Trisecant Flops, their associated K3 surfaces and the rationality of some Fano fourfolds

We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application, we present the solution of the Kuznetsov's conjecture for $d=42$ (the first open case). Our methods also show an explicit connection between the rationality of cubic fourfolds belonging to the first four admissible families $\mathcal C_d$, with $d=14,26,38$, and $42$ and some birational models of minimal K3 surfaces of degree $d$ contained in well known rational Fano fourfolds.


Introduction
The study of the rationality of higher dimensional Fano manifolds is a very active area of research. Many new and interesting contributions and conjectures appeared in the last decades, mostly concerning the irrationality of very general Fano complete intersections (see for example [44,36,28]; and also [20] and references therein). Deep recent contributions in [21] imply that the locus of geometrically rational fibers in a smooth family of projective manifolds is closed under specialisation, improving substantially our understanding of the loci of rational objects in the corresponding moduli spaces (see [17] for very significant examples in dimension four). Notwithstanding, the irrationality of the very general cubic fourfold and the complete description of the rational ones remain two of the most challenging open problems.
A great amount of recent theoretical work on cubic fourfolds (see for example the surveys [16,22]) lead to the expectation that the very general ones might be irrational and to the specification of infinitely many irreducible divisors C d of special admissible cubic fourfolds of discriminant d in the moduli space C, whose union should be the locus of rational cubic fourfolds (Kuznetsov Conjecture). According to this conjecture, the rationality of cubics in C d depends on the existence of an associated K3 surface in the sense of Hassett/Kuznetsov, see [16,22].
The first admissible values are d = 14, 26, 38, 42, 62, 74, 78, 86. Our main applications of the new methods developed here will be the theoretical explanation of the role of (non minimal) K3 surfaces in determining the rationality for d = 14, 26, 38 and 42 together with the proof of the rationality of every cubic fourfold in C 42 (the first open case of the conjecture) via the construction of a surface of degree nine and genus two with five nodes admitting a congruence of 8-secant twisted cubics and contained in a general cubic fourfold in C 42 , see Theorem 5.12. Let us recall that Fano showed the rationality of a general cubic fourfold in C 14 (see [8,5]), while every cubic fourfold in the irreducible divisors C 26 and C 38 is rational by the main results of [35] (see also [34] and Section 3 here). The proofs in [35] were achieved by constructing surfaces S d ⊂ P 5 , contained in a general cubic fourfold of C d and admitting a four dimensional family of (3e − 1)-secant curves of degree e ≥ 2 parametrized by a rational variety with the property that through a general point of P 5 there passes a unique curve of the family. Then the cubics through S d become rational sections of the universal family and hence are birational to the rational parameter space (see also Theorem 1.1 here).
This approach did not clarify the relation with the associated K3 surfaces and even the construction of explicit birational maps from general cubics X ⊂ P 5 in C 26 and in C 38 to P 4 (or to other notable rational smooth fourfolds W ) in [34] apparently did not provide a birational incarnation in P 4 (or in W ) of a K3 surface of genus 14, respectively of genus 20, determining the linear system of the inverse map. Indeed, the base loci of the linear systems of hypersurfaces of degree 3e − 1 having points of multiplicity e along the corresponding S d 's giving the birational maps µ : X P 4 are intractable reducible schemes while the base loci of the inverse maps µ −1 : P 4 X ⊂ P 5 are even worse (see [34, Table 2]). Here we study all these phenomena via Mori Theory and via the Minimal Model Program to explain the birational nature of the maps µ : X W introduced above and of their inverses which allowed us to produce a suitable birational factorization described in diagram (0.1) below. This approach provides a geometric description of the support of the base loci and, finally, illustrates the relations of these explicit birational maps with the K3 surfaces associated to X, see Subsections 3.4 and 5.5. Last but not least, computer-aided methods play a central role in some key points, also due to the complexity of the geometry involved.
Our method starts with the observation that many of the known examples of surfaces S d ⊂ P 5 used to describe the special divisors C d for small d (not only the admissible ones) have ideal generated by cubic forms defining a map ϕ : P 5 Z ⊂ P N which is birational onto its image. The restriction to a general cubic X through S d defines a birational map ϕ : X Y ⊂ P N −1 with Y a general hyperplane section of Z. In many cases, the birational morphismφ : X ′ = Bl S d X → Y is a small contraction, whose exceptional locus consists of a (union of) smooth surface(s) T ′ ⊂ X ′ ruled by the strict transforms of trisecant lines to S d . Since K X ′ is zero on the strict transforms of trisecant lines, the mapφ is a flop small contraction. In Theorem 2.6 we show that under the previous hypothesis there exists a flopψ : W ′ → Y of the surface T ′ ⊂ X ′ with W ′ a smooth projective fourfold. This trisecant flop τ : X ′ W ′ is constructed by analyzing the splitting of N * T ′ /X ′ restricted to a general strict transform of a trisecant line to S d ⊂ P 5 , see Remark 2.4. The existence of a congruence of (3e − 1)-secant rational curves to S d of degree e ≥ 2 produces an extremal ray on W ′ with divisorial locus, giving a birational morphism ν : W ′ → W with W a Qfactorial Fano variety having Pic(W ) ≃ Z, see Theorem 2.10. The birational morphism ν is (generically) the blow-up of an irreducible surface U ⊂ W , which for the admissible cases d = 14, 26, 38, 42 is a birational incarnation of the associated K3 surface to X. Moreover, the map µ = ν • τ • λ −1 : X W is given by a linear system of forms of degree 3e − 1 with points of multiplicity at least e along S d , while µ −1 : W X is given by a linear system of divisors in |O W (e · i(W ) − 1)| having points of multiplicity at least e along U , where i(W ) is the index of W . Everything is captured by the following diagram, where R ′ ⊂ W ′ is the strict transform on W ′ of the locus R ⊂ W of i(W )-secant lines to U : In the explicit examples with d = 14, 26, 38, 42 considered here, diagram (0.1) together with some standard computations of the middle cohomology of fourfolds under the blow-up of smooth surfaces (see for example the discussion in [16, pages 45-46]) shows that the (possibly singular non minimal) K3 surface U ⊂ W is a birational incarnation of the K3 surface associated to X via Hodge Theory or, equivalently, via Derived Category (see Subsection 3.4 for the complete analysis of the case d = 38). The above theoretical results (and all the examples we have constructed) point out that, for low values of the discriminant d, the birational association between admissible cubic fourfolds and K3 surfaces passes through the construction of very special (and in many cases also singular) non minimal birational models of these surfaces in the fourfolds W by mean of peculiar linear systems of hyperplane sections (often with base points of high multiplicities) on the associated K3 surfaces, exactly as in the case d = 14 considered by Fano. In particular, we are able to prove the rationality of every cubic fourfold in C d for d = 14, 26, 38, 42 via trisecant flops and to show that their inverses are determined by linear systems having a prescribed multiplicity along (non minimal) birational models of the associated K3 surfaces.
Some of these examples of non minimal K3 surfaces have been also studied by Voisin in [47, §4] to prove vanishing results related to Lehn's Conjecture and have been later considered by Fontanari and Sernesi in [10,Theorem 10]. Our constructions of K3 surfaces are based on a different geometric method developed in Section 4 and on computations, which among other things also provide explicit equations for the general K3 surfaces of genera 8, 14, 20, and 22. Since the Kodaira dimension of C d is nonnegative for d ≥ 86 admissible (see [42] and also [29,Proposition 1.3]), a description of C d via surfaces S d 's with at least two cubics containing them as the one considered here (and also in [29,35,34]) is not allowed anymore in this range because this would imply the uniruledness of C d (see the argument in the proof of Theorem 5.9). The admissible values d = 62, 74, 78 are the last ones for which the elements in C d may be arranged into homaloidal linear systems of cubics through an irreducible surface S d , simplifying the construction of U and W via the trisecant flop. For d ≥ 86 admissible one needs to consider suitable generalisations of this approach, see Remark 2.8, increasing substantially the difficulty of the conjecture.
The techniques introduced here open the way to further applications of this circle of ideas to prove the rationality of other classes of Fano fourfolds, see [34,Section 4] and [18].
Acknowledgements. This project started from an intuition of János Kollár that the maps µ : X W ⊆ P N might be useful to describe the congruences (see [ [20,Section 5,§29]). We are very indebted to him for the suggestion. We wish to thank Michele Bolognesi for his continuous support along the years and for many useful conversations on these topics. We are also very grateful to Sandro Verra for several discussions on the subject and to Brendan Hassett for his remarks on a preliminary version of the manuscript. We thank the referees for their careful reading and for their useful comments.
1. Preliminaries 1.1. Congruences of (3e−1)-secant curves of degree e to surfaces in P 5 . The following definitions have been introduced in [35, Section 1]. Let H be an irreducible proper family of (rational or of fixed arithmetic genus) curves of degree e in P 5 whose general element is irreducible. We have a diagram where π : D → H is the universal family over H and where p : D → P 5 is the tautological morphism. Suppose moreover that p is birational and that a general member [C] ∈ H is (re − 1)-secant to an irreducible surface S ⊂ P 5 , that is C ∩ S is a length re − 1 scheme, r ∈ N. We shall call such a family H (or D or π : D → H) a congruence of (re − 1)-secant curves of degree e to S. Let us remark that necessarily dim(H) = 4.
An irreducible hypersurface X ∈ |H 0 (I S (r))| is said to be transversal to the congruence H if the unique curve of the congruence passing through a general point p ∈ X is not contained in X. A crucial result is the following. Theorem 1.1. [35, Theorem 1] Let S ⊂ P 5 be a surface admitting a congruence of (re − 1)secant curves of degree e parametrized by H. If X ∈ |H 0 (I S (r))| is an irreducible hypersurface transversal to H, then X is birational to H.
Moreover, under the previous hypothesis on Φ, if a general element in |H 0 (I S (r))| is smooth, then every X ∈ |H 0 (I S (r))| with at worst rational singularities is birational to H.
Since p : D → P 5 is birational, we also have a rational map ϕ = π • p −1 : P 5 H, whose fiber through a general p ∈ P 5 , F = ϕ −1 (ϕ(p)), is the unique curve of the congruence passing through p. It is natural to ask what linear systems on P 5 give the abstract birational maps ϕ : P 5 H as above or their restrictions to a general X ∈ |H 0 (I S (r))|. The linear system |H 0 (I e S (re−1))|, when not empty, contracts the fibers of ϕ and in [34] we showed that, quite surprisingly, in many cases they can provide a birational geometric realization of ϕ for r = 3, yielding birational maps from cubic hypersurfaces through S to H with H = P 4 or with H a notable Fano fourfold. In the sequel we shall develop a theoretical framework for these phenomena in order to be able to understand also the birational maps defined by the previous linear systems.

1.2.
Divisorial contractions, small contractions and flops. We introduce some general definitions of the Minimal Model Program (MMP for short), adapting them to our setting.
Let X be a smooth projective irreducible fourfold defined over the complex field with ρ(X) = 1 (here ρ(X) denotes the Picard number of X) and let ϕ : X W be a birational map onto a smooth (or at least Q-factorial) irreducible projective fourfold, whose base locus scheme contains a surface S with at most a finite number of nodes.
Let λ : X ′ = Bl S X → X be the blow-up of S and consider the diagram: When W is smooth, the complexity of the birational map ϕ : X W depends on the base locus scheme ofφ : Bl S X W . Surely the easiest case to be considered is wheñ ϕ : Bl S X → W is a morphism, that is ϕ is a special birational map in the sense of Semple and Tyrrell (solved by a single blow-up along a smooth irreducible variety).
If X ⊂ P 5 is a cubic fourfold and if S ⊂ X is smooth, few examples of special birational maps of the above type exist. Two examples of maps of this kind were firstly considered by Fano in [8], have been revisited in modern terms in [1,5] and played a fundamental role in the formulation of Kuznetsov Conjecture. Example 1.2. Letting ϕ : X W be a special birational map with X ⊂ P 5 a cubic fourfold, letting B ⊂ W be the base locus scheme of ϕ −1 and letting U = B red , Fano's examples are the following: (i) S ⊂ P 5 is a smooth quintic del Pezzo surface, W = P 4 , ϕ is given by |H 0 (I S (2))| and U ⊂ P 4 is a surface of degree 9 and sectional genus 8 having at most a finite number of singular points corresponding to planes in X spanned by conics in S. If non singular, the surface U is the projection from a 5-secant P 3 ⊂ P 8 of a smooth K3 surface of degree 14 and genus 8 and ϕ −1 is given by |H 0 (I U (4))|. (ii) S ⊂ P 5 is a smooth quartic rational normal scroll, W = Q ⊂ P 5 is a smooth quadric hypersurface, ϕ is given by |H 0 (I S (2))| and U ⊂ P 5 is a surface of degree 10 and sectional genus 8 having at most a finite number of singular points corresponding to planes in X spanned by conics in S. If non singular, the surface U is the projection from the tangent plane of a smooth K3 surface of degree 14 and genus 8 and ϕ −1 is given by Remark 1.3. The two surfaces S ⊂ P 5 appearing in Example 1.2 are the only smooth surfaces in P 5 admitting a congruence of secant lines (r = 3 and e = 1 in the definition), see for example [32]. The lines of the congruence contained in X describe the exceptional locus E ofφ (or equivalently the exceptional locus λ(E) of ϕ : X W ) and are birationally parametrized by the surfaces U ⊂ W .
The general MMP philosophy suggests that meaningful birational properties of (rational) cubic fourfolds might be related to small contractions from X ′ . So one can start to investigate birational properties of cubic fourfolds from the point of view of the MMP and, when they exist, to consider the most elementary links in the Sarkisov Program associated to small contractions, i.e. flops and flips (one may consult [15] for results about this program in arbitrary dimension). Definition 1.4. Let X be a smooth irreducible projective variety (from now on a projective manifold) and letφ : X → Y be a small contraction, i.e.φ is a birational morphism onto a normal variety Y inducing an isomorphism in codimension one and such that ρ(X/Y ) = 1.
If K X · C = 0 for every irreducible curve contracted byφ, thenφ : X → Y is called a small flop contraction. A small flop contractionψ : W → Y with W a projective manifold is called a flop ofφ.
The resulting birational map τ =ψ −1 •φ : X W is usually called a flop if it is not an isomorphism. Since we assume ρ(X/Y ) = 1 = ρ(W/Y ), givenφ one can prove that the morphismψ, if it exists, is unique as soon as τ is not an isomorphism.
One can flop the small contractionφ : X → Y by constructing a projective manifold V and two birational morphisms σ : V → X and ω : V → W such that σ * (K X ) = ω * (K W ). This means that the exceptional locus of σ, which is divisorial by the smoothness of X, is contracted by ω and that we have a commutative diagram: First of all one may ask if there exist flops of this kind on the fourfolds X ′ = Bl S X obtained from cubic fourfolds X ⊂ P 5 by blowing-up a mildly singular surface S ⊂ X. As we shall see this is the case under some hypothesis and this occurrence is deeply related to the rationality of some special cubic fourfolds (or of other special fourfolds).

Condition K 3
and examples of small contractions on cubic fourfolds. Let us recall that, given homogeneous forms f i of degree d i ≥ 1, i = 0, . . . , M , a vector of homogenous forms (g 0 , . h for every i = 0, . . . , M , then we say that (g 0 , . . . , g M ) is a syzygy of degree h and for h = 1 we shall say that the syzygy is linear. For i < j the syzygies (0, . . . , 0, f j , 0, . . . , 0, −f i , 0, . . . 0), corresponding to the trivial identity f i f j + f j (−f i ) = 0 are called Koszul syzygies. We say that the Koszul syzygies are generated by the linear ones if they belong to the submodule generated by the linear syzygies. This is the condition K d introduced by Vermeire in [45].
The next result provides a wide class of examples of rational maps with linear fibers (hence birational under mildly natural geometrical assumptions on their base locus scheme).
is a linear space P s . For s > 0 the closure of the fiber intersects scheme theoretically the base locus scheme of ϕ along a hypersurface of degree d.
Remark 1.6. Suppose that an irreducible surface S ⊂ P 5 is scheme-theoretically defined by cubic equations satisfying condition K 3 . Then, by Proposition 1.5, every positive dimensional fiber of ϕ : P 5 Z is a linear space P s cutting S in a cubic hypersurface S ∩ P s if s > 0. In particular 0 ≤ s ≤ 2 (except some trivial cases) and s = 2 occurs only for planes spanned by cubic curves contained in S, which are mapped to a point by ϕ. Hence if condition K 3 for S ⊂ P 5 holds and if a general cubic X ⊂ P 5 through S does not contain any plane spanned by cubic curves on S, the exceptional locus T ⊂ X of the restriction of ϕ to X is ruled by proper trisecant lines. As we shall see in Section 2.1 the expected dimension of T is two so that surfaces in P 5 defined by cubic equations satisfying condition K 3 may naturally produce examples of small contractions on X ′ = Bl S X.

The trisecant flop and the extremal congruence contraction
We first introduce and study the behaviour of trisecant lines to a general non degenerate irreducible projective surface S ⊂ P 5 .
2.1. The Hilbert scheme of trisecant lines to S ⊂ P 5 . For the generalities we shall follow the treatment in [4]. Let Hilb r P 5 (respectively Hilb r S) be the Hilbert scheme of 0-dimensional length r ≥ 2 subschemes of P 5 (respectively of S ⊂ P 5 ) and let Hilb r c P 5 ⊂ Hilb r P 5 be the open non-singular subscheme consisting of curvilinear length r subschemes, that is length r subschemes which, locally around every point of their support, are contained in a smooth curve of P 5 . We can define Hilb r c S as the scheme-theoretic intersection between Hilb r S and Hilb r c P 5 inside Hilb r P 5 . Let Al r P 5 ⊂ Hilb r c P 5 denote the subscheme consisting of aligned subschems of length r, that is subschemes of length r contained in a line. Finally, the Hilbert scheme of length r aligned subscheme of S, denoted by Al r S, is the scheme-theoretic intersection of Al r P 5 with Hilb r c S. The schemes Hilb r c P 5 and Al r P 5 are smooth of dimension 5r and 8 + r, respectively. Moreover, if S ⊂ P 5 is smooth, then Hilb r c S is smooth of dimension 2r. In particular, either Al 3 S is empty or every irreducible components of Al 3 S has dimension at least dim(Al 3 P 5 ) + dim(Hilb 3 c S) − dim(Hilb 3 c P 5 ) = 2, which is therefore the expected dimension of Al 3 S. So, for an irreducible projective surface S ⊂ P 5 , one might expect that, with few exceptions, the Hilbert scheme Al 3 S of trisecant lines is of pure dimension two.
There exists a natural morphism of schemes axis : Al r S → G(1, 5), sending each length r ≥ 2 aligned subscheme of S to the unique line containing its support, that is to the multisecant line to S determined by the subscheme of points (counted with multiplicity). Let q : L → G(1, 5) be the universal family and let p : L → P 5 be the tautological morphism. Then Trisec(S) := p(q −1 (axis(Al 3 S))) ⊂ P 5 is called the trisecant locus of S ⊂ P 5 . The previous count of parameters and analysis show: that the expected dimension of Trisec(S) is three; that every irreducible component of Trisec(S) has dimension at least two; that the irreducible components of dimension two of Trisec(S) are either S (in this case S is ruled by lines) or planes cutting S along a plane curve of degree at least three, see [4]. By the Trisecant Lemma, see [33,Proposition 1.4.3], a general secant line to an irreducible non degenerate surface S ⊂ P 5 is not a trisecant line. So dim(Al 3 S) ≤ 3 and dim(Trisec(S)) ≤ 4. Very few examples of irreducible non degenerate surfaces S ⊂ P 5 having dim(Trisec(S)) = 4 are known, most of them are very singular (see [31] for a description) but a complete classification is still lacking. The smooth irreducible non degenerate surfaces S ⊂ P 5 with dim(Trisec(S)) ≤ 2 are classified in [4].
In our analysis we shall always consider the most general case dim(Trisec(S)) = 3. While the condition on the dimension is expected by the above parameter count, the generic smoothness of an irreducible component of Al 3 S is related to the dimension of the corresponding locus and to the tangential behaviour of S ⊂ P 5 at the points of intersection of a general trisecant line by [14,Proposition 4.3] (see also [6, Section 1] and [30] for spectacular generalisations). We shall specialise this general result to our setting.  A trisecant line to an irreducible surface S ⊂ P 5 having the expected trisecant behaviour is said to be general if it is the general element of an irreducible component of Al 3 S whose locus has dimension three.
We now start to study the consequences of this natural condition. Lemma 2.3. Let S ⊂ P 5 be an irreducible non degenerate projective surface with the expected trisecant behaviour and with at most a finite number of singular points, let L ⊂ P 5 be a general trisecant line and let L ′ ⊂ Bl S P 5 denote the strict transform of L. Then IfT ⊂ P 5 denotes the unique irreducible component of Trisec(S) containing L and ifT ′ denotes the strict transform ofT on Bl S P 5 , then Furthermore, ifT ′ is smooth along L ′ , then Proof. Let S reg = S \ Sing(S) be the locus of smooth points of an irreducible non degenerate surface S ⊂ P 5 and let π : Bl S P 5 → P 5 be the blow-up of P 5 along S. Then π −1 (P 5 \ Sing(S)) is a smooth variety. Since Sing(S) is zero dimensional, a general [L] ∈ Al 3 S will cut S in three smooth distinct points and L ′ will be contained in the smooth locus of Bl S P 5 . In particular, the normal bundle N L ′ / Bl S P 5 is locally free of rank four .
The strict transforms of general trisecant lines to S determine a proper family of dimension two of smooth rational curves on Bl S P 5 and the curve L ′ represents a smooth point of this family by hypothesis, yielding assures that N L ′ /T ′ is torsion free and hence locally free of rank 2. Moreover, letting N L ′ /T ′ ≃ O P 1 (a 1 ) ⊕ O P 1 (a 2 ), we deduce a i ≤ 0 for i = 1, 2. Since the curve L ′ moves in a family of dimension two insideT ′ , we have h 0 (N L ′ /T ′ ) ≥ 2 and hence a 1 = a 2 = 0. Finally, ifT ′ is smooth along L ′ , then the exact sequence (2.1) is also exact on the right and NT ′ / Bl S P 5 Remark 2.4. We are interested in studying the birational properties of smooth cubic hypersurfaces X ⊂ P 5 passing through an irreducible projective surface S ⊂ P 5 having the expected trisecant behaviour and with at most a finite number of singular points. Retain the notation of Lemma 2.3 and suppose L ⊂ X ⊂ P 5 is a general proper trisecant line to S contained in X.
and since we have the exact sequence: . If the last splitting holds, then: either the family of strict transforms of trisecant lines to S contained in X to which L ′ belongs is two dimensional (that is Trisec(S) ⊆ X) and is generically smooth; or this family is one dimensional but not generically reduced. If X ⊂ P 5 does not contain Trisec(S) and if S has the expected trisecant behaviour, then the family of trisecant lines to S contained in X is one dimensional and the corresponding locus has dimension two.
Thus when X is sufficiently general, when S has the expected trisecant behaviour and at most a finite number of singular points, the locus of trisecant lines to S contained in X is of pure dimension two and one expects that the one dimensional families of trisecant lines to S contained in X are generically smooth as subschemes of the corresponding parameter space.
The previous natural expectation/hypothesis translates into the following conditions, letting notation be as above: If T ⊂ X denotes the unique two dimensional irreducible component of the locus of trisecant lines to S contained in X to which L belongs, then Condition (2.2) is crucial. Indeed, as we shall see in the next section, it essentially says that T ′ can be flopped producing another four dimensional variety birational to Bl S X and hence to X in a very natural way.
If an irreducible projective surface S ⊂ P 5 has the expected trisecant behaviour and if S satisfies condition K 3 , then, for a general cubic through S, the expected splittings listed above hold for a general proper trisecant line to S contained in the cubic, see the proof of Theorem 2.6. There are also many other examples of different flavour for which the above conditions naturally hold and which naturally lead to flops of the trisecant locus contained in the cubic fourfold.

Assumptions and main definitions.
Assumption 1. Suppose we have a smooth irreducible projective surface (the treatment can be extended to surfaces with at most a finite number of singular points) S ⊂ P 5 , schemetheoretically defined by cubic hypersurfaces and such that the associated rational map Then the restriction of ϕ to a general X ∈ |H 0 (I S (3))| induces a birational map and our hypothesis on the defining equations of S and on the birational map ϕ : P 5 Z can be reformulated by saying that −K X ′ is a big divisor generated by its global sections. In particular, −K X ′ is nef and big.

The induced morphismφ
: is a small contraction (with very few exceptions). Indeed, the base locus scheme of ϕ is the surface S and ϕ contracts any irreducible (rational) curve C ⊂ X of degree e ≥ 1 which is 3e-secant to S, i.e. such that length(C ∩ S) = 3e (proper 3e-secant curve to S). Let us indicate by T ⊂ X the closure of the locus of proper 3e-secant curves to S contained in X. If L ′ ⊂ X ′ is the strict transform of a proper trisecant line to S contained in X, let [L ′ ] denote its numerical class in N 1 (X ′ ).
The strict transform C ′ ⊂ X ′ of a proper 3e-secant curve C ⊂ X to S of degree e ≥ 1 satisfies [C ′ ] = [eL ′ ]. Therefore on X ′ = Bl S X we have for curves C ′ ⊂ X ′ as above. Let us remark that, by definition, ifφ : X ′ → Y is a trisecant flop contraction, then the exceptional locus ofφ : X ′ → Y has dimension at most two and the irreducible components of dimension two are covered by proper 3e-secant (rational) curves (in most cases they are ruled by these curves). By Zariski's Main Theorem, a positive dimensional fiber is connected so that a general positive dimensional fiber is smooth and irreducible.
During our study of birational maps ϕ : P 5 Z of the type described above, we constructed many surfaces S ⊂ P 5 inducing trisecant flop contractions on a general cubic fourfold X through S. For example surfaces satisfying condition K 3 but not only (see Table 1).

2.3.
Existence of the trisecant flop. For simplicity we shall now assume as above that S is smooth. As always, let λ : X ′ = Bl S X → X be the blow-up of X along S, let E ⊂ X ′ be the exceptional divisor and let H ′ = λ * (H), where H ⊂ X is a hyperplane section.
The results in Subsection 2.1, in Remarks 1.6 and 2.4 suggest that, under some mild assumptions, trisecant flop contractions might exist.
We shall now construct explicitly a flop of the two dimensional irreducible components of T ruled by trisecant lines to S via ϕ as soon as S ⊂ P 5 has the expected trisecant behaviour andφ is a small contraction. When these loci exhaust the exceptional locus of a trisecant flop contraction we shall obtain a trisecant flop ofφ : X ′ → Y . Flops of this kind have been also considered in [25] in arbitrary dimension under the stronger assumption that the splitting (2.2) holds for every line of the ruling of T ′ . Theorem 2.6. (Trisecant flop) Let notation be as above, suppose that S ⊂ P 5 satisfies Assumption 1 and that it has the expected trisecant behaviour. If T ′ ⊂ X ′ denotes the exceptional locus of the associated small contractioñ then any irreducible smooth surface T ⊆ T ′ , which is ruled viaφ by the strict transforms of trisecant lines to S (that is through a general point of T ′ there passes a unique curve of this kind), can be flopped to produce a small contractionψ : W ′ → Y with W ′ a smooth projective fourfold.
In particular, under the previous assumptions, ifφ : X ′ → Y is a trisecant flop contraction and if T ′ ⊂ X ′ is a smooth irreducible surface ruled viaφ by trisecant lines, then the trisecant flop τ : X ′ W ′ exists.
Proof. First we shall first prove the second part, that is suppose that T = T ′ is a smooth irreducible surface ruled viaφ by trisecant lines and such thatφ( At the end we shall consider the general case in which T ′ is a finite union of such surfaces. The general fiber ofφ : T ′ → C is smooth and irreducible so that C generically coincides, as a scheme, with the parameter space of trisecant lines to S contained in X. In particular this parameter space is generically smooth of dimension one. Let L ′ ⊂ T ′ be a general fiber of the restriction ofφ to T ′ . By Lemma 2.3 (see also Remark 2.4): the exact sequence: and hence −K E ′ · C 2 = 2 by Adjunction Formula. From σ(C 2 ) = L ′ and from projection formula we get σ * (H ′ ) · C 2 = 1. The Hilbert scheme of curves contained in the smooth projective variety E ′ is smooth and of dimension 2 at the point [C 2 ] corresponding to C 2 ⊂ E ′ by (2.4). So [C 2 ] belongs to a unique irreducible component C of the Hilbert scheme with dim(C) = 2. Since σ * (H ′ ) · C 2 = 1 and since −K E ′ · C 2 = 2, the possible deformations of C 2 inside E ′ are: either irreducible and isomorphic to P 1 ; or they consists of two distinct smooth irreducible rational curves F 1 , F 2 ⊂ E ′ intersecting in one point and such that The last condition means that either F 2 is contracted to a point by σ and hence E ′ · F 2 = −1 by (2.3) or that σ(F 2 ) is a curve contracted to a point by λ, that is a positive dimensional fiber of the blow-up λ. The first case is excluded because it would imply ) is a line and that the projection via λ of all the fibers of T ′ → C pass through the smooth point λ(σ(F 2 )) = p ∈ T ∩ S. Then T ⊂ X would be a plane because it would coincide with its tangent plane at p. The intersection T ∩ S would contain a cubic curve since S is scheme theoretically defined by cubics and every line contained in T would be trisecant to S. The plane T would be contracted to a point by ϕ (and a fortiori byφ) so that T would not be ruled by trisecant lines to S. In conclusion the deformations of C 2 inside E ′ are all smooth, irreducible and isomorphic to P 1 , parametrized by a smooth projective surface (the splitting (2.4) necessarily holds for all the deformations of C 2 ) and the locus of these curves is E ′ . The extremal ray R + [C 2 ] determines a contraction ω ′ : E ′ → R ′ with R ′ a smooth surface and such that every fiber of ω ′ is isomorphic to P 1 , see [26, Theorem 3.5.1]. By the above analysis the surface R ′ is ruled by the curves L ′′ = ω ′ (Σ L ′ ). There exists a morphism ω : X ′′ → W ′ with W ′ a smooth irreducible projective fourfold, which is the blow-up of W ′ along the smooth surface R ′ with exceptional divisor E ′ and whose restriction to E ′ is ω ′ , see for example [27,11] and also [3].
The smooth rational curves L ′′ ⊂ R ′ are disjoint and contracted to C by the nef, big and base point free linear system | − K W ′ |, yielding a morphismψ : W ′ → Y such that C =ψ(R ′ ) and such that the surface R ′ is ruled byψ : R ′ → C.
Suppose now that T ′ = T 1 ∪ · · · ∪ T r , r ≥ 2, with T i a smooth irreducible projective surface ruled viaφ by trisecant lines to S. After applying the previous construction to T 1 we produce W 1 and we have changed . . r, are smooth irreducible surfaces which are ruled by the strict transform of trisecant lines to S and such that the restriction of N * 3). Then we can flop T ′ 2 and produce a smooth fourfold W 2 . After r steps we get a smooth fourfold W r birational to X ′ in which the smooth irreducible ruled surfaces T j have been changed with the corresponding R ′ j . The birational map X ′ W r is an isomorphism in codimension 1 but not an isomorphism and it has been factorized into a sequence of elementary flops (see also [15] for the general program of factorization of birational maps into elementary links according to Sarkisov).
We now state a useful corollary, helpful for our applications and showing that the phenomenon described above really occurs.
Corollary 2.7. Let S ⊂ P 5 be a smooth surface satisfying Assumption 1, condition K 3 and having the expected trisecant behaviour. If X ⊂ P 5 is a cubic hypersurface through S not containing any plane spanned by cubic curves on S and if T ′ ⊂ X ′ denotes the exceptional locus of the associated small contractionφ : Remark 2.8. Obviously, one might only assume that | − K X ′ | = |3H ′ − E| is generated by global sections and big (or only nef and big but not generated by global sections) without requiring that the trisecant flop contraction is necessarily given by | − K X ′ |. It is not difficult to see that in any case, for some m ≥ 1, the linear system | − mK X ′ | gives a trisecant flop contraction. We avoided this more general approach to simplify the exposition but there are examples of trisecant flops appearing also in more general settings, see Subsection 3.6 of the first arXiv version of this paper and also example (xv) of Table 1. 2.4. Trisecant flop and congruences of (3e − 1)-secant rational curves of degree e ≥ 2. The aim of this section is to relate the trisecant flop to (congruences of) (3e − 1)secant curves to S. We start by an easy but very useful result. Proposition 2.9. (Extremal ray generated by (3e − 1)-secant curves) Let notation be as above. Suppose that S ⊂ P 5 satisfies Assumption 1 and that there exists the trisecant flop If C ′ ⊂ X ′ is the strict transform on X ′ of a (3e−1)-secant curve to S of degree e contained in X, then the strict transform C ′ of C ′ on W ′ generates an extremal ray on W ′ .
Proof. By hypothesis there exists a trisecant flop of the trisecant flop contractionφ : X ′ = Bl S X → Y and hence a commutative diagram: By definition C ′ is the strict transform of a curve of degree e, which is (3e − 1)-secant to S.
Consider the possible degenerations C ′′ of C ′ ⊂ X ′ as sum of effective 1-cycles inside X ′ : The cycles C 1 and C 2 have degree e i = H ′ · C i and are β i = E · C i secant to S. In particular e 1 + e 2 = e and β 1 + β 2 = 3e − 1. Since −K X ′ is nef and since generates an extremal ray becauseφ has contracted all the The locus of the extremal ray R + [C ′ ] will determine the type of the associated elementary Mori contraction from W ′ onto a Q-factorial Fano variety. Here we shall consider only the most relevant case for our applications. Examples of fiber type contractions can be constructed as soon as |(3e − 1)H ′ − eE| = ∅ and S ⊂ P 5 has a finite number ρ ≥ 2 of (3e − 1)-secant curves of degree e ≥ 2 passing through a general point of P 5 . The dimension of the general fiber of the contraction will be ρ − 1 = 4 − dim(µ(X)), where µ is the rational map on X defined by the linear system of hypersurfaces of degree 3e − 1 having points of multiplicity at least e on S.
Theorem 2.10. (Extremal contraction of the congruence) Let notation be as above.
Suppose that S ⊂ P 5 satisfies Assumption 1, that it has the expected trisecant behaviour and that there exists the trisecant flopψ : W ′ → Y of the trisecant flop contractionφ : X ′ = Bl S X → Y with X a general cubic fourfold through S and with T ′ irreducible.
If S ⊂ P 5 admits a congruence π : D → H of (3e − 1)-secant rational curves of degree e ≥ 2, then the locus of curves of the congruence contained in X ⊂ P 5 is an irreducible divisor D ⊂ X and the following hold: (1) there exists a divisorial contraction ν : W ′ → W , with W a locally Q-factorial projective Fano variety, whose exceptional locus E is the strict transform of D on W ′ and such that ν(D) = U is an irreducible surface supporting the base locus scheme B of ν −1 . The base locus scheme B is generically smooth , irreducible and ν is generically the blow-up of the surface U . Proof. Let notation be as above and let p : D → P 5 be the tautological morphism of the congruence and let H be its parameter space. Since p is birational, the locus E ⊂ P 5 of points through which there passes more than one curve of the congruence has codimension at least two in P 5 by Zariski Main Theorem. Since the curves of the congruence are (3e − 1)secant to S and hence to X, to be contained in X imposes two conditions in H by Bézout Theorem. Putting these two facts together we deduce that D is a divisor inside X, whose irreducibility will be proved below. By hypothesis there exists a trisecant flop of the trisecant flop contractionφ : X ′ = Bl S X → Y and hence the commutative diagram (2.5). Let D ′ ⊂ X ′ be the strict transform of D on X ′ and let C ′ ⊂ D ′ be the strict transform of a general curve C of the congruence π : D → H contained in X. By definition C ′ is the strict transform of a smooth rational curve of degree e ≥ 2 which is (3e − 1)-secant to S. Let C ′ ⊂ W ′ be the strict transform of a general curve of the congruence C ′ ⊂ D ′ and let generates an extremal ray by Proposition 2.9. By construction the locus of the extremal ray R + [C ′ ] is the divisor Since the (3e − 1)-secant curves to S belong to a congruence, through a general point of D there passes a unique curve of the congruence, as recalled at the beginning of the proof. So the same holds for E and the restriction of ν to E has general fiber isomorphic to a curve C ′ , giving dim(U ) = 2. In particular there exists an open subset U 0 ⊂ U consisting of smooth points of U and such that the fiber of ν : [27,11]. In particular the base locus scheme of ν −1 , which is supported on U , coincides generically with U and hence it is generically smooth. All the assertions in (1) are now proved.
and from e ≥ 2, we get α = a(3e − 1) and β = ae with a ≥ 1. The irreducible components of T are contained in the base locus scheme of this linear system because for every strict transform of a general 3-secant line L to S contained in X. Since the map µ ′ is compatible with the trisecant flop, necessarily a = 1. Indeed, after the blow-up of T the birational map µ ′ becomes a morphism so that (2.6) [ is generated by global sections, yielding a ≤ 1 and hence a = 1, concluding the proof of (2).
We have a commutative diagram: Let H ⊂ W be as above, let H ′ be its strict transform on W ′ and, keeping notation as in the proof of Theorem 2.6, let We also have C ′ · E = 1 and H ′ · C ′ = 0. Since the birational morphism ψ : W ′ → Y is given by a linear system in |i(W )H ′ − E| as shown above, the birational map ψ =ψ • ν −1 : W Y is given by a linear system of divisors in |O W (i(W ))| vanishing on U .
sent into a curve of the congruence D, which by definition has degree e ≥ 2, we deduce Moreover, reasoning as above, the compatibility with the trisecant flop yields In conclusion, the birational map η ′ is given by a linear system in |(i(W ) · e − 1)H ′ − eE| of dimension 5 and µ −1 is given by a linear system of dimension 5 of )| having points of multiplicity at least e along U ⊂ W . The previous analysis shows that the base locus of µ −1 contains U and ν(R), which is a locus of i(W )-secant lines to U contained in W , concluding the proof of (4). Let D ′ ⊂ W be the strict transform of E ⊂ X ′ via µ, which is an irreducible divisor. Let F ⊂ E be a positive dimensional fiber of λ : X ′ → X. Then D ′ is ruled by the strict transforms of the curves F and such a general point of D ′ there passes a unique curve of this family. Moreover, This proves the last assertion in (3).
Remark 2.11. Obviously, one can also reverse the construction in Theorem 2.10 starting from suitable U ⊂ W and then producing the congruences of (3e − 1)-secant curves of degree e to a surface S ⊂ X ⊂ P 5 by taking the image of E in X and by taking S ⊂ X ⊂ P 5 as the surface describing the linear system defining the inverse map µ : X W . In practice, as soon as the trisecant flop exists, the existence of a congruence of (3e − 1)-secant lines to S is equivalent to the existence of the surface U ⊂ W , which should be an incarnation of the associated K3 surface (see the end of subsection 3.1 for the analysis of the case d = 38 to see one such explicit incarnation). From this point of view one associates to the pair (X, S) a pair (W, U ), where W is the image of X and the surface U is naturally the parameter space of the curves of the congruence D contained in X.

Associated K3 surfaces to cubic fourfolds in C 38 via the trisecant flop
In this section, as an application of previous theoretical results, we describe birational incarnations of the K3 surfaces associated to the cubic fourfolds in C 38 via Hodge Theory or via Derived Category Theory. For the sake of brevity, we omit a similar analysis of the cases of cubic fourfolds in C 14 and C 26 , which has been outlined in the first arXiv version of this paper. The case of cubic fourfolds in C 42 will be considered in Subsection 5.5.
3.1. General properties of degree 10 smooth surfaces S 38 ⊂ P 5 of sectional genus 6. Let us consider the smooth surfaces S 38 ⊂ P 5 obtained as the image of P 2 by the linear system of plane curves of degree 10 having 10 fixed triple points in general position. These surfaces are contained in a general cubic fourfold in the admissible divisor C 38 , as shown in [29]. They were also studied in [35,34] to prove that every cubic fourfold in C 38 is rational.
The Hilbert scheme S 38 parametrizing such surfaces is explicitly unirational, that is we can write out equations for the general member [S 38 ] ∈ S 38 over a pure transcendental extension of the base field. From this, one can deduce that the homogeneous ideal of S 38 is generated by 10 cubic forms, whose first syzygies are generated by the linear ones. In particular the general S 38 ⊂ P 5 satisfies condition K 3 . By Proposition 1.5 the linear system |H 0 (I S 38 (3))| defines a birational map ϕ : P 5 Z ⊂ P 9 onto its image Z. Through a general point ϕ(p) ∈ Z there passes 8 lines contained in Z. The pullbacks of these lines are seven secant lines to S 38 passing through p and a 5-secant conic to S 38 . In particular, a general S 38 ⊂ P 5 admits a congruence of 5-secant conics (see [35,Section 5] for details on this computation and also Subsection 6.1).
Moreover, we have |H 0 (I 2 S 38 (5))| = P 4 for a general S 38 ⊂ P 5 and the coefficients of the multidegree of the graph of the associated rational map µ : P 5 P 4 are (1, 5, 19, 13, 2) (see Subsection 6.1). From this one deduces that: µ is dominant since the last entry is equal to 2 (this means that the closure of a general fiber of µ, F = µ −1 (µ(p)) with p ∈ P 5 general has degree 2 and dimension 1); its base locus scheme B ⊂ P 5 has degree 6 = 5 2 − 19 and dimension three. Since the unique 5-secant conic C p to S 38 passing through p is contracted by µ to the point µ(p) (5 · 2 − 2 · 5 = 0), the fiber F coincides with C p and it is irreducible (otherwise one can argue as in Subsection 6.1, prove that F is an irreducible conic and verify that it is 5-secant to S 38 , yielding a different proof of the existence of the congruence of 5-secant conics).
The rationality of a general X through a general S 38 follows by restricting µ to X. Indeed, through a general q ∈ X there passes a unique conic C q of the congruence, which is not contained in X by the generality of q. The conic C q cuts X in q and in five points on S 38 , yielding that the general fiber of map µ = µ |X : X P 4 is a point by Bézout Theorem.

3.2.
Small contraction defined by cubics through a general S 38 ⊂ P 5 . The (closure of the) fibers of ϕ : P 5 Z are linear spaces of dimension s with 0 ≤ s ≤ 2. The two dimensional fibers ofφ : Bl S 38 P 5 → Z ⊂ P 9 are the strict transforms of planes in P 5 cutting S 38 along plane cubic curves by Proposition 1.5. Let C ⊂ S 38 ≃ Bl {p 1 ,...,p 10 } P 2 ⊂ P 5 be such a cubic and recall that the embedding is given by |10H − 10 i=1 3E i |, using the standard notation. Since the curve C is contained in a plane, it cuts each line E i ⊂ S 38 in at most one point so we deduce α = 3 and a i = 1 for nine of the ten indices. Hence C is the image of a plane cubic curve passing through 9 of the 10 general base points on P 2 . In conclusion there are 10 two dimensional fibers ofφ. The other positive dimensional fibers ofφ are the strict transforms of trisecant lines to S 38 . From the equations of the base locus scheme B of µ, we deduce that B has a unique irreducible componentB of dimension three and degree 6, which contains S 38 . In particular, B is generically reduced alongB. An explicit computation shows that the varietyB is mapped by ϕ onto an irreducible surface V ⊂ Z ⊂ P 9 , which is a Veronese surface generating a P 5 ⊂ P 9 . The general fiber of the restriction of ϕ toB has dimension one and hence it is a trisecant line to S 38 . A trisecant line to S 38 is contained in the base locus scheme B of µ because it intersects a quintic with double points along S 38 in at least six points counted with multiplicity. HenceB is the unique irreducible component of Trisec(S 38 ) of dimension 3 and S 38 has the expected trisecant behaviour because the irreducible component of Al 3 S 38 corresponding toB is birational to the smooth irreducible surface V . From this analysis and from the previous computations, we deduce that Trisec(S 38 ) consists ofB and of the 10 planes cutting S 38 along cubic curves.
Since the ten planes in Trisec(S 38 ) are mapped to ten points in Z, a general hyperplane section of Z does not pass through these 10 points. Hence a general cubic hypersurface X ⊂ P 5 through S 38 does not contain any of the 10 planes in Trisec(S 38 ). The restriction of ϕ to X ′ = Bl S 38 X induces a small contraction: Hence T is ruled by the trisecant lines to S 38 contained in X via the restriction of ϕ. The Double Point Formula yields that the singular locus of the rational scroll T , projection of a smooth rational normal scroll of degree 8 in P 9 , consists of six singular points. Its strict transform T ′ ⊂ X ′ is smooth andφ| T ′ : T ′ → C is a P 1 -bundle. The birational morphismφ is an isomorphism between X ′ \ T ′ and Y \ C and hence it is a trisecant flop contraction.
3.3. The trisecant flop determined by S 38 ⊂ P 5 . Theorem 2.6 and Theorem 2.10 assure the existence of the trisecant flopψ : W ′ → Y and of the divisorial contraction ν : W ′ → P 4 giving a factorization of the birational map µ : X P 4 (see the commutative diagram (2.7) also for recalling the notation).
The scroll R = ν(R ′ ) ⊂ P 4 has degree 6 (recall the we tensor with O P 1 (−1) performing the flop, see (2.6)) and µ −1 : P 4 X is given by a linear system in |H 0 (I 2 U (9))| by part (3) of Thereom 2.10, where U ⊂ P 4 is the support of the base locus B of ν −1 (recall that B is generically reduced so that it coincides with U generically). By part (4) of Theorem 2.10, the lines of the scroll R are 5-secant to U , the map ψ is given by a linear system in |H 0 (I U (5))| and ψ(R) =ψ(R ′ ) = C.
The cubics through S 38 restricted to X are mapped by µ onto quintics defining B as a scheme by part (3) of Theorem 2.10. Taking a basis of cubics X i through S 38 restricted to X, i = 1, . . . , 9, their images V i = µ(X i ) ⊂ P 4 determine the ideal of B. In Subsection 6.1 and in the ancillary file code_section_6.m2 we verified that B is a smooth surface of degree 12 and sectional genus 14 so that it coincides with U as scheme. Hence U ⊂ P 4 is a smooth surface of degree 12 and sectional genus 14, whose ideal is generated by 9 forms of degree five. Since ν −1 (B) = E, the universal property of the blow-up yields a birational projective morphism δ : W ′ → Bl U P 4 between smooth projective fourfolds. Since rk(Pic(W ′ )) = 2 = rk(Pic(Bl U P 4 )), we have W ′ ≃ Bl U P 4 by Zariski Main Theorem. Moreover, is given by the linear system |H 0 (I 2 U (9))| by part (3) of Theorem 2.10.
Let A = C[t 0 , . . . , t 4 ]. Using the explicit equations constructed above, we can compute the resolution of the homogenous ideal of U : and also verify that p g (U ) = 1 and q(U ) = 0, see Subsection 6.1. To conclude the analysis of U ⊂ P 4 we shall follow the arguments in [7, Section 2.7], where the authors gave a different construction of the above surface via Beilinson Spectral Sequence methods.
The intersection matrix for the sublattice H, K U of Num(U ) is where K 2 U = −11 is deduced from the Double Point Formula for a smooth surface in P 4 . The number N 6 of proper 6-secant lines to U (if finite) plus the number of exceptional lines, that is (−1)-curves L i ⊂ U such that H · L i = 1 and L 2 i = −1, is equal to 10 by Le Barz Formulas, see loc. cit.. There are no proper 6-secant lines since I U is generated by quintic forms. Let U ′ ⊂ P 14 be the first adjunction surface of U , that is the image of the birational morphism γ : U → U ′ defined by the base point free linear system |K U + H| (see the Introduction of [7] for a summary of the basic results of adjunction theory on surfaces). Since the above morphism contracts the exceptional lines on U , the morphism γ realizes U as the blow-up of U ′ in 10 distinct points and K 2 Since p g (U ′ ) = p g (U ) = 1 and since K 2 U ′ = −1, the canonical divisor is effective but not nef and the surface U ′ is non minimal. Let E ⊂ U ′ be a (−1) curve, let K U ′ = D + E (D ≥ 0) and letẼ ⊂ U be an irreducible curve such that γ(Ẽ) = E. ThenẼ 2 ≤ E 2 = −1 and we deduce H ′ · D ≤ 2 and that E is not contained in the support of D (otherwise K U ′ = 2E and K 2 U ′ = −4). Then D · E = 0, D 2 = 0 and K U ′ · D = 0. If H ′ · E = 2, then H ′ · D = 2. From (K U ′ + D) · D = 0, we deduce that D cannot be an irreducible conic. If D = 2 ·L, then H ′ ·L = 1 would be a line such (K U ′ +L) ·L = 0, which is impossible. In the same way we can exclude that H ′ ·E = 3, conclude that H ′ ·E = 4 and finally that K U ′ = E. Let ǫ : U ′ → U ′′ be the contraction of E to a point of the smooth surface U ′′ . Since K U ′′ = 0 and since q(U ′′ ) = q(U ) = 0, the surface U ′′ is a K3. The curveẼ ⊂ U is a smooth rational normal curve of degree 4, proving that the distinct points p i = γ(L i ) do not belong to E. In the next section we shall develop an alternative geometric method to obtain the whole configuration of the (−1)-curves on U in explicit examples.
Let π : U → U ′′ be the blow-up of U ′′ at the eleven distinct points described above and let H ′′ = π * (H). Then H ′′ is an ample divisor on U ′′ with a point of multiplicity four at p 11 and passing simply through p 1 , . . . , p 10 and the linear system H on U is thus given by |π * (H ′′ ) − 10 i=1 L i − 4E|. Hence (H ′′ ) 2 = (H ′ ) 2 + 10 + 16 = 12 + 10 + 16 = 38 and p a (H ′′ ) = p a (H ′ ) + 6 = 20. The generality of X through S 38 assures that the K3 surface has general moduli and that Pic(U ′′ ) ≃ Z (a direct proof of this fact will be given below). Hence the divisor H ′′ is very ample and gives and embedding U ′′ ⊂ P 20 .
3.4. The associated K3 surface to a general cubic in C 38 via the trisecant flop. Via the trisecant flop and via the contraction of the curves of the congruence contained in X, we proved that the associated surface U ⊂ P 4 to a general pair (X, S 38 ) is a birational incarnation of a general smooth K3 surface U ′′ ⊂ P 20 of degree 38 and genus 20. We now want to show that the surface U ′′ (or U ) is associated to X in the sense of Hodge Theory (or, equivalently, of Derived Category Theory) following the treatment by Hassett in [16,Section 3].
Let us recall that for an arbitrary cubic fourfold X ⊂ P 5 , letting h p,q denote the Hodge numbers of X, we have h 0,4 = h 4,0 = 0, h 1,3 = h 3,1 = 1, h 2,2 = 21. Let h denote the class of a hyperplane section of X and let . They become compatible as soon as one can find a common codimension one rank sublattice with signature (19,2).
The definition of C 38 (see [16,Section 2.3] for the general theory) and the fact that a general [X] ∈ C 38 contains a surface S 38 ⊂ P 5 (see [29]) yield for such a X. Let T X ⊂ H 4 (X, Z) denote the transcendental part of the cohomology of a cubic fourfold X ⊂ P 5 . Then in our setting has rank 21 and signature (19,2). Clearly H 2 (S 38 , Z) ≃ Z 11 , H 2 (T ′ , Z) ≃ Z 2 ≃ H 2 (R ′ , Z) and H 2 (U, Z) ≃ H 2 (U ′′ , Z) ⊕ Z 11 because we blow-up eleven points on U ′′ .
Let M be a smooth projective fourfold and let S ⊂ M be a smooth projective surface. Then We have a commutative diagram: inducing an isomorphism between For a smooth projective surface S let T S ⊂ H 2 (S, Z) denote the trascendental part of the cohomology of S.
implies rk(T Bl T ′ X ′ ) = 21. On the other hand

) and the previous isomorphisms defined via (3.2) induce an isomorphism
respecting Hodge structures. Therefore the K3 surface U ′′ is associated to X in the sense of Hodge Theory according to Hassett, see [16, Section 3.2]. The isomorphism between H 4 (Bl T ′ X ′ , Z) and H 4 (Bl R ′ W ′ , Z) sends the classes corresponding to the 10 exceptional lines on S 38 to the classes corresponding to the ten exceptional lines on U ⊂ P 4 while the rational normal curve of degree four on U correspond to the class of H 2 (T ′ , Z)(−1), T ′ ≃ P(O P 1 (4) ⊕ O P 1 (4)) → P 1 , not contacted by ω. This gives an interpretation of the fact thatφ(T ′ ) = C =ψ(R ′ ) is a rational normal curve of degree four (or, equivalently, that T ′ admits a section which is a rational normal curve of degree four) and shows that the exceptional curve of degree fourẼ ⊂ U is exactly ν(R ′ ) ∩ U .

A geometric method for detecting the exceptional (−1)-curves on some non minimal K3 surfaces
We shall now consider the problem of finding the non minimal K3 surface U ⊂ W in the base locus scheme of a the birational map µ : X W defined in the previous sections for a general cubic [X] ∈ C d with d = 14, 38 in order to develop a method to be applied later in the more difficult case d = 42. The detection of the (−1)-curves on the (smooth non minimal K3) surface U (or on its linear normalization if U has nodes) is a delicate and intriguing problem, which in some cases can lead to the explicit construction of the general K3 surface of degree 2g − 2 and genus g. When this is possible, one usually gets a direct proof that the corresponding moduli space of polarized K3 surfaces is unirational. In the previous section to analyse the case d = 38, following the traditional approach of [7], we used the smoothness of U ⊂ P 4 = W and appealed to the Le Barz Formulas together with Adjunction Theory in order to find the ten exceptional lines on U . Then, after the contraction of these ten (−1)-curves, there appeared a last exceptional curve, which is a quartic rational normal curve already contained in U .
It is very difficult to try to adapt these arguments to surfaces U ⊂ W with W a Fano fourfold lying in spaces of higher dimension. So we elaborated an entirely new geometric method, which works efficiently in the cases d = 14, 26, 38, 42 to prove that U is the blow-up of a K3 surface of degree d and genus g = d+2 2 associated to X ∈ C d . Our analysis is based on the existence of the congruence, of the map µ : P 5 W and on a careful study of the known examples studied by Fano in [8] (see also [5]).
Recall that by definition of congruence of (3e − 1)-secant curves of degree e ≥ 1 to S ⊂ P 5 , we have a diagram (1.1), where π : D → H is the universal family over the parameter space H and where p : D → P 5 is the tautological morphism, which by definition is birational, see Subsection 1.1. The Fundamental Locus of the congruence is the base locus of p −1 by Zariski Main Theorem, which also implies that E is the image of the ramification locus of p. The locus E has codimension at least two. Let E 1 , . . . , E r , r ≥ 1 be the irreducible components of dimension three of E, if any. Suppose there exists the associated rational map µ : P 5 W defined by the linear system |H 0 (I e S (3e − 1))| such that a general fiber of µ is a curve of the congruence and let C i = µ(E i ) ⊂ W , i = 1, . . . , r. These are special curves in W , defined via the congruence but without any apparent relation with cubic fourfolds through S.
Let us start by describing the surfaces considered in Example 1.2 from this perspective. Let C ⊂ S be an irreducible conic and Π = C = P 2 be its linear span. Since S is defined by quadratic equations Π ∩ E i is either empty or consists of a point (otherwise Π ∩ S would contain C and the line E i ). Let π : S → P 2 be the blow-up morphism. Then , yielding either: d = 1 and a i = 1 for only one index i; or d = 2 and a i = 1 for every index i. In conclusion C belongs to one of the five pencils described above.
Let µ : P 5 P 4 be the map associated to the congruence of secant lines to S. By Proposition 1.5 the closure of the fibers of µ are either secant lines to S or planes cutting S along a conic. Hence the Fundamental Locus E of the congruence of secant lines to S is the locus of the planes spanned by conics through S (through a point of the plane there passes infinitely many secant lines to S and if through a point of a secant line there passes another secant line, these lines span a plane by the analysis of the fibers of µ).
Example 4.2. If S ⊂ P 5 is a general quartic rational normal scroll, the Fundamental Locus E of the congruence of secant lines to S consists of the unique Segre 3-fold Σ ≃ P 1 × P 2 containing S as a divisor of type (0, 2). Indeed, S ≃ P 1 × P 1 embedded in P 5 by O P 1 ×P 1 (1, 2) so that, reasoning as above, S ≃ P 1 × C ⊂ P 1 × P 2 = Σ ⊂ P 5 with C ⊂ P 2 a conic. The conics on S are only the fibers of the first projection P 1 × C → P 1 . The fibers of µ : P 5 W with W ⊂ P 5 a smooth quadric hypersurface are all linear by Proposition 1.5. Let E ⊂ P 5 be the Fundamental Locus of the congruence of secant lines to S. If q ∈ E, then through E there pass at least two secant lines to S. So Π q = µ −1 (µ(q)) is a plane cutting S along a conic, yielding E ⊂ Σ and hence E = Σ. Then C = µ(E) ⊂ W is a smooth conic because on E we have (2, 2) − (0, 2) = (2, 0).
In general it is not easy to determine the E i 's and then calculate their images (although possible in all the examples treated here). We shall now present a key remark, which allows us to determine the C i 's without necessarily computing the E i 's in all the known examples we studied until now.
Geometric Method. Suppose there exists a congruence of (3e − 1)-secant curves of degree e ≥ 1 to S ⊂ P 5 with Fundamental Locus E = E 1 ∪ · · · ∪ E r with E i ⊂ E the irreducible components of dimension three of E, if any. Suppose there exists the associated rational map µ : P 5 W defined by the linear system |H 0 (I e S (3e − 1))| such that a general fiber of µ is a curve of the congruence and let C i = µ(E i ) ⊂ W , i = 1, . . . , r. Since through a general point of E i the fiber of µ has dimension at least two, we expect that C i ⊂ W is a curve, see Examples 4.1 and 4.2.
Let X j ⊂ P 5 , j = 1, 2, be a general cubic through S and let U j ⊂ W be the associated surface contained in the base locus of the inverse of the restriction of µ to X j . Then C i = µ(X j ∩ E i ) ⊂ U j for every i = 1, . . . , r and for j = 1, 2, so that Since U 1 and U 2 are moving surfaces in the fourfold W , one expects that C 1 ∪ · · · ∪ C r is exactly the one dimensional component C of From the equations of U 1 and U 2 we derive immediately those defining C. In the cases under consideration this allows us to verify the smoothness of C, that C = C 1 ∪ · · · ∪ C r and that all the disjoint irreducible components C i 's are rational (many components are lines) and that h 1 (N C/U ) = 0. The knowledge of the equations of the U j 's yields p g (U j ) = 1, q(U j ) = 0 (at least in the smooth cases) by direct computation. Collecting all the information one proves that U is the blow-up of a K3 surface as shown by the next crucial remark. Let C = C 1 ∪ · · · ∪ C r , r ≥ 1, be a smooth curve on U with C i a rational curve for every i = 1, . . . , r. Then: (1) C 2 i < 0 for every i = 1, . . . r.
, then the rational curves in |H 0 (O U (C i ))| would cover U and U would be uniruled contradicting p g (U ) = 1. So C 2 i < 0 for every i = 1, . . . , r. The C i 's are disjoint smooth rational curves, O C (C) |C i ≃ O P 1 (C 2 i ) and is equal to 0 if and only if C 2 i = −1 for each i = 1, . . . , r, proving (2). Since p g (U ) = 1, the canonical class is effective. From K U ·C j = −1 we deduce K U = C +D with D ≥ 0. Then H · C = H · K U = H · C + H · D yields H · D = 0 and hence D = 0. Since the C i 's are disjoint (−1)-curves, the contraction of the curves in C produces a smooth surface U ′ and a morphism π : U → U ′ which is the blow-up of r ≥ 1 distinct points on U . Then K U ′ = 0 and q(U ′ ) = q(U ) = 0 imply that U ′ is a K3 surface. The last claim about the expression of H via pull-back of H ′ is now obvious.
Example 4.4. (Application to smooth quintic del Pezzo surfaces in P 5 ) Suppose S ⊂ P 5 is a smooth del Pezzo surface of degree 5. Then U j ⊂ P 4 , j = 1, 2, are smooth surfaces of degree 9 and sectional genus 8 with p g (U j ) = 1 and q(U j ) = 0 (the invariants are determined via the explicit equations obtained via the restriction of µ : P 5 P 4 ). The one dimensional component C of U 1 ∩ U 2 is a smooth curve of degree 5 and arithmetic genus −4 from which it immediately follows that C has five irreducible components and hence is the union of five distinct lines C 1 , . . . , C 5 , as we already know.
Let U = U 1 . After verifying that h 1 (N C/U ) = 0, we deduce that every C i is a (−1)-curve on U . Since H · K U = −H 2 + 2g(H) − 2 = −9 + 16 − 2 = 5 = H · C we conclude by Lemma 4.3 that π : U → U ′ is the blow-up at five distinct points p 1 , . . . , p 5 of the K3 surface U ′ and that H ′ = π * (H) ⊂ U ′ is a very ample divisor on U ′ such that (H ′ ) 2 = 9 + 5 = 14, g(H ′ ) = 8. Indeed, H ′ is ample and the generality of X 1 and of U = U 1 imply that rk(Pic(U ′ )) = 1 (see the argument used at the end of Subsection 3.1 or compute the moduli of the K3's). Hence U ′ ⊂ P 8 is a K3 surface of degree 14 and genus 8 and the linear system |H| on U corresponds to the hyperplane sections of U ′ passing through p 1 , . . . , p 5 . In particular, we see that these points impose only four independent conditions to hyperplane sections.
Example 4.5. (Application to general quartic rational normal scrolls in P 5 ) For S ⊂ P 5 a general rational normal scroll, the corresponding surfaces U j ⊂ W are smooth, have degree 10, sectional genus 7, p g (U j ) = 1 and q(U j ) = 0. The one dimensional component C of U 1 ∩ U 2 is a smooth conic, which is a (−1)-curve on U j because h 1 (N C/U j ) = 0. Since H · K U j = 2 = H · C, we deduce K U j = C and we can apply Lemma 4.3. We conclude that U j ⊂ P 5 is obtained by blowing-up a K3 surface U ′ ⊂ P 8 of degree 14 and genus 8 at one point p ∈ U ′ and that |H| corresponds to the linear system of hyperplanes sections having a point of multiplicity at least two at p. Example 4.6. (Application to degree 10 surfaces S 38 ⊂ P 5 ) The surfaces U j ⊂ P 4 , j = 1, 2, are smooth, have degree 12, sectional genus 14, p g (U j ) = 1 and q(U j ) = 0, see Subsection 3.1. The one dimensional component C of U 1 ∩ U 2 is a smooth curve of degree 14, of arithmetic genus −10 with eleven irreducible components. By projecting C generically to P 3 (to speed computations) and to P 2 (to read more efficiently the decomposition), we get a smooth curve of degree 14, respectively a plane curveC of degree 14. The lines iñ C disappear by taking duality, that is the image ofC via the Gauss map ofC. Then, by reflexivity, the bidual curves ofC consists of the irreducible curves inC different from lines. It turns out that the bidual curve ofC is a quartic curve with three nodes from which it follows that C is the disjoint union of 10 lines and of a quartic rational normal curve, see also the ancillary file code_section_6.m2. In conclusion, C is the union of eleven smooth rational curves.
The eleven curves C i 's are (−1)-curves on U j because h 1 (N C/U j ) = 0, see code_section_6.m2. for the computation and then apply Lemma 4.3. Since H · K U = −H 2 + 2g(H) − 2 = −12 + 28 − 2 = 14 = H · C, we deduce from Lemma 4.3 that there exists a contraction π : U → U ′ of the eleven (−1)-curves C i 's to the eleven distinct points p 1 , . . . , p 11 on the smooth K3 surface U ′ . Then H ′ = π * (H) ⊂ U ′ is an ample divisor on U ′ such that (H ′ ) 2 = 12 + 16 + 10 = 38 and g(H ′ ) = 14 + 6 = 20. For a general X through S 38 we have that U ′ is a K3 surface of degree 38 and genus 20 with rk(Pic(U ′ )) = 1 (see the argument used at the end of Subsection 3.1 or compute the moduli of the K3's) so that |H ′ | gives an embedding U ′ ⊂ P 20 . The linear system |H| on U corresponds to the hyperplane section of U ′ passing through p 1 , . . . , p 10 and having a point of multiplicity at least four at p 11 . In particular, we see that these points do not impose independent conditions to hyperplane sections of U ′ .

Rationality of cubics in C 42 via congruences of 8-secant twisted cubics
In this section, we first construct a family of surfaces in P 5 to describe the divisor C 42 as the locus of cubic fourfolds containing these surfaces. This family is related to an example of a surface of degree 9 and sectional genus 2 contained in a del Pezzo fivefold, which has been discovered (computationally) for the first time in [18]. Here we give an explicit and geometric description of the complete family of these surfaces inside a del Pezzo fivefold and then use them to construct surfaces in P 5 of degree 9 and sectional genus 2 with five nodes. Finally we use this new description of C 42 to show that C 42 is unirational (see Corollary 5.10) and that every cubic fourfold in C 42 is rational (see Theorem 5.12).

Birational representations of del Pezzo fivefolds.
A del Pezzo fivefold V ⊂ P 8 is a smooth hyperplane section of G(1, 4) ⊂ P 9 . The following result is well known in classical Algebraic Geometry (see e.g. [43] and [37, Section 10]).  G(1, 4). Then: (i) the projection from Π restricted to V induces a birational map V P 5 , whose base locus scheme is Π; (ii) the inverse map P 5 V ⊂ P 8 is given by the linear system of quadric hypersurfaces through a rational normal cubic scroll contained in a hyperplane in P 5 .
Remark 5.2. A del Pezzo fivefold V = G(1, 4) ∩ P 8 ⊂ P 8 contains two distinct families of planes, F 1 and F 2 , which in the Chow ring of G(1, 4) have Schubert classes given by σ 2,2 and σ 3,1 . The family F 1 has dimension 3 and through a point of V there passes a unique plane of the family. The family F 2 has dimension 4 and through a general point of V there passes a one dimensional family of these planes.
If α : P 5 V ⊂ P 8 denotes the birational map defined by the quadrics through a rational normal cubic scroll ∆ ⊂ H ≃ P 4 ⊂ P 5 as in Proposition 5.1, then the planes through a point q = α(p), p ∈ P 5 \ H, correspond, respectively, to the unique plane generated by p and the directrix line of ∆ and to the planes generated by p and by a line of the ruling of ∆.
Since [23, p. 403]) for an irreducible conic C ⊂ V ⊂ P 8 , the Hilbert scheme Con of conics contained in V has dimension 10 = h 0 (N C/V ), a fact that can be also deduced from a simple parameter count using the previous birational representation of V .
The next result is also classical and well-known. It can also be easily verified via an explicit computation.
Proposition 5.3. Let V = G(1, 4) ∩ P 8 ⊂ P 8 be a del Pezzo fivefold and let C ⊂ V be an irreducible conic such that its linear span Π = C is not contained in V. Then: (i) the projection from Π restricted to V induces a birational map V P 5 , whose base locus scheme is C; (ii) the inverse map P 5 V ⊂ P 8 is given by the linear system of cubic hypersurfaces vanishing on a rational scroll in P 5 of dimension three and degree four, which is a projection of a smooth quartic rational normal scroll in P 6 .

5.2.
Curves of degree 8 in P 5 with a node and with geometric genus 2 contained in a projected rational scroll of degree four. Now we prove some geometrical properties about irreducible curves of degree 8 and arithmetic genus 3 in P 5 . We shall restrict ourselves to the case of curves with a node, used in the sequel, although the same proof works also for smooth curves (this case has been considered in [38]).
Proposition 5.4. Let C ⊂ P 5 be a non degenerate curve of degree 8 and arithmetic genus 3 with a node. Then: (i) the curve C ⊂ P 5 is the complete intersection of a pencil of quintic del Pezzo surfaces on a Segre 3-fold Σ = P 1 × P 2 ⊂ P 5 such that the general element of the pencil is smooth. (ii) The curve C ⊂ P 5 has ideal generated by seven quadratic forms, defining a birational map ψ : P 5 W ⊂ P 6 onto a quartic hypersurface and such that ψ(Σ) = Q ⊂ W ⊂ P 6 is a smooth quadric surface contained in the base locus scheme of ψ −1 .
(iii) The preimages on Σ of the lines of the two rulings of Q are, respectively, the planes of the ruling of Σ and the pencil of del Pezzo surfaces through C. (iv) The map ψ is an isomorphism outside Σ ∪ Sec(C) and Sec(C) is mapped onto a degree 24 surface T . The quartic hypersurface W has double points along Q and T and ψ −1 : W P 5 is given by the restriction of a linear system of quartic hypersurfaces in P 6 passing simply through T and having double points along Q.
Proof. Let C ⊂ P 5 be an irreducible curve of degree 8 and geometric genus 2 with a node. Let ν : C ′ → C be its normalization and let L = ν * (O C (1)). The linear system |H 0 (L)| has dimension six by Riemann-Roch, it is very ample and it embeds C ′ in P 6 as a smooth curve of degree 8 and genus 2. The curve C ⊂ P 5 is the projection of C ′ from a point q on the secant variety Sec(C ′ ) ⊂ P 6 of C ′ but not belonging to the tangential surface Tan(C ′ ).
Let P ′ = p ′ 1 , . . . , p ′ 4 , p ′ i ∈ C ′ , be a four secant P 3 passing through q and let P = p 1 , . . . , p 4 , with p i ∈ C the projection from q of p ′ i . Then P is a four secant plane to C and, letting D = p 1 + · · · + p 4 be the corresponding Cartier divisor on C, a general hyperplane through P cuts C in D and in other 4 points p 5 , . . . , p 8 such that Π = p 5 , . . . , p 8 = P 3 . Letting E = p 5 + · · · + p 8 , from the previous definitions and from Riemann-Roch we deduce h 0 (O C (D)) = 2 and h 0 (O C (E)) = 3. The linear system |H 0 (O C (D))| defines a morphism ξ : C → P 1 of degree 4 while |H 0 (O C (E))| defines a morphism η : C → P 2 birational onto its image. By the universal property of the product we have a morphism ξ × η : C → P 1 × P 2 which composed with the Segre embedding P 1 × P 2 ⊂ P 5 gives a morphism ǫ : . This means that the embedding C ⊂ P 5 factors through Σ ≃ P 1 × P 2 ⊂ P 5 in such a way that ξ is the composition of ǫ with the projection onto the first factor and that η is the composition of ǫ with projection onto the second factor. In particular P = p × P 2 ⊂ Σ for some p ∈ P 1 . By Riemann-Roch we deduce h 0 (I C (2)) ≥ 7 so that h 0 (I C∪P (2)) ≥ 5 (from P ∩ C = {p 1 , . . . , p 4 } we deduce that to contain P imposes only two conditions to quadrics vanishing on C). Since h 0 (I Σ (2)) = 3, there exists at least a pencil of quadrics {Q λ } λ∈P 1 vanishing on C ∪ P but not on Σ. This pencil of quadrics cuts Σ along P and along a residual pencil of divisor {S λ } λ∈P 1 of type (1, 2) containing C. The projection from P maps Σ onto a plane Π and C onto a plane quartic curve. Hence C is not contained in any divisor of type (0, 2) on Σ (they project from P onto a conic) and being non degenerate in P 5 is not contained in any divisor of type (1, 1) or (1, 0) or (0, 1), proving that every S λ is irreducible. Since the complete intersection of two distinct divisors of type (1, 2) on Σ without common irreducible components is a curve of degree 8 and arithmetic genus 3, we conclude that C is the complete intersection of two surfaces in the pencil {S λ } λ∈P 1 . The projection from P restricted to S λ resolves to a birational morphism σ λ : S λ → Π. The planes in Σ cuts on S λ a pencil of conics, whose image by σ λ is a pencil of conics having a base locus scheme Z λ ⊂ Π of length 4. The inverse map σ −1 λ : Π S λ is given by the linear system of cubics vanishing at Z λ . LetC ⊂ Π denote the projection of C from P . Clearly Z λ ⊂C and the divisors {Z λ } λ∈P 1 vary in a g 1 4 onC. Moreover, the pencil of conics through a fixed Z λ cuts onC the g 1 4 = |H 0 (OC ((σ λ ) * (D)))|. On the contrary, fixed a general divisor D µ in the last g 1 4 , the pencil of conics throughD µ cuts onC the g 1 4 = {Z λ } λ∈P 1 . Hence for general λ ∈ P 1 the scheme Z λ is smooth by Bertini Theorem and the corresponding S λ ⊂ P 5 is a smooth quintic del Pezzo surface. Thus, part (i) is proved. Let C = S ∩ S ′ ⊂ Σ be a curve of degree 8 and arithmetic genus 3, complete intersection of a pencil {S λ } λ∈P 1 of divisors of type (1, 2) on Σ, whose general member is smooth. Arguing as in [19, pp. 435-436], an iterated use of the Mapping Cone and the knowledge of the resolution of a S λ ⊂ P 5 yields that C ⊂ P 5 has ideal generated by seven quadratic equations (more precisely we can obtain the complete free resolution of the ideal of C ⊂ P 5 , see loc. cit.). Let ψ : P 5 P 6 be the map defined by |H 0 (I C (2))|. The image W = ψ(P 5 ) ⊂ P 6 is a quartic hypersurface, see [38] and [19]. The restriction of ψ to Σ is a linear system of dimension three because Σ is defined by three quadratic equations vanishing on C. Let S ⊂ Σ be a quintic del Pezzo surface through C and let π : S ≃ Bl {q 1 ,...,q 4 } P 2 → P 2 be a birational representation such that the pencil of conics on S given by the restriction of the projection onto the first factor of Σ is mapped on P 2 to the pencil of conics through q 1 , . . . , q 4 . Then π(C) is a quartic curve passing through q 1 , . . . , q 4 . Hence the free part of the restriction of |H 0 (I C (2))| to S is given by the pencil of strict transforms of conics passing through q 1 , . . . , q 4 and ψ(S) is a line contained in Q = ψ(Σ) ⊂ W . In particular, Q ⊂ P 3 ⊂ P 6 is a surface. Since each plane Π ⊂ Σ cuts C in four points, ψ maps Π onto a line in Q cutting ψ(S) in the point corresponding to ψ(Π ∩ S). In conclusion Q ⊂ P 3 is a smooth quadric surface and all the claims in (ii) and in (iii) are proved. A finer analysis of the map ψ, as in [38,Section 4] or in [19, Section 3, Lemma 3.3, Lemma 3.4], leads to the description of all the positive dimensional fibers of ψ and of all the properties listed in part (iv). We refer to [38, p. 208] for the details on these computations.
The following result will play a crucial role in our geometric constructions together with the description of the map ψ defined above.
Proposition 5.5. Let B ⊂ P 5 be a rational scroll of dimension 3 and degree 4 which is a general projection of a smooth quartic rational normal scroll B ′ ⊂ P 6 . Then: (i) there exists an irreducible family F of dimension 15 of curves of degree 8 and geometric genus 2 on B, whose general member is nodal. (ii) There exists an irreducible family D of dimension 16 of quintic del Pezzo surfaces in P 5 , whose general member is smooth and cuts B along a general curve of the family F.
Let C ′ ⊂ B ′ be a curve of degree 8 and genus 2. Then −K B ′ · C ′ = 18, deg(N C ′ /B ′ ) = 20 by Adjunction Formula and χ(N C ′ /B ′ ) = 20 + 2(1 − g(C ′ )) = 18. Since in an explicit example we verified that h 0 (N C ′ /B ′ ) = 18 and that h 1 (N C ′ /B ′ ) = 0, the general curve of degree 8 and genus 2 contained in B ′ belongs to a unique irreducible and generically smooth component of dimension 18 of the corresponding Hilbert scheme. The family of the secant varieties Sec(C ′ ) with C ′ ⊂ B ′ has dimension 21 = 18 + 3, so through a general point q ∈ P 6 there passes a family of dimension 15 of such secant varieties. The projection from q of the corresponding curves produces a 15-dimensional family of nodal curves of degree 8 and geometric genus 2 on the scroll B, projection of B ′ from q. Since one verifies that h 0 (N C/B ) = 15 (a fact which can be easily computed with Macaulay2 ), the family F of such curves is irreducible, generically smooth and of dimension 15. By Proposition 5.4, a nodal curve C ⊂ B ⊂ P 5 of degree 8 and geometric genus 2 with a node is the complete intersection of a pencil of del Pezzo surfaces on a Segre threefold Σ ≃ P 1 × P 2 ⊂ P 5 . Let C ⊂ S = S λ ⊂ Σ with λ ∈ P 1 general and let π : S = Bl {q 1 ,...,q 4 } P 2 → P 2 be the blow-up morphism. Then we can assume that π(C) =C ⊂ P 2 is a plane quartic curve with a node passing simply through q 1 , . . . , q 4 , see the proof of Proposition 5.4. The irreducible threefold B has ideal generated by one quadratic form and by three cubic forms so that it is scheme theoretically defined by nine cubic forms. To study the scheme theoretic intersection B ∩ S we restrict to S the linear system |H 0 (I B (3))|. Each cubic in this linear system cuts S along a curve D containing C and such that π(D) =D is a curve of degree nine having triple points at q 1 , . . . , q 4 . HenceD =C +Ã withÃ a quintic curve with double points at q 1 , . . . , q 4 . The linear system |Ã| has dimension eight, it is base point free and very ample, proving that B ∩ S = C as schemes for [C] ∈ F general. The linear system |Ã| is equivalent via a quadratic standard Cremona transformation centred at q 2 , q 3 , q 4 to the linear system of quartic curves having a double point at q 1 and simple base points at q 2 , q 3 , q 4 .
By varying C and recalling that any C is a complete intersection of a pencil of quintic del Pezzo surfaces inside Σ, we get a family D of dimension 16 of quintic del Pezzo surfaces cutting B scheme theoretically along a nodal curve of degree 8 and genus 2 as above.

5.3.
A rational surface of degree 9 and sectional genus 2 with 5 nodes contained in a general cubic fourfold of C 42 . We can now construct a 25-dimensional family of smooth surfaces of degree 9 and sectional genus 2 on a del Pezzo fivefold V ⊂ P 8 . This was also achieved in [18, Lemma 3.1] via a different construction of an explicit example that corresponds to a smooth point in Hilb V and which is related to a K3 surface of genus 11 in P 11 . Proposition 5.6. Let V ⊂ P 8 be a del Pezzo fivefold.
(i) There exists an irreducible and generically smooth family S ⊂ Hilb V of dimension 25 of surfaces S ⊂ V of degree 9 and sectional genus 2, whose general member is smooth. (ii) There exists an irreducible and generically smooth family S 42 ⊂ Hilb P 5 of dimension 48 of surfaces in P 5 of degree 9 and sectional genus 2 with 5 nodes, whose general member is obtained as the projection from a general plane of the family F 1 ⊂ Hilb V of a general surface of the family S ⊂ Hilb V .
Proof. Let C ⊂ V be an irreducible conic whose linear span Π = C is not contained in the del Pezzo fivefold V , and let be the inverse of the projection from Π (see Proposition 5.3). Let D be the family of quintic del Pezzo surfaces described in Proposition 5.5, and let D ∈ D be a general member. The smooth surfaces S = α(D) ⊂ V ⊂ P 8 have degree 9 and sectional genus 2 and are obtained from P 2 via the linear system of quintics having four double points (or equivalently via the linear system of quartics with a double point and three simple base points) as shown in the proof of Proposition 5.5. Since the Hilbert scheme Con of conics in V has dimension 10 (see Remark 5.2) and since the conics on such a surface S ⊂ P 8 belong to the unique pencil of conics on S (represented on the plane by the conics through the four base points of the linear system), we deduce that the above surfaces describe a family S ⊂ Hilb V of dimension at least dim(Con) − 1 + dim(D) = 10 − 1 + 16 = 25. Since we have verified in a specific example of surface [S] ∈ S that h 0 (N S/V ) = 25, we deduce that the family S is generically smooth of dimension 25. Let β : V P 5 be the birational map induced by the projection from a plane P ⊂ V of the family F 1 , see Proposition 5.3. The image via β of a general S ∈ S is a surface of degree 9, sectional genus 2, cut out by 9 cubics and having 5 nodes produced by the intersection of P with the secant variety to S, a fact which can be verified by a direct computation via Macaulay2 by following the first steps of the algorithm described in Remark 5.8 below (see also Subsection 6.2).
A surface in P 5 of degree 9, sectional genus 2 and with five nodes of the type constructed above will be denoted by S 42 . Since the cubic rational normal scrolls in P 4 depend on 18 parameters, we deduce that the cubic rational normal scrolls in P 5 depend on 23 parameters. Each rational normal scroll in P 5 determines a map β and hence a plane P ⊂ V ⊂ P 8 . The projection from P of the 25 dimensional family of surfaces of degree 9 and sectional genus 2 contained in V gives a 25 dimensional family of S 42 . By varying the scroll we deduce that the surfaces S 42 ⊂ P 5 describe a family of dimension at least 48=23+25. Since we have verified in a specific example of surface S 42 ⊂ P 5 that h 0 (N S 42 /P 5 ) = 48, we can deduce that h 0 (N S 42 /P 5 ) = 48 for a general S 42 ⊂ P 5 as above. Since we previously proved that these surfaces depend on at least 48 parameters, we conclude that there exists a unique irreducible component S 42 of the corresponding Hilbert scheme containing the surfaces S 42 ⊂ P 5 , which is generically smooth of dimension dim(S 42 ) = 48 and such that the general element of S 42 is of the kind described above.
Remark 5.7. Let V = G(1, 4) ∩ P 8 ⊂ P 8 be a del Pezzo fivefold, let F 1 ⊂ Hilb V be the 3-dimensional family of planes in V with class σ 2,2 , and let S ⊂ Hilb V and S 42 ⊂ Hilb P 5 be, respectively, the 25-dimensional family of smooth surfaces in V of degree 9 and sectional genus 2, and the 48-dimensional family of 5-nodal surfaces in P 5 of degree 9 and sectional genus 2 constructed in Proposition 5.6. To a general pair ([S], [P ]) ∈ S × F 1 , we associate a general [S 42 ] ∈ S 42 defined by S 42 = β P (S), where β P : V P 5 denotes the projection from P . The inverse map α ∆ = β −1 P : P 5 V is defined by the quadrics through a rational normal cubic scroll ∆ ⊂ P 4 ⊂ P 5 , which intersects S 42 ⊂ P 5 in a curve C ⊂ P 5 of degree 9 and arithmetic genus 7. We now illustrate how one can determine this scroll ∆ from the surface S 42 , and thus determine the pair (S, P ) via S = α ∆ (S 42 ) and P = Bs(α −1 ∆ ). Let ϕ : P 5 Z ⊂ P 8 be the rational map defined by the linear system |H 0 (I S 42 (3))| of cubic hypersurfaces through the surface S 42 ⊂ P 5 . Then ϕ is a birational map onto its image Z ⊂ P 8 and the base locus of the inverse map ϕ −1 : Z P 8 is an irreducible surface T with an immersed point q ∈ T . Then ϕ −1 (T ) is a threefold ruled by trisecant lines to S 42 while ϕ −1 (q) coincides with the scroll ∆ intersecting S 42 along a curve of degree 9 and arithmetic genus 7. 2 The previous description of ϕ assures that a general S 42 ⊂ P 5 satisfies Assumption 1 in Subsection 2.2 and that it has the expected trisecant behaviour. 2 The idea of gluing a cubic scroll ∆ ⊂ P 5 with another surface B ⊂ P 5 along some curve ∆ ∩ B and then of considering the image α∆(B) ⊂ V has been systematically applied in [41] to construct other types of surfaces in a del Pezzo fivefold.
Remark 5.8. Our construction of the surface S 42 can be easily implemented and executed in a computer program such as Macaulay2. For the convenience of the reader, we now summarize the algorithm for the construction of the general surface in the family S 42 (see also Subsection 6.2).
• Let L be a general secant line to T , H ⊃ L a general hyperplane through L, and p ∈ L a general point. Then the projection of C ′ = H ∩ T ⊂ H = P 6 to P 5 from p yields a one-nodal curve C ⊂ P 5 of degree 8 and arithmetic genus 3 contained in a singular quartic scroll threefold B ⊂ P 5 , projection from p of B ′ = P 1 × P 3 ∩ H. • The quadrics through C define a birational map ψ from P 5 into a quartic hypersurface W ⊂ P 6 . The exceptional locus of ψ contains a Segre threefold Σ ≃ P 1 × P 2 ⊂ P 5 , which is sent into a smooth quadric surface Q ⊂ P 6 . The quadric Q ⊂ W can be detected as the unique irreducible component of the base locus scheme of ψ −1 along which the base locus scheme is not generically reduced. • The inverse images via ψ of the two lines of Q passing through a general point of Q are: a plane of the ruling of Z and a smooth quintic del Pezzo surface D ⊂ P 5 , which intersects B along C. (Note that Σ and the quadric Q = ψ(Σ) are rational over their field of definition. Indeed, the syzygy matrix of the 3 quadrics defining a Segre threefold Σ ⊂ P 5 K is a 2 × 3 matrix of linearly independent linear forms on P 5 K . These 6 linear forms can be used to define an automorphism of P 5 K that sends Σ into the Segre embedding of P 1 K × P 2 K in P 5 K .) • Finally, the map α : P 5 V defined in (5.1) induces an isomorphism between D and a smooth surface S of degree 9 and sectional genus 2. The map β : V P 5 defined in Proposition 5.1 induces a birational morphism from S to our surface S 42 .
We are now ready to prove that a general cubic in C 42 contains a general surface S 42 ⊂ P 5 of degree 9 and genus 2 with five nodes in the irreducible family S 42 described above.
) at the point [S], to show that a general [X] ∈ C 42 contains a surface S 42 it is sufficient to verify that h 0 (N S 42 /X ) = 2 for a fixed S 42 and for a smooth X ∈ |H 0 (I S 42 (3))|, see also [29, pp. 284-285] for a similar argument. We verified this via Macaulay2 in an explicit example and we can conclude that a general [X] ∈ C 42 contains a surface S 42 as above.
Since at each step of the algorithm summarized in Remark 5.8 we need to introduce only new independent variables, we deduce that the family S 42 is unirational. In other words, our construction yields an explicit dominant rational map P N S 42 . As an immediate consequence of this and of Theorem 5.9, we have the following: Corollary 5.10. The irreducible divisor C 42 is unirational.
Remark 5.11. A different description of the divisor C 42 had been given in [24] as the locus of cubic fourfolds containing a rational scroll of degree 9 with 8 nodes. From this Lai deduces that C 42 is uniruled. This has been substantially refined in [ Proof. Since we have given an algorithm for computing the general surface S 42 ⊂ P 5 of the family S 42 (see Remark 5.8), we can explicitly construct it using Macaulay2 and study its geometrical properties. So let ϕ : P 5 Z ⊂ P 8 be the rational map defined by the linear system |H 0 (I S 42 (3))| of cubic hypersurfaces through a general surface S 42 ⊂ P 5 of the family S 42 . One can calculate that the map ϕ is birational onto its image Z ⊂ P 8 , which has degree 14, sectional genus 15 and ideal generated by 7 cubic forms. Through a general point q = ϕ(p) there passes 17 lines contained in Z, whose preimages via ϕ provide: nine secant lines to S 42 through p, seven 5-secant conics to S 42 through p and one 8-secant twisted cubic to S 42 through p. Then we conclude that S 42 admits a congruence of 8-secant twisted cubics. Once we have determined the congruence we have also another way of detecting it. Indeed, the linear system |H 0 (I 3 S 42 (8))| of octic hypersurfaces with triple points along S 42 defines a dominant rational map µ : P 5 W ⊂ P 7 onto a smooth linear section W of G(1, 4) ⊂ P 9 . The general fiber of µ is a twisted cubic 8-secant to S 42 so that the restriction of µ to a general cubic fourfold X through S 42 induces a birational map µ| X : X W . Since a general cubic fourfold in |H 0 (I S 42 (3))| through a general surface in S 42 is rational, we conclude that every cubic fourfold of discriminant 42 is rational by the main result in [21]. We refer to Subsection 6.2 for more details on the above calculations. 5.5. Birational model of the associated K3 surface of degree 42 and genus 22 via the trisecant flop. A general surface S 42 ⊂ P 5 satisfies Assumption 1 in Subsection 2.2 and it has the expected trisecant behaviour, see the end of Remark 5.7. By Theorem 2.6 the map µ restricted to a general cubic X ⊂ P 5 through S 42 determines a trisecant flop τ : X ′ = Bl S 42 X W ′ with W ′ a smooth fourfold. By Theorem 5.12 and by Theorem 2.10, the congruence of 8-secant twisted cubics to S 42 induces a birational morphism ν : W ′ → W , which is the blow-up of a surface B ⊂ W ⊂ P 7 .
By studying the birational map µ : X W (see Subsection 6.2), we obtain that B ⊂ W ⊂ P 7 is a smooth surface of degree 21 and sectional genus 18. Then B coincides with its support U , which is thus a smooth surface of degree 21 and sectional genus 18 with p g (U ) = 1 and q(U ) = 0. We now apply the geometric method developed in Section 4 to deduce that U is the blow-up of a K3 surface at nine distinct points and to describe the linear system giving the embedding.
Considers two surfaces U j ⊂ W , j = 1, 2 associated via µ to two general X j 's through S 42 . The one dimensional component C of U 1 ∩ U 2 is a smooth curve of degree 13, of arithmetic genus −8 with 9 irreducible components (a general projection of C to P 3 allows a quick verification of all these properties), yielding that C = C 1 ∪ · · · ∪ C 9 is the disjoint union of nine smooth rational curves. More precisely, the curve C ⊂ P 7 consists of five distinct lines, let us say C 1 , . . . , C 5 , and of four conics C 6 , . . . , C 9 (see Subsection 6.2 and the ancillary file code_section_6.m2). Since h 1 (N C/U j ) = 0, the C i 's are (−1)-curves on U j for every i = 1, . . . , 9 and for j = 1, 2 by Lemma 4.3. Let U = U 1 and let H ⊂ U be a hyperplane section. Since H · K U = −H 2 + 2g(H) − 2 = −21 + 36 − 2 = 13 = H · C we deduce from Lemma 4.3 the existence of the contraction π : U → U ′ of the C i 's to 9 distinct points p 1 , . . . , p 9 on a smooth K3 surface U ′ . The divisor H ′ = π * (H) ⊂ U ′ is ample and such that (H ′ ) 2 = 21 + 16 + 5 = 42, g(H ′ ) = 18 + 4 = 22, proving that U ′ is a K3 surface of degree 42 and genus 22. For a general X through S 42 we have that U ′ is a K3 surface of degree 42 and genus 22 with rk(Pic(U ′ )) = 1 (one can argue as at the end of Subsection 3.1 or remark that U ′ has necessarily 19 moduli equal to the moduli of a general X ∈ C 42 ) so that |H ′ | gives an embedding U ′ ⊂ P 22 . The linear system |H| on U corresponds to the hyperplane sections of U ′ passing through p 1 , . . . , p 5 and having a point of multiplicity at least two at p 6 , . . . , p 9 . In particular, we see that these points do not impose independent conditions to hyperplane sections of U ′ ⊂ P 22 .
By considering the map associated to the linear system |H +C 1 +· · ·+C 5 +2(C 6 +· · ·+C 9 )| on U , we construct its image U ′ ⊂ P 22 , which is thus a general K3 surface of degree 42 and genus 22, see the ancillary file code_section_6.m2. From this one can deduce an alternative proof of one of the main results in [9], according to which the moduli space of polarized K3 surfaces of degree 42 and genus 22 is unirational.

Explicit examples of trisecant flops in Macaulay2
We mostly used the computer algebra system Macaulay2 [13] with the packages Cremona [40] and SpecialFanoFourfolds [39] to study surfaces in P 5 admitting congruences of (3e − 1)secant curves of degree e and the rational maps given by hypersurfaces of degree 3e − 1 having points of multiplicity e along these surfaces. We refer to the documentations of these two packages for technical computational details. In particular, the first one provides tools for working with rational maps, such as computing their fibers, checking birationality and determining inverse maps. The validity of these computations relies on the fact that the irreducible components S d of the Hilbert schemes considered here are explicitly unirational. This means that we have a procedure to determine the equations of the generic member of S d as a function of a number of specific independent variables. Therefore, by adding more variables we can also take the generic point of P 5 and compute, for instance, the generic fiber of the map defined by the cubics through the generic [S d ] ∈ S d , which will depend on all these variables. In practice this is far beyond what computers can do today. Anyway, the answer we get is equivalent to the one obtained on the original field via a generic specialization of the variables and, above all, the generic specialization commutes with this type of computation.
6.1. Rationality of cubic fourfolds in C 38 . Here, we consider a specific example related to Subsection 3.1 (row (iii) of Table 1).
In the following code, we produce a surface S = S 38 ⊂ P 5 obtained as the image of P 2 by the linear system of plane curves of degree 10 having 10 randomly-chosen triple points. We work over the finite field K = F 10000019 for speed reasons. We now compute the rational map µ defined by the linear system of quintic hypersurfaces of P 5 which are singular along S. 3 From the information obtained by its projective degrees we deduce that µ is a dominant rational map onto P 4 with generic fibre of dimension 1 and degree 2 and with base locus of dimension 3 and degree 5 2 − 19 = 6.
i6 : p = point source mu; --a random point on P^5 i7 : F = mu^* mu p; It easy to verify directly that F is an irreducible 5-secant conic to S passing through p. Letting ϕ : P 5 P 9 the rational map defined by the linear system of cubics through S, one can also see that F coincides with the pull-back ϕ −1 (L) of the unique line L ⊂ ϕ(P 5 ) ⊂ P 9 passing through ϕ(p) that is not the image of a secant line to S passing through p (see [35,Section 5] for details on this computation). Finally, the following lines of code tell us that the restriction µ ′ of µ to a randomly-chosen cubic fourfold X containing S is a birational map whose inverse map is defined by forms of degree 9 and whose base locus scheme has dimension 2 and degree 9 2 − 27 = 54.
i8 : X = random(3,S); o8 : ProjectiveVariety, hypersurface in PP^5 i9 : mu' = mu|X; 3 More generally, the command rationalMap(S,d,e) returns the rational map defined by a basis of the linear system |H 0 (I e S (d))| of hypersurfaces of degree d having points of multiplicity e along S.
o9 : RationalMap (rational map from X to PP^4) i10 : projectiveDegrees mu' o10 = {3, 15, 27, 9, 1} The smooth surface U ⊂ P 4 of degree 12 and sectional genus 14 determining the inverse map of µ ′ : X P 4 can be calculated as the non-reduced part of the base locus scheme of µ ′−1 . Alternatively, one can use the same method described in the next subsection. The full code to determine U , the exceptional curves on U , and the map U U ′ ⊂ P 20 onto the K3 surface U ′ ⊂ P 20 of degree 38 and genus 20 is included in the ancillary file code_section_6.m2. We omit it here for the sake of brevity.
6.2. Rationality of cubic fourfolds in C 42 . Here, we perform similar calculations as above, but considering a specific example related to Subsection 5.3 (row (0) of Table 1).
Using the algorithm given in Remark 5.8, one can calculate the homogeneous ideal of a randomly-chosen surface S = S 42 in the 48-dimensional family S 42 constructed in Proposition 5.6. This has been implemented in the Macaulay2 package SpecialFanoFourfolds. So, to get the ideal of such a surface S and of a randomly-chosen (smooth) cubic fourfold X containing it, it is enough to run the following code: i11 : X = specialCubicFourfold("general cubic 4-fold of discriminant 42",K); o11 : ProjectiveVariety, cubic fourfold containing a surface of degree 9 and sectional genus 2 i12 : S = ideal surface X; The following is one of the ways to compute relatively quickly the rational map µ defined by the linear system of octic hypersurfaces of P 5 having triple points along S. This calculation takes about one minute.
i13 : mu = rationalMap(S^3 : first gens ring S,8); o13 : RationalMap (rational map from PP^5 to PP^7) Now, with the same code used above, we compute a special random fibre F of the map µ. Here is a practical way to get that F is a twisted cubic curve which is 8-secant to S. i16 : ? F o16 = smooth cubic curve of genus 0 in PP^5 cut out by 5 hypersurfaces of degrees (1,1,2,2,2) i17 : ? (F + S) o17 = 0-dimensional subscheme of degree 8 in PP^5 The code above also tells us that the (closure of the) image of µ is a subvariety W ⊂ P 7 of dimension 4. To get the equations of W , one can use the command image mu, but this takes a while. A faster way is to calculate the scheme of quadrics containing W , as shown below. Then, since one verifies easily that it is a smooth connected fourfold, we deduce that W coincides with this fourfold.
i18 : W = image(2,mu); The surface U ⊂ W ⊂ P 7 determining the inverse map of the restriction of µ to the cubic fourfold X can be find without computing the inverse map. Indeed, the intersection of W with a general cubic hypersurface through U is given by the image µ(X ∩ X ′ ) ⊂ W , where X ′ is a general cubic hypersurface through S. For efficiency, we suggest to calculate µ(X ∩ X ′ ) by interpolating the images via µ of several points on X ∩ X ′ . The following is a possible implementation. It gives the homogeneous ideal of U ⊂ P 7 and takes about 5 minutes.
i19 : U = trim sum(8,j->(X' = random(3,surface X); ideal take((intersect apply(90,i->mu ideal point(X * X')))_*,6))); The exceptional curves in the surface U can be determined by taking another randomlychosen cubic fourfoldX through S (for instance, as in line i8), and then by calculating the corresponding surfaceŨ (for instance, by re-executing the line i19). One expects that the exceptional curves inŨ are the same as those in U (see Section 4), so one tries to determine them as the top-dimensional components of the intersection U ∩Ũ (this can be done quickly after taking generic projections in P 3 and P 2 ). In the cases under consideration, the onedimensional components of this intersection are 9 disjoint curves: 5 lines C 1 , . . . , C 5 and 4 conics C 6 , . . . , C 9 , which are all the (−1)-curves on U (see also Subsection 5.5). Having determined these curves, one can also calculate the map f : U → U ′ ⊂ P 22 onto the K3 surface U ′ ⊂ P 22 of degree 42 and genus 22, given by the linear system |H where H denotes the hyperplane section of U . For the sake of brevity, the full code to get the map f and the equations of its image is omitted here and it is included in the ancillary file code_section_6.m2 together with all the lines of code given in this section. Note also that this kind of calculations can be automated using the function associatedK3surface provided by the package SpecialFanoFourfolds.  Table 1 in the next section. This package can be downloaded automatically from Special-FanoFourfolds. So, by typing trisecantFlop i (where i is an integer between 0 and 17), will build up a birational map µ : X W as in the i-th row of Table 1. For instance, we now consider the third example. i20 : mu = trisecantFlop 3; o20 : RationalMap (birational map from cubic fourfold containing a surface of degree 10 and sectional genus 6 to PP^4) i21 : projectiveDegrees inverse mu o21 = {1, 9, 27, 15, 3} We can obtain the smooth surface S ⊂ P 5 of degree 10 and sectional genus 6 by giving the following command.
i24 : U = surface target mu; o24 : ProjectiveVariety, surface in PP^4 i25 : (degree U, sectionalGenus U) o25 = (12,14) Finally, the following command yields an extension to P 5 of the map µ : X W = P 4 whose general fibre is a 5-secant conic to the surface S.

Summary table of examples of trisecant flops
We provide in Table 1 a list of 18 examples of maps µ : X W as in diagram (0.1), where X is a cubic fourfold in C d which contains a surface S ⊂ P 5 admitting a congruence of (3e − 1)-secant rational curves of degree e. There are some different behaviours: • [X] ∈ C d is general except in (xi) and (xii); • S ⊂ P 5 is cut out by cubics except in (xii), (xiii), (xiv), and (xvii); • the map X Y defined by the cubics through S is birational except in (xv); • the cubics through S satisfy the condition K 3 except in (0), (v), (viii), (x)-(xvii). In Table 2, we give some additional information on the examples of surfaces S ⊂ X ⊂ P 5 considered in Table 1. Most of this was achieved using the functions detectCongruence and parameterCount from the package SpecialFanoFourfolds, see also Subsection 6.3.
When the corresponding surface U ⊂ W is smooth, the proof that it is a (non-minimal) K3 surfaces follows the paths used in Section 4 (see also Subsection 5.5) by determining explicitly the exceptional curves on U . Once these exceptional curves are determined, one finds the description of the linear system on U in terms of the hyperplane sections of the K3 surface U ′ . When U ⊂ W is singular, one first determines the exceptional curves as above; then takes a linear normalization to obtain a smooth surface and finally, if necessary, follows the previous path. Rational surface of degree 9 and sectional genus 2 with 5 nodes, which is a special projection of the image of P 2 in P 8 via the linear system of quartic curves with 3 simple points and one double point Non-minimal K3 surface of degree 21 and sectional genus 18, cut out in P 7 by 5 quadrics and 8 cubics 4-fold of degree 14 in P 7 cut out by 7 cubics i 14 2 Isomorphic projection of a smooth surface in P 6 of degree 8 and sectional genus 3, obtained as the image of P 2 via the linear system of quartic curves with 8 general base points General hyperplane section of a conic bundle in P 6 of degree 13 and sectional genus 12 Complete intersection of three quadrics in P 7 Smooth non-minimal K3 surface of degree 13 and sectional genus 8, cut out by 9 quadrics Hypersurface of degree 5 in P 5 ix 14 5 General hyperplane section of a pfaffian threefold in P 6 of degree 14 and sectional genus 15 Smooth minimal K3 surface of degree 14 and sectional genus 8 embedded in P 8 ⊂ P 14 Hypersurface of degree 5 in P 5 x 38 5 Smooth surface of degree 11 and sectional genus 7, obtained as the image of P 2 via the linear system of curves of degree 12 with one general simple point, 4 general triple points, and 6 general quadruple points G(1, 5) ∩ P 10 ⊂ P 10 Smooth non-minimal K3 surface of degree 25 and sectional genus 17, cut out in P 10 by 21 quadrics Hypersurface of degree 7 in P 5 xi 38 3 Projection of an octic del Pezzo surface isomorphic to F1 from a plane intersecting the secant variety in 3 general points G(1, 3) ⊂ P 5 Non-minimal K3 surface of degree 13 and sectional genus 10, cut out in P 5 by one quadric, 9 quartics, and 3 quintics 4-fold of degree 17 in P 8 cut out by 3 quadrics and 4 cubics xii 38 3 Projection of an octic del Pezzo surface isomorphic to F0 from a plane intersecting the secant variety in 3 general points (cut out by 10 cubics and one quartic) LG3(C 6 ) ∩ P 11 ⊂ P 11 Non-minimal K3 surface of degree 26 and sectional genus 17, cut out in P 11 by 30 quadrics 4-fold of degree 18 in P 8 cut out by 2 quadrics and 8 cubics xiii 14 3 Isomorphic projection of a smooth surface in P 7 of degree 8 and sectional genus 2, obtained as the image of P 2 via the linear system of quartic curves with 4 simple base points and one double point (cut out by 10 cubics and 3 quartics) Complete intersection of 2 quadrics in P 6 Singular K3 surface of degree 14 and sectional genus 8, cut out in P 6 by 2 quadrics and 9 cubics, and having one singular point Non-minimal K3 surface of degree 14 and sectional genus 11, cut out in P 5 by one quadric, 7 quartics, and 2 quintics Complete intersection in P 6 of a quadric and a quartic xv 26 5 Surface of degree 13 and sectional genus 11 cut out by 6 cubics and with an ordinary node, which is obtained as a special projection of a minimal K3 surface of degree 26 of genus 14

Cubic fourfold
A surface of the same kind as S xvi 26 6 Surface of degree 11 and sectional genus 6 cut out by 7 cubics and with 3 non-normal nodes, which is obtained as a special projection of a smooth surface of degree 11 and sec. genus 6 in P 6 S 10 ∩ P 9 ⊂ P 9 , where S 10 ⊂ P 15 is the spinorial variety Non-minimal K3 surface of degree 21 and sectional genus 14, cut out in P 9 by 16 quadrics and one cubic Hypersurface of degree 5 in P 5 xvii 26 5 Smooth surface of degree 11 and sectional genus 7, obtained as the image of P 2 via the linear system of curves of degree 8 with 3 simple base points, 8 general double points, and 2 general triple points (cut out by 7 cubics and one quartic) Singular K3 surface of degree 15 and sectional genus 12, cut out in P 5 by one quadric and 6 quartics, and having 9 singular points Hypersurface of degree 6 in P 5 Table 1. Examples of maps µ : X W as in diagramm (0.1), where [X] ∈ C d and S ⊂ X admits a congruence of (3e − 1)-secant rational curves of degree e.