Explicit rationality of some cubic fourfolds

The rationality of smooth cubic hypersurfaces in $\mathbb{P}^{5}$ is an open problem on which a lot of new and interesting contributions and conjectures appeared in the last decades. The classical work by Fano in [Fan43], correcting some wrong assertions in [Mor40], has been the only known result about the rationality of cubic fourfolds for a long time and, together with a great amount of recent theoretical work on the subject (see for example the survey [Has16]), lead to the expectation that the very general cubic fourfold should be irrational. More precisely, in the moduli space $\mathcal{C}$ the locus $\operatorname{Rat}(\mathcal{C})$ of rational cubic fourfolds is the union of a countable family of closed subsets $T_{i}\subseteq\mathcal{C}$, $i\in\mathbb{N}$, see [dFF13, Proposition 2.1] and [KT19, Theorem 1].

Hassett defined in [Has99, Has00] (see also [Has16]) via Hodge Theory infinitely many irreducible divisors $\mathcal{C}_{d}$ in $\mathcal{C}$ and introduced the notion of admissible values $d\in\mathbb{N}$, i.e. those even integers $d>6$ not divisible by 4, by 9 and nor by any odd prime of the form $2+3m$. More recent contributions by Kuznetzsov via derived categories in [Kuz10, Kuz16] (see also [AT14, Has16]) fortified the conjecture that

The first admissible values are $d=14,26,38,42$ and Fano showed the rationality of a general cubic fourfold in $\mathcal{C}_{14}$, see [Fan43, BRS19]. The main results of [RS19a] are summarised in the following:

The geometrical definition of $\mathcal{C}_{d}$ can be also given as the (closure of the) locus of cubic fourfolds $X\subset\mathbb{P}^{5}$ containing an explicit surface $S_{d}\subset X$. These surfaces are obviously not unique and a standard count of parameters shows that in specific examples the previous locus is a divisor (the degree and self-intesection of $S_{d}$ determine the value $d$ via the formula $d=3\cdot S^{2}-\deg(S)^{2}$), see [Has99, Has00] for more details on the Hodge theoretical definition of $\mathcal{C}_{d}$ and also [YY19] for recent interesting contributions on the divisors $\mathcal{C}_{d}$. For example, the divisor $\mathcal{C}_{14}$ can be described either as the closure of the locus of cubic fourfolds containing a smooth quintic del Pezzo surface or, equivalently, a smooth quartic rational normal scroll, see [Fan43, BRS19] and also [Nue15] for other descriptions with $12\leq d\leq 44$, $d\neq 42$ or [Lai17] for $d=42$.

Fano proved that the restriction of the linear system of quadrics through a smooth quintic del Pezzo surface, respectively a smooth quartic rational normal scroll, to a general cubic through the surface defines a birational map to $\mathbb{P}^{4}$, respectively onto a smooth four dimensional quadric hypersurface. Indeed, a general fiber of the map $\mathbb{P}^{5}\dasharrow\mathbb{P}^{4}$ given by quadrics through a quintic del Pezzo surface is a secant line to it, yielding the birationality of the restriction to $X$ (the other case is similar). The extension of this explicit geometrical approach to rationality for other (admissible) values $d$ appeared to be impossible because there are no other irreducible surfaces with one apparent double point contained in a cubic fourfold, see [CR11, BRS19].

In [RS19a] we discovered irreducible surfaces $S_{d}\subset\mathbb{P}^{5}$ admitting a four-dimensional family of 5-secant conics such that through a general point of $\mathbb{P}^{5}$ there passes a unique conic of the family (congruences of 5-secant conics to $S_{d}$) for $d$=14, 26 and 38. From this we deduced the rationality of a general cubic in $|H^{0}(\mathcal{I}_{S_{d}}(3))|$, showing that it is a rational section of the universal family of the congruence of 5-secant conics.

Here we come back to Fano’s method and we propose an explicit realisation of the previous abstract approach. Some simplifying hypotheses, based on the known examples in [RS19a], suggest that rationality might be related to linear systems of hypersurfaces of degree $3e-1$ having points of multiplicity $e\geq 1$ along a right surface $S_{d}$ contained in the general $X\in\mathcal{C}_{d}$, see Section 1. Clearly the problem is to find the right $S_{d}$, prove that the above map is birational and, if the dimension of the linear system is bigger than four, describe the image (which might be highly non trivial). This expectation is motivated by the remark, due to János Kollár, that if the surface $S_{d}\subset\mathbb{P}^{5}$ admits a congruence of $(3e-1)$-secant curves of degree $e\geq 1$ generically transversal to the cubics through $S_{d}$ (see Section 1 for precise definitions), then the above linear systems contract the curves of the congruence. So if the general fiber of the map is a curve of the congruence, then a general cubic through $S_{d}$ is birational to the image of the associated map. We shall see that, quite surprisingly, this really occurs for the first three admissible values $d=14,26,38$ and, even more surprisingly, that these linear systems provide by restriction explicit birational maps from a general cubic fourfold in $\mathcal{C}_{d}$ for $d=14,26,38$ to $\mathbb{P}^{4}$ (and also to a four dimensional linear section of a $\mathbb{G}(1,3)$, respectively $\mathbb{G}(1,4)$, respectively $\mathbb{G}(1,5)$), a fact which was not known before at least for $d=26,38$ (see also [Kol96, Section 5, §29]). The theoretical framework explaining the birationality of these maps, the relations with the theory of congruences to the surfaces $S_{d}$ and to the associated K3 surfaces has been developed recently in [RS19b].

To analyze the algebraic and geometric properties of the surfaces $S_{d}\subset\mathbb{P}^{5}$ involved as well the particular linear systems of hypersurfaces of degree $3e-1$ having points of multiplicity $e$ along the $S_{d}$’s we used Macaulay2 [GS19] together with some standard semicontinuity arguments to pass from a particular verification to the general case.

Acknowledgements. We wish to thank János Kollár for asking about the maps defined by the linear systems of quintics singular along surfaces admitting a congruence of 5-secant conics and for his interest in our subsequent results.

Let us recall the following definitions introduced in [RS19a, Section 1]. Let $\mathcal{H}$ be an irreducible proper family of (rational or of fixed arithmetic genus) curves of degree $e$ in $\mathbb{P}^{5}$ whose general element is irreducible. We have a diagram

where $\pi:\mathcal{D}\to\mathcal{H}$ is the universal family over $\mathcal{H}$ and where $\psi:\mathcal{D}\to\mathbb{P}^{5}$ is the tautological morphism. Suppose moreover that $\psi$ is birational and that a general member $[C]\in\mathcal{H}$ is ($re-1$)-secant to an irreducible surface $S\subset\mathbb{P}^{5}$, that is $C\cap S$ is a length $re-1$ scheme, $r\in\mathbb{N}$. We shall call such a family $\mathcal{H}$ a congruence of ($re-1$)-secant curves of degree $e$ to $S$. Let us remark that necessarily $\dim(\mathcal{H})=4$.

An irreducible hypersurface $X\in|H^{0}(\mathcal{I}_{S}(r))|$ is said to be transversal to the congruence $\mathcal{H}$ if the unique curve of the congruence passing through a general point $p\in X$ is not contained in $X$. A crucial result is the following.

Thus, when a surface $S\subset\mathbb{P}^{5}$ admits a conguence of $(3e-1)$-secant curves of degree $e\geq 1$ which is transversal to a general cubic fourfold through it, such a cubic fourfold is birational to $\mathcal{H}$ via $\pi\circ\psi^{-1}_{|X}$, see the proof of the above result. Obviously, the difficult key point is to find a congruence as above with $\mathcal{H}$ rational (or irrational, depending on the application).

Since $\psi:\mathcal{D}\to\mathbb{P}^{5}$ is birational, we also have a rational map

whose general fiber through $p\in\mathbb{P}^{5}$, $F=\overline{\varphi^{-1}(\varphi(p))}$, is the unique curve of the congruence passing through $p$. On the contrary, if there exists a map $\varphi:\mathbb{P}^{5}\dasharrow Y\subseteq\mathbb{P}^{N}$ with $Y$ a four dimensional variety, whose general fiber $F$ is an irreducible curve of degree $e$ which is $(3e-1)$-secant to a surface $S\subset\mathbb{P}^{5}$, then we have found a congruence for $S$, a birational realization of the variety $\mathcal{H}$ and a concrete representation of the abstract map $\pi\circ\psi^{-1}$.

It is natural to ask what linear systems on $\mathbb{P}^{5}$ can give maps $\varphi:\mathbb{P}^{5}\dasharrow Y$ as above. Since a linear system in $|H^{0}(\mathcal{I}_{S}^{e}(3e-1))|$ contracts the fibers of $\varphi$, these (complete) linear systems appear as natural potential candidates, as remarked by János Kollár. A posteriori we shall see that, quite surprisingly, this really occurs with $Y=\mathbb{P}^{4}$ (or with $Y$ a linear section of a Grassmannian of lines) for the first three admissible values $d=14,26,38$ and, even more surprisingly, that these linear systems provide the explicit rationality of a general cubic fourfold in $\mathcal{C}_{d}$ for $d=14,26,38$.

Let $E\subset\operatorname{Bl}_{S}\mathbb{P}^{5}$ be the exceptional divisor of the blow-up of $\mathbb{P}^{5}$ along $S$, let $H\subset\operatorname{Bl}_{S}\mathbb{P}^{5}$ be the pull back of a hyperplane in $\mathbb{P}^{5}$, let $F^{\prime}$ be the strict transform of $F$ and let $\tilde{\varphi}:\operatorname{Bl}_{S}\mathbb{P}^{5}\dasharrow\mathcal{H}$ be the rational map induced by $\varphi$. Then $E\cdot F^{\prime}=3e-1$ by hypothesis and $(3H-E)\cdot F^{\prime}=1$. The last condition translates both that a general cubic through $S$ is mapped birationally onto $\mathcal{H}$ by $\varphi$, both the fact that the linear system $|H^{0}(\mathcal{I}_{S}(3))|$ sends a general $F$ into a line contained in the image of the corresponding map $\psi:\mathbb{P}^{5}\dasharrow\mathbb{P}(H^{0}(\mathcal{I}_{S}(3)))$ and, last but not least, also that the congruence is transversal to a general cubic through $S$ (see [RS19a] for a systematic use of these key remarks).

For $e=1$ one should consider linear systems of quadric hypersurfaces through $S$; for $e=2$ quintics having double points along $S$; for $e=3$ hypersurfaces of degree 8 having triple points along $S$ and so on.

For $e=1$ we have a unique secant line to $S$ passing through a general point of $\mathbb{P}^{5}$, which is a very strong restriction. Indeed, such a $S$ is a so called surface with one apparent double point. These surfaces are completely classified in [CR11] and those contained in a cubic fourfold are only quintic del Pezzo’s and smooth quartic rational normal scrolls. Cubic fourfolds through these surfaces describe the divisor $\mathcal{C}_{14}$ as it was firstly remarked by Fano in [Fan43] (see also [BRS19, Theorem 3.7] for a modern account of Fano’s original arguments using deformations of quartic scrolls).

Let $D\subset\mathbb{P}^{5}$ be an arbitrary smooth quintic del Pezzo surface. Then $|H^{0}(\mathcal{I}_{D}(2))|=\mathbb{P}^{4}$ and this linear system determines a dominant rational map $\varphi:\mathbb{P}^{5}\dasharrow\mathbb{P}^{4}$, whose general fiber $F$ is a secant line to $D$. Then the restriction of $\varphi$ to a cubic fourfold through $D$ yields a birational map $\psi:X\dasharrow\mathbb{P}^{4}$ and hence the rationality of a general $X\in\mathcal{C}_{14}$, as firstly remarked by Fano in [Fan43].

Let $T\subset\mathbb{P}^{4}$ be a smooth quartic rational normal scroll. Then $|H^{0}(\mathcal{I}_{T}(2))|=\mathbb{P}^{5}$ and this linear system determines a dominant rational map $\varphi:\mathbb{P}^{5}\dasharrow Q\subset\mathbb{P}^{5}$, whose general fiber $F$ is a secant line to $T$ and with $Q$ a smooth quadric hypersurface. Then the restriction of $\varphi$ to a cubic fourfold through $T$ yields a birational map $\psi:X\dasharrow Q$ and another proof of the rationality of a general $X\in\mathcal{C}_{14}$, see [Fan43].

We shall mainly consider the surfaces $S_{d}\subset\mathbb{P}^{5}$ admitting a congruence of 5-secant conics parametrised by a rational variety studied in [RS19a] (but also other new examples) to determine explicitly the rationality of a general $X\in\mathcal{C}_{d}$ for $d=14,26,38$ with $e=2$ (or also for other values $e\geq 3$). To this aim we shall summarise some well known facts in the next subsection.

Let $S_{14}\subset\mathbb{P}^{5}$ be an isomorphic projection of an octic smooth surface $S\subset\mathbb{P}^{6}$ of sectional genus 3 in $\mathbb{P}^{6}$, obtained as the image of $\mathbb{P}^{2}$ via the linear system of quartic curves with 8 general base points. Let $\mathcal{S}_{14}$ be the irreducible component of the Hilbert scheme parametrizing surfaces $S_{14}\subset\mathbb{P}^{5}$ as above. In [RS19a, Theorem 2] we verified that a general $X\in\mathcal{C}_{14}$ contains a surface $S_{14}$ and that each $S_{14}$ admits a congruence of 5-secant conics (birationally) parametrized by its symmetric product and transversal to $X$.

Let $S_{26}\subset\mathbb{P}^{5}$ be a rational septimic scroll with three nodes recently considered by Farkas and Verra in [FV18], where they also proved that a general $X\in\mathcal{C}_{26}$ contains a surface of this kind. Also these surfaces admit a congruence of 5-secant conics transversal to $X$ and parametrized by a rational variety, see [RS19a, Remark 6].

Let $S_{38}\subset\mathbb{P}^{5}$ be a general degree 10 smooth surface of sectional genus 6 obtained as the image of $\mathbb{P}^{2}$ by the linear system of plane curves of degree 10 having 10 fixed triple points. As shown by Nuer in [Nue15], these surfaces are contained in a general $[X]\in\mathcal{C}_{38}$. In [RS19a, Theorem 4] we proved that a general $S_{38}$ admits a congruence of 5-secant conics transversal to $X$ and parametrised by a rational variety.

In this section we analyse some examples and look at them as suitable generalisation of those considered by Fano. A smooth quintic del Pezzo surface $D\subset\mathbb{P}^{5}$ can be realized as a divisor of type $(1,2)$ on the Segre 3-fold $Z=\mathbb{P}^{1}\times\mathbb{P}^{2}\subset\mathbb{P}^{5}$ while a smooth quartic rational normal scroll $T\subset\mathbb{P}^{5}$ can be realized (also) as a divisor of type $(2,1)$. Moreover, a general cubic through $D$ will cut $Z$ along $D$ and a smooth divisor of type $(2,1)$, that is a smooth rational normal scroll $T$ (and viceversa).

One might wonder if something similar happens for the next admissible values $d=26$ and $d=38$ or if, at least, also in these cases there exist surfaces $S_{d}$ giving explicit birational maps to four dimensional (smooth) linear sections of $\mathbb{G}(1,3+k)$, $d=14+12k$. We shall see that very surprisingly this is the case although the linkage phenomenon described above appears again only for $d=38$.

Let $S\subset\mathbb{P}^{6}$ be a septimic surface with a node, which is the projection of a smooth del Pezzo surface of degree seven in $\mathbb{P}^{7}$ from a general point on its secant variety. Let $S^{\prime}_{26}\subset\mathbb{P}^{5}$ be the projection of $S$ from a general point outside the secant variety $\operatorname{Sec}(S)\subset\mathbb{P}^{6}$. These surfaces admit a congruence of 5-secant conics parametrised by a rational variety and a general cubic in $\mathcal{C}_{26}$ contains such a surface, see [RS19a, Theorem 4].

The two surfaces $S_{26}$ and $S^{\prime}_{26}$ are not linked in a variety of dimension three via cubics because the sum of their degrees is 14, which is not divisible by 3.

Studying the rational map $\varphi:\mathbb{P}^{5}\dasharrow\mathbb{P}^{4}$ treated in Theorem 4 we realized that its base locus contains an irreducible component of dimension three $B\subset\mathbb{P}^{5}$ of degree 6 and sectional genus 3. This variety has 7 singular points and it has homogeneous ideal generated by four cubics. So $B\subset\mathbb{P}^{5}$ is a degeneration of the so called Bordiga scroll, which is a threefold given by the maximal minors of a general $3\times 4$ matrix of linear forms on $\mathbb{P}^{5}$. The variety $B$ contains the surface $S_{38}\subset\mathbb{P}^{5}$ and a general cubic through $S_{38}$ cuts $B$ along $S_{38}$ and an octic rational scroll $S^{\prime}_{38}\subset\mathbb{P}^{5}$ with 6 nodes belonging to the singular locus of $B$. The scroll $S^{\prime}_{38}$ is a projection of a smooth octic rational normal scroll $S\subset\mathbb{P}^{9}$ from a special $\mathbb{P}^{3}$ cutting the secant variety to $S$ in six points.

As far as we know this octic rational scroll with six nodes has not been constructed before and, in this context, it is the right generalization of the smooth quartic rational normal scroll considered by Fano. Moreover, it is remarkable also because it does not come from the diagonal construction via the associated $K3$ surfaces as for the Farkas-Verra and Lai scrolls (see [FV18, Lai17] also for more details on this construction). Let us now describe some geometrical properties of the octic rational scroll $S^{\prime}_{38}\subset\mathbb{P}^{5}$.

The aim of this section is to show how one can ascertain the contents of Theorems 2, 3, 4, and similar results in specific examples using the computer algebra system Macaulay2 [GS19].

We begin to observe that given the defining homogeneous ideal of a subvariety $X\subset\mathbb{P}^{n}$, the computation of a basis for the linear system $|H^{0}(\mathcal{I}_{X}^{e}(d))|$ of hypersurfaces of degree $d$ with points of multiplicity at least $e$ along $X$ can be perfomed using pure linear algebra. This approach is implemented in the Macaulay2 package Cremona (see [Sta18]), which turns out to be effective for small values of $d$ and $e$. In practice, in any Macaulay2 session with the Cremona package loaded, if I is a variable containing the ideal of $X$, we get a rational map defined by a basis of $|H^{0}(\mathcal{I}_{X}^{e}(d))|$ by the command¹¹1For all the examples treated in this paper, the linear system of hypersurfaces of degree $d$ with points of multiplicity at least $e$ along $X\subset{\mathbb{P}}^{n}$ coincides with the homogeneous component of degree $d$ of the saturation with respect to the irrelevant ideal of ${\mathbb{P}}^{n}$ of the $e$-power of the homogeneous ideal of $X$. So one can also compute it using the code: gens image basis(d,saturate(I^e)). rationalMap(I,d,e).

Now we consider a specific example related to Theorem 4. In the following code, we produce a pair $(f,\varphi)$ of rational maps: $f:{\mathbb{P}}^{2}\dashrightarrow{\mathbb{P}}^{5}$ is a birational parameterization of a smooth surface $S=S_{38}\subset{\mathbb{P}}^{5}$ of degree $10$ and sectional genus $6$ as in Theorem 4, and $\varphi:{\mathbb{P}}^{5}\dashrightarrow{\mathbb{P}}^{9}$ is a rational map defined by a basis of cubic hypersurfaces containing $S$ (see also Section 5 of [RS19a]). Here we work over the finite field $\mathbb{F}_{10000019}$ for speed reasons.

Macaulay2, version 1.14with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,               LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone, Truncationsi1 : needsPackage "Cremona";i2 : f = rationalMap(ZZ/10000019[vars(0..2)],{10,0,0,10});o2 : RationalMap (rational map from PP^2 to PP^5)i3 : S = image f;i4 : phi = rationalMap S;o4 : RationalMap (cubic rational map from PP^5 to PP^9) We now compute the rational map $\psi$ defined by the linear system of quintic hypersurfaces of ${\mathbb{P}}^{5}$ which are singular along $S$. From the information obtained by its projective degrees we deduce that $\psi$ is a dominant rational map onto ${\mathbb{P}}^{4}$ with generic fibre of dimension $1$ and degree $2$ and with base locus of dimension $3$ and degree $5^{2}-19=6$.

i5 : time psi = rationalMap(S,5,2);     -- used 9.07309 secondso5 : RationalMap (rational map from PP^5 to PP^4)i6 : projectiveDegrees psio6 = {1, 5, 19, 13, 2, 0} Next we compute a special random fibre $F$ of the map $\psi$.

i7 : p = point source psi;  -- a random point on P^5i8 : F = psi^*(psi(p)); It easy to verify directly that $F$ is an irreducible $5$-secant conic to $S$ passing through $p$. One can also see that $F$ coincides with the pull-back $\overline{\varphi^{-1}(L)}$ of the unique line $L\subset\overline{\varphi({\mathbb{P}}^{5})}\subset{\mathbb{P}}^{9}$ passing through $\varphi(p)$ that is not the image of a secant line to $S$ passing through $p$ (see [RS19a] for details on this computation). Finally, the following lines of code tell us that the restriction of $\psi$ to a random cubic fourfold containing $S$ is a birational map whose inverse map is defined by forms of degree $9$ and has base locus scheme of dimension $2$ and degree $9^{2}-27=54$.

i9 : psi’ = psi|sum(S_*,i->random(ZZ/10000019)*i);o9 : RationalMap (rational map from hypersurface in PP^5 to PP^4)i10 : projectiveDegrees psi’o10 = {3, 15, 27, 9, 1}

For the convenience of the reader, we have included in a Macaulay2 package (named ExplicitRationality and provided as an ancillary file to our arXiv submission) the examples listed in Table 2 of birational maps between cubic fourfolds and other rational fourfolds. The two examples with $d=38$ can be obtained as follows:

i11 : needsPackage "ExplicitRationality";i12 : time example38();      -- used 1.2428 seconds The above command produces the following $6$ rational maps:

1. (1)

   a parameterization $f:{\mathbb{P}}^{2}\dashrightarrow{\mathbb{P}}^{5}$ as that obtained above of a surface $S=S_{38}\subset{\mathbb{P}}^{5}$ of degree $10$ and sectional genus $6$;

2. (2)

   the rational map $\psi:{\mathbb{P}}^{5}\dashrightarrow{\mathbb{P}}^{4}$ defined by the quintic hypersurfaces with double points along $S$;

3. (3)

   the restriction of $\psi$ to a cubic fourfold $X$ containing $S$, which is a birational map;

4. (4)

   the linear projection of a smooth scroll surface of degree $8$ in ${\mathbb{P}}^{9}$ from a linear $3$-dimensional subspace intersecting the secant variety of the scroll in $6$ points, so that the image is a scroll surface $T\subset{\mathbb{P}}^{5}$ of degree $8$ with $6$ nodes; moreover, we have the relation: $T\cup S=X\cap\mathrm{top}(\mathrm{Bs}(\psi))$, where $\mathrm{top}(\mathrm{Bs}(\psi))$ denotes the top component of the base locus of $\psi$;

5. (5)

   the rational map $\eta:{\mathbb{P}}^{5}\dashrightarrow Z\subset{\mathbb{P}}^{10}$ defined by the octic hypersurfaces with triple points along $T$, and where $Z\subset{\mathbb{P}}^{10}$ is a $4$-dimensional linear section of $\mathbb{G}(1,5)\subset{\mathbb{P}}^{14}$;

6. (6)

   the restriction of $\eta$ to the cubic fourfold $X$, which is a birational map onto $Z$.

Now we can quickly get information on the maps, e.g. on the inverse of the last one:

i13 : g = last oo;o13 : RationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^10)i14 : describe inverse go14 = rational map defined by forms of degree 5      source variety: 4-dimensional variety of degree 14 in PP^10 cut out by 15                      hypersurfaces of degree 2      target variety: smooth cubic hypersurface in PP^5      birationality: true      projective degrees: {14, 70, 80, 24, 3}