New examples of rational Gushel-Mukai fourfolds

We construct new examples of rational Gushel-Mukai fourfolds, giving more evidence for the analogous of the Kuznetsov's Conjecture for cubic fourfolds: a Gushel-Mukai fourfold is rational if and only if it admits an associated K3 surface.


Introduction
A Gushel-Mukai fourfold is a prime Fano fourfold X ⊂ P 8 of degree 10 and index 2 (see [Muk89]). These fourfolds are parametrized by a coarse moduli space X 10 of dimension 24, and the general fourfold [X] ∈ X 10 is a smooth quadratic section of a smooth hyperplane section of the Grassmannian G(1, 4) ⊂ P 9 of lines in P 4 .
In [DIM15] (see also [DK18a,DK16,DK18b]), similarly to Hassett's analysis of cubic fourfolds (see [Has99,Has00]), the authors studied Gushel-Mukai fourfolds via Hodge theory and via the period map. In particular, they showed that inside X 10 there is a countable union d GM d of (not necessarily irreducible) hypersurfaces parametrizing special Gushel-Mukai fourfolds, that is fourfolds [X] ∈ X 10 that contain a surface S whose cohomology class does not come from the Grassmannian G(1, 4). The index d is called discriminant of the fourfold and it runs over all positive integers congruent to 0, 2, or 4 modulo 8 (see [DIM15]). However, as far as the authors know, the explicit geometric description (of at least some irreducible component) of GM d is unknown for d > 12. In Theorem 2.3 we shall provide a description of an irreducible component of Similarly to the case of cubic fourfolds, all the fourfolds [X] ∈ X 10 are unirational, some rational examples are classical and easy to construct, and no example has yet been proven to be irrational. Furthermore, there are values of the discriminant d such that a fourfold [X] ∈ GM d admits an associated K3 surface of degree d. For instance, this occurs for d = 10 and d = 20. The hypersurface GM 10 has (at least) two irreducible components, and the general fourfold in each of these two components is rational (see [DIM15,Propositions 7.4 and 7.7], and see also Examples 1.1 and 1.2). Some of these fourfolds was already studied by Roth in [Rot49]. As far as the authors know, there are no more examples of rational Gushel-Mukai fourfolds. In Theorem 2.4, we shall provide new examples of rational Gushel-Mukai fourfolds which belong to GM 20 .
Recall that a classical and still open question in algebraic geometry is the rationality of smooth cubic hypersurfaces in P 5 (cubic fourfolds for short). An important conjecture, known as Kuznetsov's Conjecture (see [AT14,Kuz10,Kuz16,Has16]) asserts that a cubic fourfold is rational if and only if it admits an associated K3 surface in the sense of Hassett/Kuznetsov. This condition can be expressed by saying that the rational cubic fourfolds are parametrized by a countable union d C d of irreducible hypersurfaces inside the 20-dimensional coarse moduli space of cubic fourfolds, where d runs over the so-called admissible values (the first ones are d = 14, 26, 38, 42, 62). The rationality for the cubic fourfolds in C 14 has been classically proved by Fano in [Fan43] (see also [BRS19]), while in [RS19a] it has been proved the rationality in the case of C 26 and C 38 . Very recently, in [RS19b] it has been also proved the rationality in the case of C 42 . We point out that the proof of this last result shows a close relationship between the cubic fourfolds in C 42 and the Gushel-Mukai fourfolds in GM 20 constructed in this paper. We would like to remark that this beautiful geometry was discovered with the help of Macaulay2 [GS19].

Generality on Gushel-Mukai fourfolds
In this section, we recall some general facts about the theory of Gushel-Mukai fourfolds which have been proved in [DIM15] (see also [DK18a,DK16,DK18b]).
A Gushel-Mukai fourfold X ⊂ P 8 , GM fourfold for short, is a degree-10 Fano fourfold with Pic(X) = Z O X (1) and K X ∈ |O X (−2)|. Equivalently, X is a quadratic section of a 5-dimensional linear section of the cone in P 10 over the Grassmannian G(1, 4) ⊂ P 9 of lines in P 4 . There are two types of GM fourfolds: • quadratic sections of hyperplane sections of G(1, 4) ⊂ P 9 (Mukai or ordinary fourfolds, [Muk89]); • double covers of G(1, 4) ∩ P 7 branched along its intersection with a quadric (Gushel fourfold, [Gus82]). There exists a 24-dimensional coarse moduli space X 10 of GM fourfolds, where the locus of Gushel fourfolds is of codimension 2. Moreover, we have a period map p : X 10 → D onto a 20-dimensional quasi-projective variety D, which is dominant with 4-dimensional smooth fibers (see [DIM15]).
For a very general GM fourfold [X] ∈ X 10 , the natural inclusion of the middle Hodge classes is an equality. A GM fourfold X is said to be special if the inequality (1.1) is strict. This means that the fourfold contains a surface whose cohomology class "does not come" from the Grassmannian G(1, 4). The special GM fourfolds are parametrized by a countable union of hypersurfaces d GM d ⊂ X 10 , labelled by the positive integers d ≡ 0, 2, or 4 In some cases, the value of d can be explicitly computed (see [DIM15,Section 7]). Indeed, let X ⊂ P 8 be an ordinary GM fourfold containing a smooth surface S such that [S] ∈ A(X) \ A (G(1, 4)). We may write [S] = aσ 3,1 + bσ 2,2 in terms of Schubert cycles in G(1, 4) for some integers a and b. We then have that For some values of d, the non-special cohomology of the GM fourfold [X] ∈ GM d looks like the primitive cohomology of a K3 surface. In this case, analogously to the case of cubic fourfolds, one says that X has an associated K3 surface. The first values of d that satisfy the condition for the existence of an associated K3 surface are: 2, 4, 10, 20, 26, 34. We refer to [DIM15, Section 6.2] for precise definitions and results.
In Examples 1.1 and 1.2 below, we recall all the known examples of rational GM fourfolds, which all have discriminant 10. In Section 2 we shall construct rational GM fourfolds of discriminant 20. 4) is a linear section of G(1, 3) ⊂ G(1, 4); its class is σ 2 1 · σ 1,1 = σ 3,1 + σ 2,2 . In [DIM15, Proposition 7.4] it has been proved that the closure D ′ 10 ⊂ X 10 of the family of fourfolds containing a τ -quadric surface is an irreducible component of p −1 (D ′ 10 ), and that the general member of D ′ 10 is rational. In [RS19b, Theorem 5.3] it has been given a different description of D ′ 10 and another proof of the rationality of its general member.

A special family of Gushel-Mukai fourfolds
Let S ⊂ P 8 be the image of P 2 via the linear system of quartic curves through three simple points and one double point in general position. Then S is a smooth surface of degree 9 and sectional genus 2 cut out in P 8 by 19 quadrics.
Moreover, there exists an irreducible component of the Hilbert scheme parameterizing such surfaces in Y which is generically smooth of dimension 25.
Proof. Using Macaulay2 [GS19] (see Section 3), we constructed a specific example of a surface S ⊂ P 8 as above which is embedded in a del Pezzo fivefold Y ⊂ P 8 and satisfies (2.1). Moreover we verified in our example that h 0 (N S,Y ) = 25 and h 1 (N S,Y ) = 0.
Remark 2.2. After the construction, in a preliminary version of this paper, of an explicit example of a surface as in Lemma 2.1, in [RS19b, Section 4] has been provided the explicit geometric description of an irreducible 25-dimensional family of these surfaces inside a del Pezzo fivefold, confirming the claim of Lemma 2.1.
Theorem 2.3. Inside X 10 , the closure D 20 of the family of GM fourfolds containing a surface S ⊂ P 8 as in Lemma 2.1 forms an irreducible hypersurface, which is contained in p −1 (D 20 ).
Proof. The component S of the Hilbert scheme parameterizing rational surfaces in P 8 of degree 9 and sectional genus 2 as above is generically smooth of dimension 80, and two such general surfaces are projectively equivalent. By Lemma 2.1, there exists a smooth del Pezzo fivefold Y = G(1, 4) ∩ P 8 ⊂ P 8 containing a 25-dimensional family F Y of surfaces S ∈ S satisfying (2.1) in the Chow ring of G(1, 4). By the rigidity of Y , we can conclude that this happens for the general del Pezzo fivefold Y ∈ DP, where DP denotes the Hilbert scheme parameterizing smooth del Pezzo fivefolds in P 8 . Let us consider the incidence correspondence: S DP be the two natural projections. By the previous analysis, we have that p 1 and p 2 are two surjective morphisms, and the general fiber of p 2 is irreducible of dimension 25. Moreover, since dim S = 80 and dim DP = 65, we deduce that the general fibre of p 1 has dimension 10. Let where GM denotes the Hilbert scheme parameterizing GM fourfolds X ⊂ P 8 , and let be the two natural projections. For a general S ∈ S, since dim p −1 1 (S) = 10 and the homogeneous ideal of S is generated by 19 quadrics, we deduce that dim π −1 1 (S) = 23 = 10 + 19 − 5 − 1, so that dim J = 103 = 80 + 23. In a specific example we verified that h 0 (N S,X ) = 0. Thus, by semicontinuity, it follows that π 2 is a generically finite morphism and that dim π 2 (J) = 103. In other words, since dim GM = 104, we have that the closure of the family of GM fourfolds containing a surface S ∈ S as in Lemma 2.1 forms an irreducible hypersurface in GM (as well as in X 10 ). Finally, we apply (1.2) and (1.3) to deduce that a general [X] ∈ π 2 (J) lies in p −1 (D 20 ).
Theorem 2.4. Every GM fourfold belonging to the family D 20 described in Theorem 2.3 is rational.
Proof. Let Y ⊂ P 8 be a del Pezzo fivefold, and let S ⊂ Y be a general rational surface of degree 9 and sectional genus 2 belonging to the 25-dimensional family described in Lemma 2.1 and Remark 2.2.
The restriction to Y of the linear system of cubic hypersurfaces with double points along S gives a dominant rational map ψ : Y P 4 , whose general fibre is an irreducible conic curve which intersects S at three points. Thus S admits inside Y a congruence of 3-secant conic curves. This implies that the restriction of ψ to a general GM fourfold X containing S and contained in Y is a birational map to P 4 . The existence of the congruence of 3-secant conics can be also verified in this way. Remark 2.5. The inverse map of the birational map ψ : X P 4 described in the proof of Theorem 2.4 is defined by the linear system of hypersurfaces of degree 9 having double points along an internal projection to P 4 of a smooth surface T ⊂ P 5 of degree 11 and sectional genus 6 cut out by 9 cubics. This surface T is an internal triple projection of a smooth minimal K3 surface of degree 20 and genus 11 in P 11 .
Actually, this has been the starting point for this work. In fact, from the results contained in [RS19b], it could be suspected that a triple internal projection of a minimal K3 surface of degree 20 and genus 11 could be related to a GM fourfold of discriminant 20.

Explicit computations
In the proof of Lemma 2.1, we claimed that there exists an example of a rational surface S ⊂ P 8 of degree 9 and sectional genus 2 which is also embedded in G(1, 4) and satisfies [S] = 6 σ 3,1 + 3 σ 2,2 . In an ancillary file, we provide the explicit homogeneous ideal of such a surface which contains the ideal generated by the Plücker relations of G(1, 4). The class [S] in terms of the Schubert cycles σ 3,1 , σ 2,2 can be easily calculated using, for instance, the Macaulay2 package SpecialFanoFourfolds.
In the following, we explain the main steps of the procedure we followed to construct the surface in G(1, 4). We start by taking a general nodal hyperplane section of a smooth Mukai threefold of degree 22 and sectional genus 12 in P 13 (see [MU83,Sch01]). The projection of this surface from its node yields a smooth K3 surface T ⊂ P 11 of degree 20 and sectional genus 11 which contains a conic (see [Kap18]). Then we take a general triple projection of T in P 5 , which is a smooth surface of degree 11 and sectional genus 8 (this follows from [Voi19, Proposition 4.1] and [FS19, Theorem 10] in the case when the K3 surface T is general). Let T ′ ⊂ P 4 be a general internal projection of this surface in P 5 . Then T ′ is a singular surface of degree 10 and sectional genus 8, cut out by 13 quintics. The linear system of hypersurfaces of degree 9 having double points along T ′ gives a birational map η : P 4 X ⊂ P 8 onto a GM fourfold X, whose inverse map is defined by the restriction to X of the linear system of cubic hypersurfaces having double points along a smooth surface S ⊂ X of degree 9 and sectional genus 2. Finally, to determine explicitly the surface S, one can exploit the fact that the generic quintic hypersurface corresponds via η to the generic quadric hypersurface (inside X) containing S. Indeed, behind the scenes, we have an occurrence of a flop, similar to the Trisecant Flop considered in [RS19b]. In particular, we have a commutative diagram where m 1 and m 2 are birational maps defined, respectively, by the linear system of quintics through T ′ and by the linear system of quadrics through S; M is a fourfold of degree 33 in P 12 cut out by 21 quadrics.