The cotangent bundle of K3 surfaces of degree two

. K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface S of degree two. In particular, we describe the geometry of a surface D S ⊂ P ( Ω S ) that plays a similar role to the surface of bitangents for a quartic in P 3 .


Motivation
The cotangent bundle of a K3 surface S is well understood from the point of view of stability theory: we know that Ω S is stable for every polarisation L. Moreover there are effective bounds guaranteeing that the restriction Ω S ⊗ O C is stable for every irreducible curve C ∈ |dL|; see [Hei06,Fey16]. Nevertheless these stability results do not provide a complete description of the positivity properties of Ω S . In fact, the cotangent bundle of a K3 surface is never pseudoeffective (cf. [Nak04, BDP + 13], and [HP19] for a more general result).
A way of measuring the negativity of the cotangent bundle is to describe the pseudoeffective cone of the projectivised cotangent bundle π : P(Ω S ) → S. Denote by ζ S → P(Ω S ) the tautological class, and let α S be a Kähler class on S. In our recent paper [AH21] we showed that if α 2 S ≥ 8, then ζ S + π * α S is pseudoeffective, the bound being optimal for a very general non-projective K3 surface; see [AH21,Theorem 1.5]. The bound is also optimal for infinitely many, but not all, families of projective K3 surfaces with Picard number one, see [GO20,Theorem B], which leads us to a more geometric question.
Question 1.1. Let S be a projective K3 surface, and let L be an ample Cartier divisor on S. Assume that ζ S + λπ * L is pseudoeffective for some λ < 8 L 2 . Can we relate the pseudoeffectivity of ζ S + λπ * L with the projective geometry of (S, L)?
In view of the arguments used by Lazić and Peternell [LP20], we might ask even more boldly for a relation between Pseff(P(Ω S )) and families of elliptic curves on S. Gounelas and Ottem [GO20] give a natural framework for these questions: the embedding P (Ω S ) ⊂ S [2] in the Hilbert square allows one to use results on the pseudoeffective cone of S [2] by Bayer and Macrì [BM14]. For example this approach allows one to recover the following classical result. Theorem 1.2 (cf. [Tih80,Wel81], [GO20,Section 4]). Let S ⊂ P 3 be a smooth quartic surface with Picard number one, and denote by L the restriction of the hyperplane class. Then the surface of bitangents U ⊂ P(Ω S ) has class 6ζ S + 8π * L and generates an extremal ray of Pseff(P(Ω S )).
The goal of this paper is to give an analogue of Theorem 1.2 in the case of a very general polarised K3 surface (S, L) of degree two; i.e. S is a K3 surface with Picard number one obtained as a two-to-one cover f : S −→ P 2 2 with ramification divisor a smooth curve of genus ten. Oguiso and Peternell [OP96] observed that in this case, the projectivised cotangent bundle should have some exceptional properties that are not representative for general K3 surfaces. The approach of Gounelas and Ottem allows one to determine the nef cone of P(Ω S ) [GO20, Section 4.1] but yields only that ζ S + π * 2L is pseudoeffective without determining the extremality in Pseff(P(Ω S ). Note that α S := 2L has α 2 S = 8, so this situation corresponds exactly to the set-up of Question 1.1.

Main results
Let S be a very general K3 surface of degree two, and let f : S → P 2 be the double cover. Let d ⊂ P 2 be a line that is a simple tangent of the branch divisor B; then its preimage C := f −1 (d) has a unique node, so the normalisation n :C → C is a smooth elliptic curve. The natural surjection n * Ω S −↠ ΩC determines a morphismC → P(Ω S ) which we call the canonical lifting of C ∈ |L| (cf. Section 3.3). The surface D S itself is very singular; the description of the normalisation occupies the larger part of this paper. We will see that D S contains a lot of geometric information about the K3 surface; for example it contains a curve isomorphic to the branch divisor B in its non-normal locus. While the surface D S plays a similar role to the surface of bitangents for the quartic surfaces, it does not generate an extremal ray in the pseudoeffective cone. with λ ≤ 1.7952024. (1) Moreover let Z S ⊂ P(Ω S ) be a prime divisor with this property. Then the canonical liftings of C node,S ⊂ P(Ω S ) of the 324 rational curves C in |L| are contained in Z S .
From a numerical point of view, both parts of this statement come as a surprise: • The non-nef locus of the divisor ζ S + 1.8π * L consists of a unique curve R S ⊂ P(Ω S ). If we denote by µ S : Y → P(Ω S ) the blow-up of this curve (cf. Section 3.1), the strict transform D ⊂ Y of D S generates an extremal ray in Pseff(Y ). Nevertheless D S itself is a big, non-nef divisor with D 3 S < 0. • The divisor Z S satisfies Z S · C node,S > 0, so we would not expect these curves to be contained in the stable base locus. This property will be a consequence of our investigation of the threefold Y → P(Ω S ). While the class of the surface D S does not generate an extremal ray in Pseff(P(Ω S )), the detailed geometric study involved in the proof of Theorem 1.3 allows one to give a pretty sharp estimate for the extremal ray.
(1) The exact statement is

Notation and general set-up
We work over C. For general definitions, we refer to [Har77] and [Voi02]. All the schemes appearing in this paper are projective; manifolds and normal varieties will always be supposed to be irreducible. For notions of positivity of divisors and vector bundles, we refer to Lazarsfeld's books [Laz04a,Laz04b]. Given two Cartier divisors D 1 , D 2 on a projective variety X, we denote by D 1 ≃ D 2 (resp. D 1 ≡ D 2 ) the linear equivalence (resp. numerical equivalence) of the Cartier divisor classes, while D 1 = D 2 is used for an equality of the cycles. We will frequently identify an effective divisor with its cohomology class in N 1 (X), where N 1 (X) := NS(X) ⊗ R is the Néron-Severi space of R-divisors.
In the whole paper we will work in the following setting.
Set-up 2.1. Let S be a very general polarised K3 surface of degree two, and let L be the primitive polarisation on S. We denote by f : S −→ P 2 the double cover defined by the linear system |L|, so we have L ≃ f * H 1 , where H 1 is the hyperplane class on P 2 . Since |L| = f * |H 1 |, every curve C ∈ |L| is a double cover of a line d ⊂ P 2 . We denote by B ⊂ P 2 the branch locus of f and by R ⊂ S the ramification divisor. By the ramification formula we know that B is a sextic curve, so Since f * B = 2R, we see that the ramification divisor R is an element of |3L|. Denote by π : P(Ω S ) −→ S the projectivisation of the cotangent bundle and by ζ S the tautological class. We denote by l ⊂ P(Ω S ) a fibre of π.
This set-up will become increasingly rich through a series of geometric constructions which we will summarise in Diagrams (3.4) and (3.8).
Remark. Many of our arguments are valid for an arbitrary smooth polarised K3 surface of degree two. However the assumption that S is very general is necessary to ensure that the Picard number of S is one. Moreover this implies that the branch curve B ⊂ P 2 satisfies the assumptions of Plücker's theorem, which is crucial in Section 3.2 and in the proof of Theorem 3.18.

Elementary transform of Ω S
Consider the projective plane P 2 and its hyperplane class H 1 . We denote by p 1 : P(Ω P 2 ) → P 2 the projectivisation and by ζ P 2 its tautological class. Let us recall some elementary facts: twisting the Euler sequence by O P 2 (−1) and recalling that T P 2 ≃ Ω P 2 ⊗ K * P 2 ≃ Ω P 2 ⊗ O P 2 (3), we obtain an exact sequence Thus the vector bundle Ω P 2 ⊗ O P 2 (2) is globally generated, and h 0 (P 2 , Ω P 2 ⊗ O P 2 (2)) = 3.
The sequence above shows that there is an inclusion P(Ω P 2 ) ⊂ P 2 × (P 2 ) ∨ , and if we denote by p 2 : P(Ω P 2 ) → (P 2 ) ∨ the fibration defined by the projection on the second factor, it allows us to identify P(Ω P 2 ) with the universal family of lines on P 2 ; i.e. for a point y ∈ (P 2 ) ∨ , the 5 curve p 1 (p −1 2 (y)) ⊂ P 2 is the line corresponding to the point y. The divisor class ζ P 2 + 2p * 1 H 1 defines a base-point-free linear system; in fact we have where H 2 is the hyperplane class on (P 2 ) ∨ . From now on, we work in the setting of Set-up 2.1: let f : S → P 2 be the double cover, and denote bỹ the induced cover and by p : P(f * Ω P 2 ) → S the natural fibration. We set for the tautological class. The linear system defined by ζ P + p * 2L ≃f * (ζ P 2 + 2p * 1 H 1 ) is base-point-free and defines a fibration Since the pull-backs of lines d ∈ |O(1)| correspond exactly to the elements of the linear system |L|, the fibration q : P(f * Ω P 2 ) → (P 2 ) ∨ is the universal family of curves in the linear system |L|.
The key to our investigation is the birational map which can be explicitly described as follows: since R is the ramification divisor of the double cover f , we have and the relative cotangent sheaf Ω f is isomorphic to O R (−R). Now consider the exact sequence The vector bundles f * Ω P 2 and Ω S are isomorphic in the complement of R, and we will see that the indeterminacy locus of P(f * Ω P 2 ) P(Ω S ) (resp. its inverse) is also a curve R ≃ R P ⊂ P(f * Ω P 2 ) (resp. R ≃ R S ⊂ P(Ω S )). Geometrically the curve R P corresponds to the cotangent bundle of the ramification divisor, while R S corresponds to the conormal bundle.
More formally, we make the following construction: since Ω f is a rank one bundle with support on the Cartier divisor R, we can see f * Ω P 2 as the strict transform of Ω S along Ω f (cf. [Mar72, Theorem 1.3] for the terminology). Let be the curve defined by the surjection Ω S → Ω f . Then denote by µ S : Y → P(Ω S ) the blow-up along this curve and by E S its exceptional divisor. Denote by E P the strict transform of the divisor π −1 (R) ⊂ P(Ω S ). Note that since P(Ω f ) ⊂ π −1 (R) is a Cartier divisor, we have where in the last step, we used (3.1). By the relative contraction theorem applied to the morphism π • µ S , there exists a birational morphism contracting the divisor E P onto a curve. By [Mar72, Theorem 1.3] this contraction can be described more explicitly: the image of f * Ω P 2 ⊗ O R → Ω S ⊗ O R is isomorphic to ω R , so we have a surjection f * Ω P 2 → ω R . Thus is a curve, and the blow-up along R P gives a morphism µ P : Y → P(f * Ω P 2 ) such that the following diagram commutes: Arguing as above we see that where in the last step, we used that f | R is an isomorphism onto B. We deduce from [Mar72, Theorem 1.1] that Lemma 3.1. In the situation summarised by Diagram 3.4, we have the following intersection numbers: Remark. We denote by (R) 2 E S (resp. (R) 2 E P ) the self-intersection of the curveR, seen as a divisor in E S (resp. E P ).
Proof. We describe E S and E P in terms of the blow-up: by construction one has E S ≃ P(N * R S /P(Ω S ) ) and

Thus we have an extension
Now recall that by (3.5), we also have E S ≃ P(f * Ω P 2 ⊗ O R ), so there exists a line bundle M → R such that Now recall that µ P is the blow-up along the curve R P corresponding to the quotient f * Ω P 2 ⊗ O R → ω R ≃ O R (3L). Then the curveR can be identified with R P under the isomorphism µ P | E S : E S → P(f * Ω P 2 ⊗ O R ), and we have We repeat the argument for E P : by construction one has E P ≃ P(N * R P /P(f * Ω P 2 ) ) and . Now recall that µ S is the blow-up along the curve R S corresponding to the quotient Then the curveR is identified with R S under the isomorphism µ S | E P : E P → P(Ω S ⊗ O R ), and we have In order to determine the self-intersection ofR, recall that the self-intersection of a curve is invariant under isomorphism. The isomorphism µ P | E S : E S → P(f * Ω P 2 ⊗ O R ) sendsR to R P ; thus we obtain Then the pseudoeffective cone of E S coincides with the nef cone, and its extremal rays are generated by the classes l S andR − 27l S . In particular if τ : E S → W is a generically finite morphism onto a surface W , it is finite.
Proof. By (3.5) we have E S ≃ P(Ω P 2 ⊗ O B ). Since the K3 surface S is general, the sextic curve B is general in its linear system, so by [Fle84, Theorem 1.2] the restricted vector bundle Ω P 2 ⊗ O B is semistable. Thus by [Laz04a, Section 1.5.A] the pseudoeffective cone of E S coincides with the nef cone. Since a nef divisor with positive self-intersection is big, this also shows that the nef cone coincides with the positive cone of E S . The curveR ⊂ E S is a section, and by Lemma 3.1 we have (R) 2 E S = 54. ThusR − 27l S is a generator of the second extremal ray of the positive cone.
The last statement follows by observing that a curve contracted by a generically finite morphism of surfaces has negative self-intersection. □ Remark 3.3. We saw at the beginning of this subsection that the vector bundle f * Ω P 2 ⊗ O S (2L) is globally generated and h 0 (S, f * Ω P 2 ⊗ O S (2L)) ≥ 3. Using the projection formula and f * O S ≃ O P 2 ⊕ O P 2 (−3), we see that equality holds. Twisting the tangent sequence (3.2) by O S (2L) and using that , we obtain that h 0 (P(Ω S ), ζ S + π * 2L) = h 0 (S, Ω S ⊗ O S (2L)) = 3, and the linear system |ζ S + π * 2L| is globally generated in the complement of R S = P(ω f ). In fact since the tangent map has rank one along R, we see that the scheme-theoretic base locus of |ζ S + π * 2L| is the reduced curve R S . Thus the blow-up R S resolves the base locus, and where H 2 is the hyperplane class on (P 2 ) ∨ .

Lemma 3.4. In the situation summarised by Diagram 3.4, we set
where H 2 is the hyperplane class on (P 2 ) ∨ . Then we have the following intersection numbers: The result follows from Lemma 3.1 and standard computations involving the geometric construction. In particular observe that As an example, let us show that M 2 · E S = 6: note that µ P maps E S isomorphically onto P(f * Ω P 2 ⊗ O R ). A fibre of p maps onto a line in (P 2 ) ∨ , so M has degree one on the fibres of E S → S. The curvẽ R ≃ R P maps birationally onto R ∨ ⊂ (P 2 ) ∨ . Since R ∨ has degree 30 by Plücker's formula (see Section 3.2 below), we obtain M ·R = 30. Since (R) 2 E S = 54 by Lemma 3.1, this shows that M| E S ≡R − 24l S , and thus M 2 · E S = (M| E S ) 2 = 6. □

Remark.
For the convenience of the reader, let us summarise that by the argument above and Lemma 3.1, we have

The surface D
Let B ∨ ⊂ (P 2 ) ∨ be the dual curve of the sextic curve B ⊂ P 2 . Since B is a general sextic, we can apply Plücker's formulas to see that deg B ∨ = 30 and B ∨ has exactly 324 nodes (resp. 72 simple cusps) corresponding to bitangent lines (resp. inflection lines) and no other singularities. By definition the dual curve parametrises lines d ⊂ P 2 that are tangent to B in at 9 least one point. Since the preimage f −1 (d) of a line is singular if and only if it is tangent to B in at least one point, we see that B ∨ naturally parametrises the singular elements of |L|. We denote by Since the elements of |L| have arithmetic genus two. it is not difficult to see that • a smooth point t ∈ R ∨ parametrises a curve C t with exactly one node, so the normalisation is an elliptic curve; • a node t ∈ R ∨ parametrises a curve C t with exactly two nodes, so the normalisation is a rational curve; • a cusp t ∈ R ∨ parametrises a curve C t with one simple cusp, so the normalisation is an elliptic curve.
The surface q −1 (R ∨ ) ⊂ P(f * Ω P 2 ) identifies with the universal family of singular elements in |L|; in particular, this surface is not normal.
Lemma 3.5. We have R P ⊂ (q −1 (R ∨ )) sing , and R P is the unique irreducible component of (q −1 (R ∨ )) sing mapping onto R ∨ . Moreover q −1 (R ∨ ) has a nodal singularity in a point of R P that maps onto a smooth point on R ∨ .
Proof. For a smooth point t ∈ R ∨ , the curve C t ⊂ |L| has a unique singular point x, so it is clear that Proof of the claim. Let d ⊂ P 2 be a line, and letd ⊂ P(Ω P 2 ) be the lifting defined by the canonical quotient Let B P ⊂ P(Ω P 2 ) be the curve defined by the canonical quotient Ω P 2 ⊗ O B → ω B . Since the curve R P corresponds to the quotient Thus the curvesd and B P intersect over f (x). This proves the claim.
In order to see that q −1 (R ∨ ) has a nodal singularity over the smooth points of R ∨ , we just observe that the statement follows by considering the intersection of a general P 1 with the branch divisor of the double coverf (see the proof of Theorem 3.18 for a more refined description of the singularities). □ We now introduce the main object of our study.
Proposition 3.6. Let D ⊂ Y be the strict transform of the surface q −1 (R ∨ ) ⊂ P(f * Ω P 2 ). Using the notation of Lemma 3.4, we have Proof. Since R ∨ ⊂ |L| has degree 30, we have (q • µ P ) * R ∨ = 30M. By Lemma 3.5 the surface q −1 (R ∨ ) has multiplicity two along the curve R P ; thus its strict transform has class 30M − 2E P . □ Since D = 30M − 2E P the following statement follows from Lemma 3.4 by elementary computations.

Lemma 3.7. In the situation summarised by Diagram 3.4, we have the following intersection numbers:
The prime divisors E P , E S , D form a basis of N 1 (Y ) that we will use for our computations in the later sections. The class M = ψ * c 1 (O (P 2 ) ∨ (1)) will also be useful due to its simple geometric interpretation. For the convenience of the reader, we summarise their intersections with a dual base.
Lemma 3.8. In the situation summarised by Diagram 3.4, denote by l P (resp. l S ) an exceptional curve of the blow-up µ P (resp. µ S ), and by l D a general fibre of the fibration Then we have the following intersection numbers: Proof. First note that the intersection numbers involving D follow from the other numbers and (3.7). Also note that the intersection numbers for the curves l S and l P with E S , E P , M are straightforward from the construction of the elementary transform, summarised in Diagram (3.4).
Thus we are left to compute the intersection numbers of l D . Recall that, by definition, The fibre C t has a nodal singularity in its intersection point with the curve R P ; the natural map P(f * Ω P 2 ) → S maps R P (resp. C t ) isomorphically onto R (resp. the corresponding curve in |L|). Thus each branch of C t meets R P transversally since this holds for the branches' images in S. This shows that E P · l D = 2. Since E P + E S = (p • µ P ) * 3L and p(l D ) ∈ |L|, we deduce that E S · l D = 4. □

Canonical liftings
In this short subsection, we relate the surface D to the surface D S appearing in the introduction.
Definition 3.9. Let S be a smooth projective surface, and let C ⊂ S be an irreducible curve. Denote by n :C → C the normalisation and by The canonical lifting of C to P(Ω S ) is the image of the morphismñ :C → P(Ω S ) corresponding to the line bundle Q C . We denote this curve by C S ⊂ P(Ω S ).
(a) If the curve C is singular, the birational map C S → C is not necessarily an isomorphism.
is a smooth curve: since π •ñ = n, the morphism is immersive, so we only have to show that it is one-to-one. Recalling that P(Ω S ) = P(T S ) (the projective bundle of lines in T S ), we see from the definition of the canonical lifting thatñ maps a point t ∈C to the point [T n(T C,t )] ∈ P (T S,n(t) ), where T n is the tangent map. Yet if t 1 , t 2 are two points inC such that n(t 1 ) = n(t 2 ), then [T n(T C,t 1 )] [T n(T C,t 2 )] since the curve is nodal. Thusñ(t 1 ) ñ(t 2 ). This also shows that C S meets the fibre π −1 (n(t i )) ≃ P(T S,n(t i ) ) transversally in two points.
(d) If C ⊂ S is a curve such that the unique singular point is a simple cusp p, the canonical lifting C S ⊂ P(Ω S ) is a smooth curve that has tangency order two with the fibre π −1 (p): the claim is local, so we can consider that C is given by a parametrisation Denote by x, y the coordinates on C 2 ; then we have induced coordinates (x, y), The tangent map T n is given by so we see that Q C ≃ I 0 ⊗ ω ∆ andñ is given by This map is well defined in the origin and maps it onto the point ((0, 0), [2 : 0]). In the affine chart u x 0 the mapñ is given by In the coordinates (x, y, z) of the affine chart, the image of this map is the smooth curve cut out by 9x = z 2 , 27y = z 3 . The intersection of this curve with the fibre x = 0, y = 0 is a finite non-reduced scheme of length two.
Notation 3.11. We will denote by C node (resp. C cusp ) an element of |L| that has exactly two nodes (resp. a cuspidal point).
(2) We will denote by C node,S (resp. C cusp,S ) the canonical lifting of the curve C node (resp. C cusp ) to P(Ω S ).
Lemma 3.12. In the situation summarised by Diagram 3.4, let C ⊂ P(f * Ω P 2 ) be a fibre of the universal family P(f * Ω P 2 ) → |L|, and let C S ⊂ P(Ω S ) be the canonical lifting of the curve [C] ∈ |L|. Then C S is the strict transform of C under the birational map µ S • µ −1 P . In particular let be the irreducible surface obtained by canonical liftings of singular elements of |L|. Then we have D S = µ S (D).
Proof. Since D is the strict transform of q −1 (R ∨ ), which is the universal family of singular elements of |L|, the second statement follows from the first.
For the proof of the first statement, recall (cf. the proof of Lemma 3.5) that the fibres of p 2 : P(Ω P 2 ) → (P 2 ) ∨ are given by liftings corresponding to canonical quotients Ω P 2 ⊗ O d → ω d of lines d ⊂ P 2 . Since q is obtained by base change from p 2 , we see that the fibres of q are given by liftings corresponding to quotients In the complement of the ramification divisor, the vector bundle f * Ω P 2 ⊗ O C (resp. (f | C ) * ω d ) coincides with Ω S ⊗ O C (resp. ω C ). Thus the two liftings coincide in the complement of R; hence the curves are strict transforms of each other. □

Geometry of the surfaceD
This subsection contains the technical core of our study. We will use the notation introduced in the Sections 3.1 and 3.2, in particular Diagram (3.4). We still denote by p 1 : P(Ω P 2 ) −→ P 2 , p 2 : P(Ω P 2 ) −→ (P 2 ) ∨ the projections and by H 1 (resp. H 2 ) the hyperplane classes on P 2 (resp. (P 2 ) ∨ ). In order to simplify the notation, we will identify Restricting the universal family q : P(f * Ω P 2 ) → (P 2 ) ∨ over the dual curve, we obtain a flat fibration The singularities of the source and target make it difficult to analyse this fibration. Therefore we will normalise both spaces to obtain a fibration q :D −→ R and show the following.
Remark 3.14. Let R → R ∨ be the normalisation. The fibre product is a family of curves that satisfies the assumptions of Teissier's simultaneous normalisation theorem [DPT80, Section I.1.3.2, Theorem 1]. Thus the fibrationD → R is smooth near the bisection determined by the preimage of the rational section R P ⊂ q −1 (R ∨ ). Note that Teissier's theorem does not imply the smoothness ofD in the 648 remaining nodes.
The following commutative diagram will guide the reader through the construction. The varieties and morphisms in columns 3 to 5 have been introduced in the Sections 3.1 and 3.2. The second column is obtained from the third column by normalisation. The curves in the first column will be successively introduced in this subsection. (3.8) be the curve defined by the canonical quotient Ω P 2 ⊗ O R → ω R (here we use the identification B = R). The curve R P 2 maps birationally onto R (resp. R ∨ ), so its class in P(Ω P 2 ) is (3.9) R P 2 = 30l 1 + 6l 2 , 13 where l i is a fibre of the fibration p i . Let n R : R → R ∨ be the normalisation, and let In fact we have T ≃ P(n * R Ω (P 2 ) ∨ ), and we set for the tautological class. The curve R P 2 ⊂ p −1 2 (R ∨ ) is not contained in the singular locus of p −1 2 (R ∨ ), and its strict transform is a section R T ⊂ T .
Lemma 3.15. In the situation summarised by Diagram 3.8, we have where l T is a fibre of q T . Moreover we have Since R T is a q T -section, we conclude that The description of the Mori cone and the nef cone is now standard; cf. [Har77, Section V.2]. The canonical class of T is given by Moreover, one has KD ≡ 72f * D l T .
14 Note that at this point, we do not know thatD is smooth. However, being a cyclic double cover of a smooth surface, it is Gorenstein.
Since R ≡ 6H 1 , by (3.9) we have ν * T p * 1 R · R T = p * 1 R · R P 2 = 36. Also note that ν * T p * 1 R has degree six on the fibres of q T since a fibre maps onto a line in P 2 . Thus using (3.10), we deduce that ν * T p * 1 R ≡ 6ζ T + 360l T . If l T is a general q T -fibre, we can identify it to the corresponding p 2 -fibre. Hence we know that this fibre intersects p * 1 R transversally in four points and with multiplicity two in the point which will give the node. This last point is on the curve R P 2 , so we see that the effective divisor ν * T p * 1 R contains its strict transform R T with multiplicity two. Thus we can write ramifies exactly over p * 1 R ∩ p −1 2 (R ∨ ) and the surface q −1 (R ∨ ) has a nodal singularity in the generic point of R P ≃ R P 2 ≃ R T (cf. Lemma 3.5), the branch locus offD is exactly B T . Now we can compute the canonical class ofD: sincefD is a double cover ramified along B T , we have KD ≡f * We will now analyse the singularities ofD via the double coverfD: the morphismfD is the degree two cyclic cover associated to a line bundle isomorphic to 1 2 B T ≡ 2ζ T + 144l T , so by [Laz04a, Section 4.1.B], given a point p ∈ T and a local equation g(x, y) for B T near p, a local equation ofD is

In particularD is singular if and only if B T is singular in p.
Proof of Theorem 3.13. By Lemma 3.16 we know that the canonical class is nef, so the surface is minimal. We will show that the branch locus B T is smooth; henceD is smooth.
Let n R : R → R ∨ be the normalisation. Fix a point r ∈ R, and denote by T r the q T -fibre over the point r. The double coverf −1 d (T r ) → T r is determined by the double cover f | C : C → d of the line d ⊂ P 2 corresponding to the point n R (r). We make a case distinction: • If n R (r) is smooth, the corresponding cover C → d ramifies in the four points where d is not tangent to the branch divisor B. Since by Lemma 3.16, the curve B T does not contain a fibre of the ruling and has degree four on every fibre, we see that B T ∩ T r is smooth. Thus B T is smooth near T r . • If n R (r) is a cusp, the corresponding cover C → d ramifies in the three points where d is not tangent to B. Thus the finite scheme B T ∩ T r has at least three smooth points. Since the scheme has length four, it is smooth. Thus B T is smooth near T r . • If n R (r) is a node, the corresponding cover C → d ramifies in the two points where d is not tangent to B. Since the normalisationC → C is a rational curve, the double coverC → l has exactly two ramification points. This shows that B T ∩ T r has two smooth points and a point t ∈ T r with multiplicity two supported on one of the two points where d is tangent to B (the other point is on R T , whence the asymmetry). Thus B T ∩ T r is not smooth; nevertheless, we claim that B T is smooth near T r : the fibration p 2 is the universal family of lines on P 2 , so by construction the line p 1 (p −1 2 (n R (r))) 15 is tangent to R. Thus p −1 2 (n R (r)) is tangent to p * 1 R. Since B T is the unique component of ν * T p * 1 R passing through t, we obtain that B T is tangent to the fibre T r . Since the local intersection number in the point t is two, we deduce (e.g. by [Per08, p.225, Axiom 5)]) that B T is smooth near this point.
The analysis of the branch locus also allows us to determine the singular fibres ofq :D → R: since q T is a smooth fibration, a fibre ofq = q T •fD is singular if and only if the restriction of the branch locus B T to the fibre T r is singular. We have seen that this happens if and only if n R (r) is a node. Since R ∨ has 324 nodes, we see that there are 648 singular fibres. Since the intersection B T ∩ T r has two reduced points and one point with multiplicity two, the double cover is a nodal cubic. □ For later use, we note the following.

Geometry of the surfaceD
The first goal of this subsection is to describe the normalisation of the surface D. Proof. For simplicity's sake we denote by the restriction of the fibration q over the curve R ∨ (resp. the restriction of ψ to D). Given a point r ∈ R ∨ we will describe the map D → q −1 (R ∨ ) in a neighbourhood of the fibre q −1 (r). This local description will then allow us to show thatD is smooth. Case 1: r ∈ R ∨ is a smooth point. We know by Lemma 3.5 that q −1 (R ∨ ) has nodal singularities near q −1 (r), so the blow-up along R P coincides with the normalisation, and D is smooth near ψ −1 (r).
Case 2: r ∈ R ∨ is a node. Denote by ∆ ⊂ R ∨ and ∆ ′ ⊂ R ∨ the two local branches through r. Then ∆ is smooth, and we can suppose that it is a small disc that contains no other singular points of R ∨ . The preimage is an analytic hypersurface, so Gorenstein, and generically reduced, hence reduced. We have seen in Section 3.2 that the fibres of for t 0 are curves with exactly one node and the central fibre has exactly two nodes. Denote by x 1 , x 2 the nodes of the central fibre and by x t the unique node of the fibre over t 0. Then up to renumbering, x 1 is in the closure of ∪ t 0 x t , so these points form a curve R ∆ that is a section of q −1 (∆) −→ ∆, and the curve R ∆ is in the non-normal locus of q −1 (∆). Since R ∆ is a section over a smooth curve, the point x 2 is not in R ∆ . Since ∆ is smooth and q −1 (∆) → ∆ has smooth fibres in the complement of R ∆ ∪ x 2 , we see that the singular locus of q −1 (∆) is contained in R ∆ ∪ x 2 . Since x 2 is an isolated singularity and q −1 (∆) is Gorenstein, we see that x 2 is in the normal locus of q −1 (∆). Let q −1 (∆) → q −1 (∆) be the normalisation.
By the universal property of the normalisation, we have an embedding q −1 (∆) →D. SinceD is smooth by Theorem 3.13, we see that q −1 (∆) is smooth; in particular x 2 is a smooth point q −1 (∆).
We can now describe the blow-up of q −1 (∆) along the scheme q −1 (∆) ∩ R P . By construction R P contains all the nodes over R ∨ , so we see that set-theoretically, Since q −1 (∆) has nodal singularities along R ∆ ⊂ q −1 (∆) ∩ R P , the blow-up coincides with the normalisation. We also observe that the curve R P meets q −1 (∆) transversally in x 2 : if the intersection is not transversal, the intersection of the images q(q −1 (∆)) and q(R ∆ ′ ) is not transversal. But q(q −1 (∆)) = ∆ and q(R ∆ ′ ) = ∆ ′ , so they intersect transversally. Thus near the point x 2 , the blow-up of q −1 (∆) along the scheme R P ∩ q −1 (∆) coincides with the blow-up of the reduced point x 2 ∈ q −1 (∆); hence the blow-up is smooth.
We can now conclude as follows: near the fibre q −1 (r) the surface q −1 (R ∨ ) is reducible with irreducible components q −1 (∆) and q −1 (∆ ′ ). The strict transform of q −1 (R ∨ ) in Y coincides with the union of the strict transforms of q −1 (∆) and q −1 (∆ ′ ) in Y . We have seen that the strict transforms of q −1 (∆) and q −1 (∆ ′ ) are smooth. Thus in a neighbourhood of ψ −1 (r), the normalisationD → D consists just of separating the two irreducible components.
This finishes the discussion of this case. Note that the argument above also shows that near one of the 648 nodal fibres of the elliptic fibrationq :D → R (cf. Theorem 3.13), the birational mapD → D is given by the blow-up of the node of the fibre.
Case 3: r ∈ R ∨ is a cusp. Let x ∈ q −1 (r) be the unique singular point. The statement is clear in the complement of x, so a local computation near x will allow us to conclude: the universal family of lines P(T P 2 ) ⊂ P 2 × (P 2 ) ∨ can be given by the equation where the x i are coordinates on P 2 and y i on (P 2 ) ∨ . Identifying x ∈ q −1 (r) ⊂ S to its image in S, we can assume that f (x) = [1 : 0 : 0] and work in the affine chart x 0 = 1 on P 2 . Since the branch curve B has a simple inflection line in (0, 0), we can suppose that it is given in the local chart by x 2 = x 3 1 . The tangent line to B in the origin is given by x 2 = 0, so it corresponds to the point [0 : 0 : 1] ∈ (P 2 ) ∨ . Therefore, we choose the affine chart y 2 = 1 for (P 2 ) ∨ , so the universal family of lines is given by y 0 + x 1 y 1 + x 2 = 0.
Let C cusp,Y be the strict transform of the cuspidal curve q −1 (r). Computing the Jacobian matrix for the system of equations (3.13), we see that D is singular along C cusp,Y , but its unique singular point on the exceptional divisor u = 0 is the origin. Since the origin is contained in C cusp,Y , it is a non-normal point of D. Thus we are done if we show that the normalisation is smooth near the preimage of C cusp,Y : note that C cusp,Y can locally be parametrised by gives a local chart for the blow-up of C 4 along this curve (the other charts are less interesting for the proof and left to the reader). The strict transform of the hypersurface v + tw + u = 0 under this blow-up is given by and the strict transform of −9t 2 w 2 + 4uw 3 − 108t 3 + 54twu + 27u 2 = 0 is −108β + 27δ 2 + 54αδ + 27α 2 = 0.
It is straightforward to check that D ′ is smooth and the morphism D ′ → D is finite. By the universal property of the normalisationD → D, we obtain an embedding D ′ →D, soD is smooth. □ We are finally ready to prove our first main result.
Proof of Theorem 1.3. We have shown in Lemma 3.12 that D S = µ S (D). We claim that the birational morphism D → D S is finite. In particular the normalisation of D S is given by the surfaceD, so the statement follows from Theorem 3.18. For the proof of the claim, recall that D is the strict transform of q −1 (R ∨ ) ⊂ P(f * Ω P 2 ) and the curves l S contracted by µ S are the strict transforms of the fibres of P(f * Ω P 2 ⊗ O R ) → R S (cf. Corollary 3.2). Thus it is sufficient that q −1 (R ∨ ) does not contain any p-fibres; yet this is clear: a p-fibre is mapped by q onto a line in (P 2 ) ∨ , so it is not contained in the irreducible curve R ∨ .
In order to determine the class of D S , it is sufficient to compute the push-forward of the class D ≡ 30M − 2E P (cf. Proposition 3.6).
Since µ * S (ζ S + π * 2L) ≡ M + E S (cf. Remark 3.3), we have (µ S ) * M ≡ ζ S + π * 2L. We saw in Section 3.1 that µ S (E P ) = π −1 (R). Since R ≃ 3L we obtain The computation for the other charts is analogous, but simpler. 18 Theorem 3.18 gives a complete description of the elliptic fibrationD → R. In order to determine an intersection table forD, we are left to describe a curve that is horizontal with respect to the fibration. • if t ∈ R is a point mapping onto a node in R ∨ , the intersection of ND with the fibreq −1 (t) consists of two smooth points; in particular, ND is disjoint from the node inq −1 (t); • if t ∈ R is a point mapping onto a cusp in R ∨ , the intersection of ND with the fibreq −1 (t) is non-reduced, its support being the unique point ofq −1 (t) mapping onto the cusp q −1 (ν(t)).
Proof. We know by (3.12) that the pull-back of p * 1 R to R T decomposes as 2R T + B T , where B T is the branch divisor offD. In particularfD is étale in the generic point of R T , so the pull-backf * D R T is a reduced bisection, and by Lemma 3.15 one has In order to see that ND is smooth, we observe, e.g. by using the description in the proof of Theorem 3.13, that R T is disjoint from the branch divisor B T in the complement of the fibres corresponding to cusps. Thus its preimage ND is smooth in the complement of the fibres corresponding to cusps. For a fibre over a cusp, the section R T passes exactly through the inflection point. Thus, analogously to the proof of Theorem 3.18, a local computation shows that ND is also smooth near these fibres.
Finally observe that for a point t ∈ R mapping onto a node in R ∨ , the curve R T is disjoint from the branch curve B T , so there are two distinct points ND mapping onto R T ∩ q −1 T (t). For a point t ∈ R mapping onto a cusp in R ∨ , the intersection R T ∩ q −1 T (t) is contained in B T , so the set-theoretic preimage is a single point. □ Proposition 3.20. In the situation of Theorem 3.18, let ND be the strict transform of ND (cf. Lemma 3.19 above) under the birational mapμ P :D →D. Moreover • if t ∈ R is a point mapping onto a node in R ∨ , denote by C node,t ⊂D the strict transform of the nodal cubic q −1 (t) and by l node,t ⊂D the exceptional curve mapping onto the node; • if t ∈ R is a point mapping onto a cusp in R ∨ , denote by C cusp,t ⊂D the strict transform of the smooth elliptic curve q −1 (t) and by l cusp,t the exceptional curve mapping onto a point in q −1 (t).

Then we have the following intersection numbers:
C node,t · l node,t = 2, C 2 node,t = −4, l 2 node,t = −1, C cusp,t · l cusp,t = 1, C 2 cusp,t = −1, l 2 cusp,t = −1, ND · l node,t = 0, ND · C node,t = 2, ND · l cusp,t = 1, ND · C cusp,t = 1, Proof. The intersection numbers that do not involve ND are immediate consequences of Theorem 3.18. The intersection numbers involving ND follow from Lemma 3.19. □ The final step of our computations will be to use the intersection table in Proposition 3.20 to further investigate the divisor classes on Y . As a preparation, we consider the image of the curve ND in Y : by construction this image is a curve N ⊂ E P which is a bisection of the ruling E P → R. We can make this description more geometric: recall that and by Lemma 3.12 the fibres of the universal family map birationally onto the canonically lifted curves. Also recall that for a point x ∈ R, there exists a unique element of |L| that is singular in x (the preimage of the tangent line of B ≃ R in x); denote this element by C x . Using the notion of a tangent line of a plane singular curve as explained in [Per08, Section V. 4.8], we obtain that Lemma 3.21. The class of N ⊂ E P is 2R + 72l p , where l p is a fibre of the ruling E P → R P . Moreover we have the following intersection numbers: Proof. We know that N ⊂ E P is a bisection, so we have N ≡ 2R + λl p . We claim that N is disjoint from the negative sectionR ⊂ E P . By Lemma 3.1, we have (R) 2 E P = −36, so the claim determines the class of N ⊂ E P . The intersection numbers then follow from Lemma 3.1 and the fact that the map (q • µ P )| N : N → R ∨ has degree two and R ∨ = B ∨ has degree 30 (cf. Section 3.2).
Proof of the claim. Recall that the tangent map f * Ω P 2 → Ω S has rank one along R, and its image defines a canonical inclusion Ω R → Ω S ⊗O R (cf. Section 3.1). The negative sectionR is the section corresponding to the induced quotient bundle Ω S ⊗O R ↠ (Ω S ⊗O R /Ω R ). If we write f in local coordinates as (x, y) → (x, y 2 ) (so the ramification divisor is y = 0), the image of f * Ω P 2 ⊗ O R → Ω S ⊗ O R is generated by dx, so the quotient is given by dy. Dually the image of the inclusion (Ω S ⊗ O R /Ω R ) * → T S | R is generated by ∂ ∂y . Thus it is sufficient to verify that the tangent lines of the curves C x are not collinear to ∂ ∂y . If x ∈ R ≃ B is an inflection point, the curve C x is cuspidal and has a unique tangent line which coincides with the tangent line of the ramification divisor R. Thus it is generated by ∂ ∂x . If x ∈ R ≃ B is not an inflection point, the curve C x is nodal with local equation y 2 − x 2 = (y − x)(y + x), so none of the tangent lines is generated by ∂ ∂y . □ Remark 3.22. We conclude this subsection with an observation indispensable for the proof of Theorem 1.5: we saw in Section 3.4 that the intersection of the surfaces p −1 2 (B ∨ ) = p −1 2 (R ∨ ) and p −1 1 (B) has two components, which in the ruled surface T correspond to the curves R T and B T . By Corollary 3.17 the curve B T is irreducible.
Since the double coverf is bijective along its ramification divisor p −1 (R), this shows that the intersection of q −1 (R ∨ ) and p −1 (R) has two irreducible components, the curves R P andf −1 (B T ). In the generic point of R P , each branch of q −1 (R ∨ ) meets p −1 (R) transversally, so their strict transforms in Y , i.e. the surfaces D and E S , intersect only along the strict transform of the irreducible curvef −1 (B T ). The curvef −1 (B T ) is not contained in the singular locus of D, so the pull-back is an irreducible curve in the normalisationD.

Numerical restrictions on pseudoeffective classes
Our goal in this subsection is to use the description of the surfaceD to obtain additional information on the effective divisors in Y . Somewhat abusively we denote by  N 1 (D) is contained in the subspace generated by the classes C node,t , l node,t , C cusp,t , l cusp,t and ND (cf. Proposition 3.20 for the notation).
Proof. Denote by i : S → S the involution induced by the double cover f . Then i acts via push-forward on the linear system |L| = (P 2 ) ∨ . Since all the elements of |L| are pull-backs from P 2 , this action is trivial. Thus we have a natural involution (i, i * ) : S × |L| → S × |L| which preserves the universal family of |L|, i.e. the subvariety P(f * Ω P 2 ) ⊂ S × |L|. We denote by i P : P(f * Ω P 2 ) → P(f * Ω P 2 ) the induced involution and note that by construction, the fibrations p and q are equivariant with respect to the action of i P . Since R P ⊂ P(f * Ω P 2 ) is the locus where q is not smooth, it is is preserved by i P . Thus i P lifts to an involution i Y : Y → Y . Since the action of i * preserves the curve R ∨ , the involution i Y leaves the surface D invariant, so it lifts to an involutionĩ :D →D. Taking the quotient by these involutions, we obtain a commutative diagramD The threefold Y /⟨i Y ⟩ is a (weighted) blow-up of P(Ω P 2 ), so it has Picard number three. Since Y also has Picard number three, this shows that the pull-back τ * : is an isomorphism. Thus it is sufficient to show that the image of the pull-back is generated by the classes τ D (C node,t ), τ D (l node,t ), τ D (C cusp,t ), τ D (l cusp,t ) and τ D (ND). But the surfaceD/⟨ĩ⟩ is a (weighted) blow-up of the ruled surface T with exceptional locus the curves τ D (l node,t ) and τ D (l cusp,t ), so its Néron-Severi space is generated by these classes.
Before we give the proof of the lemma, let us recall the geometry of the situation: the curves l node,t and l cusp,t are the exceptional curves of the blow-upμ P :D →D (cf. Theorem 3.18). This blow-up is induced by the blow-up µ P : Y → P(f * Ω P 2 ), and we have ν * l node,t = l P = ν * l cusp,t , where l P is an exceptional curve of µ P . The general fibre of ψD :D → R maps isomorphically onto the general fibre of q • µ P : D → R ∨ . This justifies that we denote them by the same letter l D (cf. Lemma 3.8).
Proof of Lemma 3.24. By Lemma 3.23 we know that the pull-backs can be written as linear combinations of the classes C node,t , l node,t , C cusp,t , l cusp,t and ND. These classes are not linearly independent, but by Theorem 3.18 one has l D ≡ C node,t + 2l node,t ≡ C cusp,t + l cusp,t .
Thus it is straightforward to see that the classes ND, l D , l node,t , l cusp,t form a basis of the vector space. In particular we can verify the formulas by computing the intersection numbers with the elements of the basis. By the discussion before this proof, we see that the intersection numbers with the left-hand sides are determined by Lemmas 3.8 and 3.21 and the projection formula. The intersections with the right-hand sides are determined by Proposition 3.20. □ Remark 3.25. The Mori cone of the surfaceD is quite complex: the irreducible curves ND, C node,t , l node,t , C cusp,t , l cusp,t have negative self-intersection, so they generate extremal rays in the Mori cone; cf. [KM98, Lemma 1.22]. By Remark 3.22 the effective divisor ν * E S is irreducible, and by Equation (3.16) the class of this divisor is not in the cone generated by the classes ND, C node,t , l node,t , C cusp,t , l cusp,t . Since (ν * E S ) 2 = E 2 S · D = −288 < 0 by Lemma 3.7, the irreducible curve ν * E S thus generates another extremal ray in Mori cone.
With the description of the geometry of E S and D in mind, we obtain strong obstructions to the effectivity of a class in P(Ω S ).
Proof of Theorem 1.5. Let Z S ⊂ P(Ω S ) be a prime divisor, and let Z ⊂ Y be its strict transform. The statement is clear for Z = D and Z = E P , so we assume that Z is distinct from these two surfaces. Note that we can assume that the class of Z is not a linear combination Z = xD + yE P : computing the intersections with l D and l P (see Lemma 3.8), we obtain that x ≥ 0 and y ≥ 0. Thus we have λ ≥ 1.8. Thus, up to a multiple, we can denote by η := E S + xD + yE P the cohomology class of Z in Y . Since E S is µ S -exceptional, the prime divisor Z is not equal to E S , and its restriction to E S is effective. By Corollary 3.2, this tells that Z| E S is also nef, and in particular η 2 · E S ≥ 0. Using Lemmas 3.4 and 3.7 one can easily see that this implies that We claim that α = ν * E S + 0.5C node is nef.
Proof of the claim. Using Equation (3.16) and Proposition 3.20, one sees that α·C i node = 0 and α·ν * E S = 360. Since by Remark 3.22 the curve ν * E S is irreducible, the class α is non-negative along the irreducible components of its support. This implies the claim.
Combining this condition with the inequality (3.19) gives y x ≥ − 3 11 . Since (µ S ) * E S = 0, (µ S ) * E P ≡ π * 3L and (µ S ) * M ≡ ζ S + π * 2L (cf. Remark 3.3), a straightforward computation (cf. Lemma 4.1) gives Since the ratio y 10x is bounded from below by − 3 110 , the cohomology class of the prime divisor Z S in P(Ω S ) can thus be written as [Z S ] ≡ a(ζ S + λπ * L) with λ ≥ 9 5 − 3 110 = 39 22 , as desired. □ Remark. The argument above and thus the bound in Theorem 1.5 depend on our choice of the class α, so it likely that a full description of the nef cone ofD would lead to a better estimate. However, it is not clear that even such an optimised necessary condition determines the pseudoeffective threshold.

Positivity results
In this whole section we work in the same set-up as in Section 3.1 (cf. the summary in Diagram (3.4)). We denote by ϕ : Y −→ S 22 the composition π • µ S = p • µ P . The pseudoeffective cone of the threefold P(f * Ω P 2 ) is generated by the classes p * L and q * H 2 , where H 2 is the hyperplane class on (P 2 ) ∨ . The picture becomes significantly less trivial when we consider the pseudoeffective cone of P(Ω S ) or Y . Given a divisor class Z on Y , we write If Z ⊂ Y is an effective R-divisor such that the support does not contain the exceptional divisor E P , its image Z P ⊂ P(f * Ω P 2 ) has class Z P ≡ aq * H 2 + bp * L with a ≥ 0, b ≥ 0. Moreover, the effective divisor Z P has multiplicity m along the curve R P .

Necessary conditions
In this subsection we collect some basic necessary conditions. As an application we prove the second part of Theorem 1.4. Lemma 4.1. Let Z ⊂ Y be an effective R-divisor such that the support does not contain the exceptional divisors E P and E S . Let Z S ⊂ P(Ω S ) be the image of Z. Then, using the notation (4.1), one has

Corollary 4.2.
Let Z S ⊂ P(Ω S ) be an effective R-divisor such that the class of Z S is not contained in the cone generated by D S and π * L. Then, using the notation (4.1), we have Assume furthermore that there exists a canonical lifting C node,S of a rational nodal curve C ∈ |L| (cf. Notation 3.11 ) that is not contained in Z S . Then one has As preparation let us note that the class of the curve R P ⊂ p −1 (R) is given by Indeed since p(R P ) = R (resp. q(R P ) = R ∨ ), we know that p * L · R P = 6 and q * H 2 · R P = 30. By Lemma 3. 4 we have (q * H 2 ) 2 · p −1 (R) = M 2 · E S = 6, which implies the result.
Proof of Lemma 4.3. The support of Z does not contain E S , so the support of its image Z P does not contain µ P (E S ) = p −1 (R). Since Z P has multiplicity m along R P , the intersection Z P ∩ p −1 (R) contains the curve R P with multiplicity at least m. Thus if f is a general fibre of the ruling p −1 (R) → R, then (Z P ∩ p −1 (R)) · f ≥ m.
Since Z P ≡ aq * H 2 + bp * L we obtain that a ≥ m.
By what precedes the class is effective. The isomorphism E S → p −1 (R) maps the curveR onto R P , so the classR − 27l S maps onto the class (q * H 2 − 1 2 p * L)| p −1 (R) . By Corollary 3.2 the classR − 27l S is nef, so Using the intersection numbers from Lemma 3.4, we obtain the second inequality in (4.3).
For the proof of the last statement, we recall that by Lemma 3.12, the curve C node,S is the strict transform of the corresponding fibre C node of the universal family P(f * Ω P 2 ) → |L| ≃ (P 2 ) ∨ . Since the generic point of C node,S is contained in the locus where µ P • µ −1 S is an isomorphism, its strict transform C node is not contained in Z P . The curve C node has two singular points, which are both on the curve R P (cf. the proof of Lemma 3.5). Since Z P has multiplicity m along R P , we obtain that Z P ·C node ≥ 4m. Since Z P ≡ aq * H 2 +bp * L and C node is contracted by q, this yields the inequality (4.4). □ Proof. Since Z is modified nef, it is in the closure of the mobile cone. Thus we can find a sequence of effective mobile Q-divisors Z t such that their classes converge to Z. Since the decomposition of the divisor class in the basis M, ϕ * L, E P is unique, this implies that the coefficients a t , b t , m t converge to a, b, m. The divisor Z t being mobile, its support does not contain the exceptional divisors E P and E S . Thus Lemma 4.3 yields a t ≥ m t and 1 2 a t + b t ≥ 9 2 m t . We conclude by passing to the limit. □ Corollary 4.5. Let Z S ⊂ P(Ω S ) be an effective R-divisor such that Z S = a(ζ + λπ * L) with λ < 1.8. Let C S ⊂ P(Ω S ) be a canonical lifting of a rational nodal curve C ∈ |L|. Then C S ⊂ Z S .

Sufficient conditions
In this subsection we will use Boucksom's divisorial Zariski decomposition [Bou04, Theorem 4.8]: given a pseudoeffective R-divisor class Z on Y , there exists a decomposition Z + + Z − such that Z + is modified nef, cf. [Bou04, Definition 2.2], and Z − is an effective R-divisor. Lemma 4.6. Let Z be a pseudoeffective R-divisor class on Y , and let Z + +Z − be its divisorial Zariski decomposition. Assume that the support of Z − does not contain the exceptional divisors E P and E S . Then one has Remark 4.7. We will need the following elementary remark: let Z ′ ⊂ Y be a prime divisor that surjects onto the surface S. For d ≫ 0 let A ∈ |dL| be a general element. Then the intersection Z ′ ∩ ϕ −1 (A) is an irreducible curve. Namely the linear system ϕ| * Z ′ |dL| is globally generated, so a general element does not have an irreducible component that is in the non-normal locus of Z ′ . Thus we can assume without loss of generality that Z ′ is normal. Yet then a general element is normal by [BS95, Theorem 1.7.1], hence smooth. Since it is connected by the Kawamata-Viehweg vanishing theorem, a general element is irreducible.
Proof. We first prove the statement in the case where Z is a prime divisor. By our assumption on Z − , the divisor Z is then distinct from E S and E P . We use the notation (4.1); i.e. we write Note that if a = 0, then Z · l s = −m. Since Z is distinct from E S , we obtain that m = 0, so Z ≡ bϕ * L. Thus we can assume that a > 0.
Since a > 0 we have a 2 + ab > a · ( 1 2 a + b). Applying (4.3) twice, we obtain a · ( 1 2 a + b) ≥ 9 2 m 2 . Thus we have a 2 + ab − 3m 2 > 1.5m 2 ≥ 0. The same argument holds if Z is a pseudoeffective divisor class that is modified nef. Indeed the argument above only uses (4.3). By Corollary 4.4, this inequality also holds for pseudoeffective divisors classes that are modified nef. Now we consider the general case: we write Z = Z + + λ i Z i , where λ i > 0 and the Z i are the irreducible components of the negative part. Fix d ≫ 0 such that for a very general element A ∈ |dL|, the intersection C i := Z i ∩ ϕ −1 (A) is irreducible (this is possible by Remark 4.7) and Z + | ϕ −1 (A) is nef (this is possible since the non-nef locus of Z + consists of at most countably many curves; cf. [Bou04, Definition 3.3]).
Since the divisors Z i are distinct, the irreducible curves C i = Z i ∩ ϕ −1 (A) are distinct. Thus we have By what precedes all the terms are non-negative, and the sum is positive unless all the classes are pull-backs from S. □ We can now give a bigness criterion on Y . For the proof of Theorem 1.4, we will need to work with R-divisor classes. This leads to some additional technical effort.
Lemma 4.8. Let Z ⊂ Y be a pseudoeffective R-divisor class, and let Z + +Z − be its divisorial Zariski decomposition. Assume that the support of Z − does not contain the exceptional divisors E P and E S . Assume that there exists a rational number ε > 0 such that (Z + εE S ) · l S > 0 and (Z + εE S ) 3 > 0. Then Z + εE S is big. Proof.
Step 1. Positivity properties of Z + εE S . The relative Picard number of the conic bundle ϕ : Y → S is two, and the relative Mori cone is generated by the curve classes l P and l S . By assumption,we have (Z + εE S ) · l S > 0. Since E P is not contained in the support of Z − and the deformations of l P cover a divisor, we have Z · l P ≥ Z − · l P ≥ 0. Thus we also have (Z + εE S ) · l P > 0, and the divisor class Z + εE S is ϕ-ample.
Let A ∈ |dL| be a very general element for some d ≫ 0. Set Y A := ϕ −1 (A), and denote by ϕ A : Y A → A the restriction of ϕ. We denote the restriction of a divisor class T to Y A by T A . We claim that (Z + εE S ) A is an ample R-divisor class.
Proof of the claim. Note that we do not have Z ≡ bϕ * L since otherwise (Z + εE S ) · l S = −ε < 0. Thus we can apply Lemma 4.6 and obtain ϕ * L · Z 2 > 0. Since the class of A is a positive multiple of L, this implies Z 2 A > 0. By construction Y A is the blow-up of P(Ω S ⊗O A ) in the finite set R S ∩π −1 (A). Since E S is the exceptional divisor of µ S , the exceptional locus of is equal to the support of E S ∩ Y A , which is a disjoint union of curves numerically equivalent to l S . Since by assumption (Z + εE S ) · l S > 0, we obtain that Thus we have where in the last inequality, we used again that E S is not in the negative part of Z. Since Z 2 A > 0 this finally shows that (4.5) (Z + εE S ) 2 A > 0. Now let Z = Z + + λ i Z i be the divisorial Zariski decomposition. By assumption none the prime divisors Z i coincides with E S or E P . Thus the intersection Z i ∩ ϕ −1 (A) is either a union of general ϕ A -fibres or an irreducible curve C i (cf. Remark 4.7). Moreover by Lemma 4.6 we have , and equality holds if and only if Z i ≡ b i ϕ * L. In both cases, the curve C i is a nef divisor on Y A . Since Z + is modified nef and A is very general, the restriction (Z + ) A is nef . In conclusion we obtain that In view of (4.5) and the Nakai-Moishezon criterion, cf. [Laz04a, Theorem 2.3.18], the claim follows if we show that (Z + εE S ) A · G > 0 for every irreducible curve G ⊂ Y A . Arguing by contradiction, we assume that there exists an irreducible curve G such that (Z + εE S ) A · G ≤ 0.
Since (Z + εE S ) · l S > 0 we have G E S . Since Z A is nef and (Z + εE S ) A · G ≤ 0, we obtain that E S · G = 0. Hence G is disjoint from the exceptional locus of µ S,A . Thus we have 0 ≥ (Z + εE S ) A · G = µ S,A * (Z + εE S ) A · (µ S,A ) * G. = µ S,A * Z A · (µ S,A ) * G.
We claim that (µ S,A ) * Z A is an ample class; this yields the desired contradiction.
For the proof of the claim, note that the push-forward (µ S ) * Z is pseudoeffective and not a pull-back from S, so it is collinear to ζ S + λπ * L for some λ ∈ Q. Since ζ S is not pseudoeffective, we have λ > 0. Yet the cotangent bundle Ω S is stable, so if A is a sufficiently positive hyperplane section, the restricted vector bundle Ω S ⊗ O A is stable with trivial determinant. Thus the restriction of the tautological class ζ S to π −1 (A) is nef. Since λ > 0 this shows that the restriction of ζ S + λπ * L to π −1 (A) is ample. This proves the claim.
We claim that such a Q-divisor class is big. Since δ 1 ϕ * L + δ 2 E P + δ 3 E S is effective, this will finish the proof of the lemma.
Proof of the claim. Our goal is to show that for m sufficiently high and divisible, we have H j (Y , O Y (m(Z δ + εE S ))) = 0 for j = 2, 3. Then we have so the bigness of Z δ + εE S follows from the asymptotic Riemann-Roch theorem and the property (Z δ + εE S ) 3 > 0.
Since Z δ + εE S is ϕ-ample, we know by relative Serre vanishing that R 1 ϕ * O Y (m(Z δ + εE S )) = 0 for sufficiently high and divisible m. By cohomology and base change, cf. [Har77, Theorem II.12.11], the direct image sheaf ϕ * O Y (m(Z δ + εE S )) is locally free and commutes with base change.
Hence the vanishing follows if we show that Since the direct image sheaf has the base-change property, we have ).
Yet (Z δ + εE S ) A is ample, so the direct image sheaf (ϕ A ) * O Y A (m(Z δ + εE S )) is ample for m ≫ 0 by [Anc82, Corollary 2.11, Theorem 3.1.]. By [Laz04b, Example 6.1.4], this implies that its dual (ϕ * O Y (m(Z δ +εE S ))) * ⊗O A does not have global sections. □ The next statement is an immediate application of Lemma 4.8 and a first step towards Theorem 1.4.
Proof. We are done if we verify the conditions of Lemma 4.8: using the formulas in Lemma 3.7, we obtain that (D + 4E S ) 3 = 10242 > 0. The divisor D is effective and prime, so the exceptional divisors E P and E S are not in the support of the negative part of its divisorial Zariski decomposition. Finally by Lemma 3.8 one has (D + 4E S ) · l S = 24 > 0. □ Proof of Theorem 1.4. For simplicity of notation, we will work with the rounded value 1.7952024. We argue by contradiction and assume that λ ≥ 1.7952024 for all prime divisors in P(Ω S ). Let Z be a pseudoeffective divisor class on Y such that (µ S ) * Z := Z S generates the second extremal ray of the pseudoeffective cone of P(Ω S ). Since (µ S ) * Z is not big, the divisor class Z + εE S is not big for every ε ≥ 0.
First assume that the negative part the divisorial Zariski decomposition of Z contains the exceptional divisor E P or E S ; then we can write Z = Z ′ + αE P + βE S with Z ′ pseudoeffective and α ≥ 0 and β ≥ 0. Since (µ S ) * Z ≡ (µ S ) * Z ′ + α(µ S ) * E P ≡ (µ S ) * Z ′ + 3απ * L generates an extremal ray, we see that α = 0. Since (µ S ) * Z = (µ S ) * Z ′ , we can replace Z with Z ′ and assume without loss of generality that the negative part of Z does not contain E P or E S . 27 Since µ S contracts the extremal ray generated by l S , we have Z · l S = 0 if and only if Z = µ * S Z S . Since Z S = (µ S ) * Z generates the second extremal ray of the pseudoeffective cone, we have Z S ≡ a(ζ S + λπ * L) with λ < 3. In particular, Z S · R S < 0, so Z| E S = (µ S | E S ) * Z S | R S is not pseudoeffective, which gives a contradiction.
Thus we have Z · l S > 0. We claim that there exists an ε > 0 such that (Z + εl S ) > 0 and (Z + εE S ) 3 > 0. Then we know by Lemma 4.8 that Z + εE S is big, which gives a contradiction.
Proof of the claim. A positive multiple of Z + εE S can be written as E S + xD + ηxE P .
Set y = ηx; Figure 1 shows the planar cubic defined by this polynomial. Hence for η ≥ −0.047976, we can find an x ≥ 0 that satisfies both conditions. (4) Figure 1. Vanishing set of (4.6) in red (note that since E 3 S > 0, the origin lies in a connected component of the plane where (4.6) is positive), line x = 0.03577 in blue and line y = −0.047976x in green. □ Remark 4.10. Theorem 1.4 may be a rather small improvement of Corollary 4.9, yet in view of Theorem 1.5 it is also clear that there is not much space left. In fact more should be true: by Corollary 4.9 the divisor D + 4E S (which in Figure 1 corresponds to the point x = 0.25, y = 0) is big. If it is also nef, the estimate in Theorem 1.5 can be improved to λ ≥ 25 14 ≈ 1.7857.