism of Hodge structures, where the maps are defined by In the present note, we investigate some additional properties of F 2 (X).
In the first section, we establish the following cotangent bundle exact sequence.
Theorem 1.2.Let X ⊂ P(V * ) be a smooth cubic 5-fold for which F 2 (X) is a smooth irreducible surface.Then the cotangent bundle Ω F 2 (X) fits in the exact sequence where the tautological rank 3 quotient bundle E 3 and the other bundle appear in the exact sequence and the first map (of (1.1)) is the contraction with an equation eq X ∈ Sym 3 V * defining X, i.e. for any [P ] ∈ F 2 (X), v → eq X (v, •, •)| P .
In the second section of the note, we prove the following.
Theorem 1.3.The Albanese map is an embedding.In particular, the Gauss map is defined everywhere.Moreover, G is an embedding, and its composition with the Plücker embedding is the composition of the degree 3 Veronese of the natural embedding F 2 (X) ⊂ G(3, V ) ⊂ P( 3 V * ) followed by a linear projection.
The last section is concerned with some properties of the variety of osculating planes of a cubic 4-fold, namely where Z ⊂ P(H * ) ≃ P 5 is a smooth cubic 4-fold containing no plane.This variety admits a natural projection to the variety of lines F 1 (Z) of Z whose image (under that projection) has been studied, for example, in [GK21].The interest of the authors there for the variety F 0 (Z) stems from its image in F 1 (Z) being the fixed locus of the Voisin self-map of F 1 (Z) (see [Voi04]), a map that plays an important role in the understanding of algebraic cycles on the hyper-Kähler 4-fold F 1 (Z) (see for example [SV16]).
In [GK21], it is proven that for Z general, F 0 (Z) is a smooth irreducible surface, and some of its invariants are computed.
We compute some more invariants of F 0 (Z) using its link with the variety of planes F 2 (X Z ) of the associated cyclic cubic 5-fold: to a smooth cubic 4-fold Z = {eq Z = 0} ⊂ P 5 , one can associate the cubic 5-fold X Z = {X 3 6 + eq Z (X 0 , . . ., X 5 )} which (by linear projection) is the degree 3 cyclic cover of P 5 ramified over Z.
Theorem 1.4.For Z general, F 0 (Z) is a smooth irreducible surface, and Remark 1.5.As mentioned by the referee and Frank Gounelas, in [GK21], it is proven that , where Z ∩ H is a cubic 3-fold obtained as a general hyperplane section, which implies that [ℑ(F 0 (Z) → F 1 (Z))] is Lagrangian (see [Huy23,Lemma 6.4.5], for example).
I would like to thank Hsueh-Yung Lin for pointing me to the article [IM08] some years ago.I would like to also thank Pieter Belmans for explaining how to use Sage to decompose the tensor powers of E 3 into irreducible modules and the anonymous referee for their remarks.
Finally, I am grateful to the gracious Lord for His care.

Cotangent bundle exact sequence
Let X ⊂ P(V * ) ≃ P 6 be a smooth cubic 5-fold.Its variety of planes F 2 (X) ⊂ G(3, V ) is the zero locus of the section of Sym 3 E 3 (where E 3 is defined by (1.2)) induced by an equation eq X ∈ H 0 (O P 6 (3)) of X.
Let us gather some basic properties of F 2 (X) before proving Theorem 1.2.It is proven in [Col86, Proposition 1.8] that F 2 (X) is connected for any X, so that by Bertini-type theorems, for X general, F 2 (X) is a smooth irreducible surface.

and as
Associated to X, there is also its variety of lines F 1 (X) ⊂ G(2, V ).It is a smooth Fano variety of dimension 6 which is cut out by a regular section of Sym 3 E 2 , where E 2 is the tautological rank 2 quotient bundle appearing in an exact sequence

Let us examine the relation between the two auxiliary varieties by introducing the flag variety
On Fl(2, 3, V ), the relation between the two tautological bundles is given by the exact sequence We can restrict the flag bundle to get F 2 (X).
We have the following property.
Proposition 2.1.The tangent map T e F 2 of e F 2 is injective; i.e., e F 2 is an immersion.Moreover, the "normal bundle" N P F 2 /F 1 (X) := e * F 2 T F 1 (X) /T P F 2 of P F 2 admits the following description: (1) Let us first prove that e F 2 is an immersion.Let us recall the natural isomorphism between the two presentations of the tangent space of Fl(2, 3, V ): looking at t, we can write and looking at e, we have where we denote by ⟨K⟩ ⊂ V the linear subspace whose projectivisation is K ⊂ P(V * ).For a given decomposition ⟨P ⟩ ≃ ⟨ℓ⟩ ⊕ ⟨P ⟩/⟨ℓ⟩, the isomorphism takes the following form: Notice that, by definition, we have ℑ(f ) ∩ ℑ(g) = {0}, so that in proving that T ([ℓ],[P ]) e F 2 is injective, we can examine the two components separately.
The second summand is readily seen to inject into T F 1 (X),⟨ℓ⟩ by (2.4).Next, we have the exact sequence To prove that T ([ℓ],[P ]) e F 2 is injective, it is thus sufficient to prove that H 0 (N P /X (−1)) = 0. Consider the exact sequence Up to a projective transformation, we can assume P = {X 0 = • • • = X 3 = 0}, so that eq X has the following form: (2.7) where R is a homogeneous cubic polynomial, the D i , 4 ≤ i ≤ 6, are homogeneous quadratic polynomials in the variables (X k ) k≤3 and the Q i , 0 ≤ i ≤ 3, are homogeneous quadratic polynomials in (X i ) 4≤i≤6 .With this notation, X is smooth along P if and only if Span((Q i | P ) i=0,...,3 ) is base-point-free.We recall the following result found in [Col86, Proposition 1.2 and Corollary 1.4].
Proposition 2.2.For X smooth along P , the following properties are equivalent: (1) The variety F 2 (X) is smooth at [P ].
(2) We want now to establish the exact sequence (2.3).Pulling back the natural exact sequence of locally free sheaves, we get the commutative diagram which by the snake lemma yields By the definition of the normal bundle, we get coker(T e F 2 ) ≃ N P F 2 /F 1 (X) .The restriction of the exact sequence of locally free sheaves . The relative tangent bundle appears in the exact sequence: . Next, taking the symmetric power of (2.2) we get the exact sequence Putting everything together, we get the desired exact sequence.□ For any plane P 0 ⊂ X, looking for example at the associated quadric bundle where 31 of these quadrics (where L has to be thought of as a formal square root of O P 3 (1)).
In particular, the locus , there are finitely many lines that belong to another planes P ′ ⊂ X).
To any hyperplane H ⊂ P(V * ) such that Y := X ∩ H is a smooth cubic 4-fold containing no plane, we can attach the morphism j H : As a result, as noticed in [IM08, Proposition 7] (the published version corrects the preprint, in which it is wrongly claimed that j H is an embedding, as underlined in [Huy23]), j H : The following diagram is commutative: As Z H ⊂ P F 2 is the zero locus of a regular section of e * | Z H , so that the last vertical arrow in the diagram is an isomorphism.As the second vertical arrow is injective by Proposition 2.1, the first is injective as well.So the snake lemma gives In particular, outside a codimension 2 subset of F 2 (X), we have As both sheaves are locally free, the isomorphism holds globally; i.e., (2.9) We can now prove Theorem 1.2 Proof of Theorem 1.2.Looking at (2.9) and (2.3), we see that we only have to check that O e (1) To get the result, we will show more generally that ϕ * O Gr(2,⟨H⟩) (−1) ⊗ O Gr(3,V ) (1) restricts to the trivial line bundle on the open set where ϕ is defined, i.e., on Gr(3, V )\ Gr(3, ⟨H⟩).

Gauss map of F 2 (X)
Let X ⊂ P(V * ) ≃ P 6 be a smooth cubic hypersurface such that F 2 (X) is a smooth (irreducible) surface.We begin this section with the following.
Theorem 3.1.The following sequence is exact: where ϕ eq X is defined to be e i + e j → eq X (e i , e j , •).

Moreover, we have an inclusion
), which by Hodge symmetry yields Proof.As O F 2 (X) admits the Koszul resolution (2.1), to understand the cohomology groups H i (O F 2 (X) ), we can use the spectral sequence . As a reminder, we borrow from [Jia12] (see also [Spa03]) the following elementary presentation of the Borel-Weil-Bott theorem for a G(3, W ) with dim(W ) = d.
For any vector space L of dimension f and any decreasing sequence of integers a = (a 1 , . . ., a f ), there is an irreducible GL(L)-representation (Weyl module) denoted by Γ (a 1 ,...,a f ) L.
To Theorem 3.2.We have , where Q 3 and E 3 are defined by (1.2) and Γ ψ(a,b) W = 0 if ψ(a, b) is not decreasing.Now, we want to apply this theorem to compute the E p,q 1 of the spectral sequence.Using Sage with the code R=WeylCharacterRing("A2") V=R(1,0,0) for k in range(11): print k, V.symmetric_power(3).exterior_power(k) we get the decompositions into irreducible modules of ∧ k Sym 3 E * 3 .Then by the Borel-Weil-Bott theorem, we have To understand H 1 (O F 2 (X) ), we have to examine the E −i,i+1 ∞ for i = 0, . . ., 10.As E −i,i+1 1 = 0 for any i 3, we get E −i,i+1 ∞ = 0 for i 3. On the other hand, for r ≥ 2, E −3,4 r is defined as the (middle) cohomology of From the above computations, we see that ; i.e., the following sequence is exact: 1 is given by contracting with the section defined by eq X , so that, choosing a basis (e 0 , . . ., e 6 ) of V , we have (e i + e j ) ⊗ (e 0 ∧ • • • ∧ e 6 ) −→ k eq X (e i , e j , e k ) e k = eq X (e i , e j , If this map is not surjective, we can choose the basis so that e * 0 ⊗ (e 0 ∧ • • • ∧ e 6 ) ℑ(d −3,4

1
).Then we get eq X (e i , e j , e 0 ) = 0 for any i, j, which means that the cubic hypersurface X is a cone with vertex [e 0 ].
Proposition 3.3.We have and the following sequence is exact: ), we use again the Koszul resolution (2.1) tensored by Q * 3 .We have the spectral sequence . We again use the Borel-Weil-Bott theorem 3.2 to compute the cohomology groups on G(3, V ).The decompositions of the ∧ i Sym E * 3 's into irreducible modules have already been obtained in Theorem 3.1.So we get The graded pieces of the filtration on ) are given by E −i,i ∞ , i = 0, . . ., 10. From the above calculations, we see that Using Sage with the code R=WeylCharacterRing("A2") V=R(1,0,0) W=R(0,0,-1) for k in range(11): print k, W.symmetric_power(2)*V.symmetric_power(3).exterior_power(k) → and the Borel-Weil-Bott theorem 3.2, we get The graded pieces of the filtration on H 0 (Sym 2 E 3 | F 2 (X) ) are given by the E −i,i ∞ .We have E −i,i ∞ = 0 for any i 0, 4 since E −i,i 1 = 0 for i 0, 4. As E a,b r = 0 for any a > 0 and E −r,r−1 ) = 21 (see Theorem 3.1).So the exactness of the sequence ) and the surjectivity of the last map.□ | 2 H 0 (Ω F 2 (X) )| is the composition of the degree 3 Veronese of the natural embedding F 2 (X) ⊂ G(3, V ) followed by a linear projection.Moreover, we have the following.Lemma 3.4.
(1) The canonical bundle K F 2 (X) is generated by the sections in 2 H 0 (Ω F 2 (X) ) ⊂ H 0 (K F 2 (X) ).In particular, (2) For any [P ] ∈ F 2 (X), the following sequence is exact: where Proof.(1) As E 3 | F 2 (X) is globally generated (as a restriction of E 3 , which is globally generated, by (1.2)), is also globally generated.The same holds for Q * 3 | F 2 (X) (by (1.2)).So applying the evaluation to (3.3), we get the commutative diagram where the bottom row is (1.1).As ev 2 is surjective, we get that ev 3 is also surjective; i.e., Ω F 2 (X) is globally generated.Then taking the exterior square of ev 3 , we get that ∧ 2 ev 3 is surjective: Now a base point of | 2 H 0 (Ω F 2 (X) )| would be a point where ∧ 2 ev 3 fails to be surjective.So , so the snake lemma gives the exact sequence.□ Now, let us come back to the Gauss map of F 2 (X), that we have defined to be where alb It is defined on the smooth locus of alb F 2 (F 2 (X)).
According to [Col86, Section (III)], T alb F 2 is injective.So the indeterminacies of G are resolved by the pre-composition with alb F 2 , i.e., We have the Plücker embedding and the commutative diagram The following proposition completes the proof of Theorem 1.3.
Proposition 3.5.The morphism ρ is an embedding, which implies that alb F 2 is an isomorphism unto its image and G is an embedding.
Proof.Let us denote by J X the Jacobian ideal of X, i.e., the ideal of the polynomial ring generated by and by J X,2 its homogeneous part of degree 2. By Proposition 2.2, for any [P ] ∈ F 2 (X), dim(J X,2 | P ) = 4, so that dim(J X ∩ {Q ∈ H 0 (O P 6 (2)), P ⊂ {Q = 0}}) = 3.We have the following.
By definition, the quadrics of the Jacobian ideal are ∂ eq X ∂X i , and according to Proposition 2.2, ∂ eq X ∂X i | P i=0,...,3 are linearly independent, so that .
which, when restricted to P ′ , gives , ∂ eq X ∂X 6 ∈ L 2 P ,P ′ ∩ J X .For X general, those two quadric polynomials are independent.
Now, given a [P ] ∈ F 2 (X), we recall that (the first order of eq X (x + u(x), x + u(x), x + u(x)) is 0 for all x ∈ ⟨P ⟩).
Moreover, given a u ∈ T [P ] F 2 (X), we have □

Variety of osculating planes of a cubic 4-fold
In (1.3), we have previously introduced, for a smooth cubic 4-fold containing no plane Z ⊂ P(H * ) ≃ P 5 , the variety of osculating planes F 0 (Z) : The variety F 0 (Z) lives naturally in Fl(2, 3, H), i.e., and from the exact sequence (2.2): ) ∈ Fl(2, 3, H), the bundle of equations of ℓ ⊂ P .As a result, F 0 (Z) is the zero locus on Fl(2, 3, H) of a section of the rank 9 vector bundle F defined by the exact sequence In particular (since F is globally generated by the sections induced by H 0 (t * Sym 3 E 3 )), by Bertini-type theorems, for Z general, F 0 (Z) is a smooth surface with . Its link to the surface of planes of a cubic 5-fold is the following.
(1) The point p 0 does not belong to X Z .In particular, any [P ] ∈ F 2 (X Z ) is sent by π p 0 : P(V * ) P(H * ) to a plane in P(H * ), where V = H ⊕ C • p 0 .The restriction of π p 0 (also denoted by π p 0 ) to X is a degree 3 cyclic cover of P 5 ramified over Z.Let us denote by τ : [a 0 : with ξ a primitive third root of 1, the cover automorphism.
Conversely, for any ([ℓ], [P ]) ∈ F 0 (Z), π −1 p 0 | X Z (P ) → P is a degree 3 cyclic cover ramified over {ℓ} 3 , so it consists of three surfaces isomorphic each to P , i.e., three planes.To make it even more explicit, is defined in π −1 p 0 (P ) ≃ Span(P , p 0 ) ≃ P 3 by X 3 6 − aX 3 3 for some a 0 (since Z contains no plane), and we have (2) The equation eq Z defines a section σ eq Z ∈ H 0 (t * Sym 3 E 3 ) ≃ H 0 (Sym 3 E 3 ) and by projection in (4.1) a section σ eq Z of F whose zero locus is F 0 (Z).Restricting (4.1) to F 0 (Z), we see that σ eq Z induces a section of , we would be able to define three distinct sections of π : Moreover, we readily see that for any has a nowhere-vanishing section, hence is trivial. ( In [GK21], the interest for the image e(F 0 (Z)) ⊂ F 1 (Z) stems from e(F 0 (Z)) being the fixed locus of a rational self-map of the hyper-Kähler 4-fold F 1 (Z) defined by Voisin (cf.[Voi04]).Proposition 4.3.For Z general, the tangent map of e F 0 := e| F 0 (Z) : F 0 (Z) → F 1 (Z) is injective, and e F 0 is the normalisation of e F 0 (F 0 (Z)) and is an isomorphism unto its image outside a finite subset of F 0 (Z).
(1) That e F 0 is injective outside a finite number of points follows from a simple dimension count: let us introduce ) are isomorphic to each other and are sub-linear systems of |O P 5 (3)|.
Notice that, since F 0 (Z) is a surface for Z general, we know that dim(I) = dim(|O P 5 (3)| + 2.
Let us analyse the fiber of p 2 .To do so, we can assume where the Q i (X 0 , X 1 , X 2 ) are quadratic forms in X 0 , X 1 , X 2 , the D i are quadratic forms in X 3 , X 4 , X 5 and R is a cubic form in X 3 , X 4 , X 5 .Notice that this is the general form of a member of the fiber The additional condition Z ∩ P 2 = ℓ implies that Q 3 (X 0 , X 1 , 0) = 0, D 0 (X 3 , 0, 0) = 0, D 1 (X 3 , 0, 0) = 0, which gives 3+1+1 = 5 constraints.So dim(p such that there are at least two planes P 1 , P 2 ⊂ P 5 such that Z ∩ P i = ℓ, i = 1, 2, i.e., there is a finite set γ ⊂ F 0 (Z) such that e| F 0 : F 0 (Z)\γ → F 1 (Z) is a bijection unto its image.
As for Z general, e F 0 is an immersion, N F 0 (Z)/F 1 (Z) := e * F 0 T F 1 (Z) /T F 0 (Z) is locally free.Moreover, since e F 0 is, outside a codimension 2 subset of F 0 (Z), an isomorphism unto its image and that image is a Lagrangian subvariety of F 1 (Z), we get (outside a codimension 2 subset, thus globally) an isomorphism Ω F 0 (Z) ≃ N F 0 (Z)/F 1 (Z) .
Proof.We have seen that F 0 (Z) ⊂ Fl(2, 3, H) is the zero locus of a section of F appearing in the sequence (4.1).Taking the symmetric power of (2.2), we have the following commutative diagram with exact rows: / / e * Sym 2 E 2 / / 0.
The projection to e * Sym 2 E 2 of the section σ eq Z ∈ H 0 (t * Sym 3 E 3 ) induced by eq Z vanishes on F 1 (Z) by the definition of F 1 (Z).So it induces a section of / / e * F 0 T F 1 (Z) / / N F 0 (Z)/F 1 (Z) / / 0 and the description of the relative tangent bundle of e F 1 give the following.
Proposition 4.6.The following sequence is exact: We finish this section by computing the Hodge numbers of F 0 (Z).
Proof.For the universal variety of planes r univ : F 2 (X ) → |O As a consequence, the Abel-Jacobi isomorphism q * p * : H 3 (F 2 (X), Q) ∼ − → H 5 (X, Q) given by the result of Collino (Theorem 1.1) for general X extends to the case of the general cyclic cubic 5-fold.