Remarks on the positivity of the cotangent bundle of a K3 surface

Using recent results of Bayer-Macr\`i, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth curves on which the restriction of the cotangent bundle is not semistable (and hence not nef). In particular, this leads to a counterexample to a question of Campana-Peternell.


Introduction
Miyaoka proved that the cotangent bundle of a non-uniruled variety is generically nef, in the sense that its restriction to a sufficiently ample and general complete intersection curve is a nef vector bundle [MR84]. This in turn has many interesting geometric consequences; see [CP11,Pet11] for more general properties of such vector bundles, in particular for the tangent and cotangent bundles. The starting point of the present paper was the following question of Campana and Peternell: Question 1.1 (Question 1.6 in [CP11]). Let f : X → Y be a birational morphism of smooth projective non-uniruled varieties. Is f * Ω 1 Y generically nef?
We answer this question in the negative, by constructing a certain blow-up f : X → S of a K3 surface. Let us sketch the construction. Let S be a generic quartic surface and let P(Ω 1 S ) denote the projectivisation of the cotangent bundle. Let L = O P(Ω 1 S ) (1) be the relative hyperplane bundle and H the pullback of the polarisation on S. Then one finds that D = L + 2H is a base-point free ample divisor (see Proposition 3.1), so a generic element X ∈ |D| is a smooth surface and the projection π : X → S is birational. Moreover, as L · D 2 < 0, the canonical quotient π * Ω 1 S → L → 0, shows that π * Ω 1 S is not nef restricted to any smooth curve C ⊂ X in the (ample) linear system |mD| X |. In particular π * Ω 1 S is not generically nef. This example has other interesting features. For instance, the curve C maps isomorphically to a (movable) smooth curve D ⊂ S with the property that Ω 1 S | D is not nef, and hence not stable. This addresses [CP19,Example 4.3], which asks about explicit examples of curves destabilising the cotangent bundle of a K3. We show more generally the following: Theorem A. Let (S, O S (1)) be a general polarised K3 surface of degree d.
(i) If d = 2 then there is a positive-dimensional family of smooth curves C in |O S (6)| so that Ω 1 S | C is not semistable.
(ii) If d = 4, 6 there are families as above in |O S (3)| (and another in |O S (4)| if d = 4). (iii) If d = 8 there is a family in |O S (3)| so that Ω 1 S | C is strictly semistable. These results should be compared with the results of Bogomolov and Hein (see Theorem 2.2), which essentially say that if either the degree of the polarisation or the multiple of the polarisation in which the curve lies is high, then there can be no smooth destabilising curves, and also that a general curve in any multiple of the polarisation is not destabilising.
More generally, one can ask how the sets of all curves and effective divisors on P(Ω 1 S ) reflect the geometric properties of S. In particular this is closely related to describing the nef and effective cones on P(Ω 1 S ), and in turn the existence of sections of the twisted symmetric differentials S a (Ω 1 S ) ⊗ O S (b). These subjects have a long history, going back to the work of Kobayashi [Kob80], who showed that a simply connected Calabi-Yau variety has no symmetric differentials, i.e., H 0 (X, S m Ω 1 X ) = 0 for all m ≥ 1. For K3 surfaces, this was extended by Nakayama [Nak04], who showed that in fact L = O P(Ω 1 S ) (1) is not even pseudoeffective on P(Ω 1 S ) (i.e., its class is not a limit of effective classes). Very recently this was extended to simply connected Calabi-Yau threefolds by Druel [Dru18,Theorem 6.1] and by Höring-Peternell [HP19] to all dimensions.
In general computing these cones explicitly seems like a difficult problem. In some low-degree cases one can use the well-known projective models to study the cohomology of the symmetric powers of the cotangent bundle, but in general the cones seem to depend on the degree d in a rather subtle way. We are able to solve the problem at least for infinitely many d, using results of Bayer-Macrì.
The starting observation is that P(Ω 1 S ) embeds in the Hilbert scheme S [2] as the exceptional divisor of the Hilbert-Chow morphism, so one can try restricting extremal divisors from the pseudoeffective and nef cones of S [2] to P(Ω 1 S ), which are known from the results of Bayer-Macrì [BM14]: the effective cone Eff(S [2] ) (resp. nef cone) is spanned by B (resp. H) and one more extremal ray which in what follows we call the second extremal ray (see Theorem 2.4). We prove that the restriction of this second ray is indeed extremal for infinitely many d, but also that this is not always the case. In what follows, we will consider the Pell-type equation (1.1) x 2 − 4ty 2 = 5.
Next, the restriction of the second extremal ray of Eff(S [2] ) is extremal on Eff(P(Ω 1 S )) if the following three conditions hold (i) t is not a square, (ii) (1.1) has no solutions, (iii) the minimal solution (a, b) of x 2 − ty 2 = 1 has b even. Finally, the restriction of the second extremal ray of Nef(S [2] ) is extremal on Nef(P(Ω 1 S )) if (1.1) has a solution. In particular we know the effective cone of P(Ω We emphasise that in the above cases the cones are explicitly computable, the slopes given in terms of solutions to Pell-type equations. We defer to Section 3 for particulars. As mentioned above, we also prove that the above trick does not work in general: we show that the restriction of the nef cone of S [2] is a strictly smaller subcone of Nef(P(Ω 1 S )) in the degree four and six cases (see Section 4). The cases in which this restriction is extremal depend on the geometry of the minimal models of S [2] , and one can find results and a classification in work of Bayer-Macrì, Hassett-Tschinkel, Markman and others.
In the final section we analyse in more detail these contractions in the cases of K3s of degree d = 2, 4, 6, 8, improving bounds on, or computing the aforementioned slopes, and describe the geometry of the extremal divisors.
Notation. We work over the complex numbers. For G a vector bundle on a variety X we say that G is pseudoeffective, big, nef, ample if O P(G) (1) is a pseudoeffective, big, nef, ample line bundle on P(G) respectively. We use Grothendieck notation throughout, so that P(G) parameterises one-dimensional quotients of G. In particular if π : P(G) → X is the projection, we have a universal quotient line bundle π * G → O P(G) (1) → 0, and for a surjection G 1 → G 2 → 0 we have an induced inclusion P(G 2 ) ⊆ P(G 1 ) so that O P(G 1 ) (1)| P(G 2 ) = O P(G 2 ) (1).

Acknowledgements.
We would like to thank Yohan Brunebarbe and Mike Roth for numerous conversations which got this project started. Also thanks to N. Addington, B. Bakker, J. Guéré, B. Hassett, H.-Y. Lin, E. Macrì, M. Mauri, G. Mongardi, K. O'Grady. We would also like to thank the Mathematisches Forschungsinstitut of Oberwolfach for providing excellent working conditions when this project began during a "Research in Pairs" of the authors.

Preliminaries
Throughout the paper we will let S denote a K3 surface with a primitive ample line bundle O S (1) of degree d = O S (1) 2 , which is always an even number, so we will sometimes write d = 2t. We will mostly assume that Pic(S) = Z, generated by O S (1). Let E = P(Ω 1 S ) and π : E → S the projection. In this case the Picard group of E is generated by L = O P(Ω 1 S ) (1) and H = π * O S (1). We have the following intersection numbers: L 3 = s 2 (Ω 1 S ) = −24, L · H 2 = d, We denote by α e,d and α n,d the (positive) real numbers which are the slopes of the pseudoeffective and nef cones of E respectively. In other words, where the first follows from the fact that pr * 1 O S (1) ⊗ pr * 2 O S (1) restricts to O S (2) on the diagonal ∆ ⊂ S × S.

Stability of restrictions
As mentioned in the introduction, the positivity of Ω 1 S on a curve is closely related to stability. This is because of Hartshorne's theorem [Laz04,§6.4.B] which implies that on a smooth curve, a vector bundle of degree zero is nef if and only if it is semistable. In particular, Lemma 2.1. Let (S, O S (1)) be a K3 surface and C ⊂ S be a smooth curve. Then Ω 1 S | C is nef if and only if it is semistable.
A consequence of Yau's Theorem is that Ω 1 S is stable with respect to any polarisation. This does not imply that the restriction to each curve C is stable, as there can be negative quotients Ω 1 S | C → Q → 0 which do not come from quotients of Ω 1 S . On the other hand, one can apply results of Bogomolov to show that for any stable vector bundle G on a surface S, if we take a smooth curve in a (computable) high enough multiple of the polarisation, then the restriction G| C is stable. In our case, Bogomolov's theorem [HL10,Theorem 7.3.5] implies that if C ∈ |O S (n)| for n > 49 then Ω 1 S | C is stable. In fact, if S is a K3 surface this has been extended in various directions by Hein. Hein's results apply to any K3 surface, but we only state them in the case of Picard number one.
Theorem 2.2 (Hein). Let (S, O S (1)) be a smooth projective K3 surface of degree 2t and Picard number one.
(1) If t(2m − 1) > 48 then for any smooth curve C ∈ |O S (m)| we have that Ω 1 S | C is a stable vector bundle.
Proof. The first statement follows directly from [Hei06, Theorem 2.8]. Then (ii) follows from [Hei96, Korollar 3.11], and the fact that a general member of |O S (m)| is not hyperelliptic for these values of t and m.
To be more concrete, for m = 1 we obtain t > 48, for m = 2, t > 16 and m = 3 gives t > 9. On the other hand, for t ≥ 1 we obtain that m ≥ 25, which is an improvement on the bound of Bogomolov mentioned above.
In other words finding curves which destabilise the cotangent bundle is limited to low degree K3s and low degree multiples of the polarisation, and even more so only to small parts of the linear systems considered. Constructing such curves in general seems out of reach, so in Section 4 we proceed on a case by case basis.
An initial natural attempt is to consider the ramification curves of a generic projection S → P 2 . The starting point here is the fact that in the degree two case, the ramification curve C of the degree two cover f : S → P 2 is smooth, and the restriction of Ω 1 S to it is not semistable (see Section 4.1). As explained in the proof below, these curves lie in |O S (3)|, so Hein's Theorem 2.2 implies that they cannot be destabilising if t > 9.
Proof. From [CF11] we know that since Λ is general, the projection S → P 2 is a morphism and R Λ is a smooth irreducible curve in |O S (3)|. Consider the natural morphism φ : P(Ω 1 S ) → Gr(2, t + 2) (see Section 3). This factors as where U is the universal subbundle on the Grassmannian and p is the bundle projection. Let us consider σ 2 ⊂ Gr(2, t + 1) the Schubert cycle parameterising lines meeting the (t − 1)-plane Λ. The class φ * σ 2 is represented by a smooth curve C ⊂ P(Ω 1 S ) whose image in S is a curve whose points correspond to lines meeting S with multiplicity at least two that also meet Λ, i.e. the curve R Λ . We claim that L is negative on this curve.
To compute the class of C, note that σ 2 = c 2 1 (U ∨ ) − c 2 (U ∨ ). On P(U ∨ ), the line bundle O P(U ∨ ) (1) corresponds to H and g = L + 2H corresponds to the pullback of the Plücker polarisation. Using the Grothendieck relation on P(U ∨ ) we find that φ * σ 2 = L 2 + 3L · H + 3H 2 . This gives L · φ * σ 2 = 6t − 24 which is negative if t ≤ 3, so L · C < 0. Since R Λ is smooth, we have from Lemma 2.1 that Ω 1 S | R Λ is not semistable for t ≤ 3. If t = 4, we get a degree zero quotient of the vector bundle π * Ω 1 S restricted to the curve C, and hence by pushing down a degree zero quotient on R Λ . Hence Ω 1 S | R Λ is semistable as it is an extension of degree zero line bundles, but not stable.

Results of Bayer-Macrì
To approach the computation of the pseudoeffective and nef cones of P(Ω 1 S ), we will repeatedly be using the results of Bayer-Macrì [BM14, Section 13]. Note that the pseudoeffective and movable cones are dual under the Beauville-Bogomolov-Fujiki form, which allows the computation of the former from the results loc. cit. on the latter (see [BM14,p.570]). (1) Assume the equation x 2 − 4ty 2 = 5 has no solutions.
(a) If t is a perfect square then Nef( If t is not a perfect square then x 2 − ty 2 = 1 has a minimal solution 1 (a, b) and the cones are given by whereas Mov(S [2] ) and Eff(S [2] ) are as in Case (1).
The above results in fact give a complete picture of the birational models of S [2] . We give a short description here as they will be important in what follows. First note that one extremal ray of the movable cone of S [2] is given by H, and it induces the Hilbert-Chow contraction to Sym 2 S. We now describe the other ray of the movable cone, which we will often call the second extremal ray. Firstly, assume that the Pell-type equation does not have a solution. If t is a square we have a Lagrangian fibration S [2] → P 2 , and hence the second boundaries of Nef(S [2] ) and Eff(S [2] ) agree. If t is not a square then (see [DM17, Proposition 3.6]) there is a divisorial contraction S [2] → Y contracting an irreducible divisor D to a smooth K3 surface T ⊂ Y and the restriction to D is a P 1 -fibration D → T (in the sense that the general fibre is P 1 ). We note that like in the example of the Hilbert-Chow contraction to Sym 2 S, the variety Y need not be smooth.
If on the other hand the above Pell-type equation does have a solution, then there is a finite sequence of Mukai flops to a smooth irreducible holomorphic symplectic variety S [2] X + which, as above, either admits a divisorial contraction X + → Y , contracting an irreducible divisor D → T to a smooth K3 surface T (if t is not a square), or has a Lagrangian fibration X + → P 2 (if t is a square).

The cones of divisors of P(Ω 1 S )
We begin with some simple bounds for the nef and pseudoeffective cones on P(Ω 1 X ) for a smooth projective variety X ⊂ P n . A convenient ingredient here is the morphism which associates a tangent vector to the corresponding point in the Grassmannian. For L the tautological line bundle on P(Ω 1 X ) and H the polarisation on X as in Section 2, one checks that f * O Gr(2,n+1) (1) = L + 2H. When f maps to a lower-dimensional variety, L+2H is extremal in the nef cone as well as the pseudoeffective cone.
Proposition 3.1. The vector bundle Ω 1 P n (m) is nef if and only if m ≥ 2. More precisely, Nef(P(Ω 1 P n )) = Eff(P(Ω 1 P n )) = H, L + 2H . Moreover, if X ⊆ P n is a smooth subvariety, then L + 2H is nef, and ample if and only if X does not contain a line.
Proof. The equality of the cones follows from the preceding paragraph. The surjection Ω 1 P n | X → Ω 1 X allows us to factor f as P(Ω 1 X ) ⊂ P(Ω 1 P n | X ) ⊂ P(Ω 1 P n ) → Gr(2, n + 1). So L + 2H is base-point free, and ample if and only if f is finite, i.e., when X does not contain a line.
Thus, if X ⊂ P n a smooth subvariety, the slope of the nef cone of P(Ω 1 X ) (with respect to O P n (1)| X ) is at most 2. Using results of Section 2.2, we can improve this in the case of K3 surfaces. Note first that if S a K3 surface of degree 2t, then the nef cone is contained in the positive cone {D | D 3 ≥ 0}. However, the boundary divisor L + 2 √ t H of this cone can in general fail to be nef (see Section 4). Nevertheless, the following shows these two cones approximate each other if t is large.
There are similar lower bounds for α n,2t , α e,2t , showing that they both limit to 2 √ t for large t.
Proof. From Theorem 2.4 the nef cone of S [2] is determined by the fundamental solution to the Pell-type equation x 2 − 4ty 2 = 5. Assume first that this has no solutions. In this case the nef and movable cones agree. First, if t is a square then the divisor H − √ tB is the boundary of the nef cone. If t is not a square, for a solution (x, y) of x 2 − ty 2 = 1, we must have x ≥ √ t. Rewriting the equation as t , and so we find that the divisor H − √ t − 1B is nef. If x 2 − 4ty 2 = 5 has a solution, then for such a solution (x 1 , y 1 ) with x 1 > 0 minimal and y 1 > 0, the boundary of the nef cone is given by t from which the Pell-type equation gives This implies that H − aB is nef for a = √ 4t−5 2 . From this we see that regardless of whether x 2 − 4ty 2 = 5 has a solution or not, the divisor ( We will see below that the bound for α e,2t is actually attained for infinitely many d. First, we prove the following result for the nef cone: Theorem 3.3. Let (S, O S (1)) be a K3 surface of degree 2t and Picard number one. Assume that either t is a square or that x 2 − 4ty 2 = 5 has a solution. Then the restriction map is an isomorphism.
Proof The conditions on t imply in particular that we are not in case (iii), so we consider the other two. In case (i) we have D| 3 E = 0 and so D is also extremal on E, meaning the restriction map (3.2) is indeed an isomorphism.
In case (ii), note first that P ∩ E ∅: If not then the class of a line would have to also satisfy H · = 0, but this does not happen though as the Hilbert-Chow morphism does not contract . This means that P and E intersect along a curve R, and equals a multiple of i * R.
Remark 3.4. In case (iii), the map (3.2) is not always an isomorphism. This is because the divisorial contraction S [2] → Z may restrict to a finite map on E. This happens for instance when the degree is four or six (see Sections 4.2 and 4.3).
We now turn to the problem of computing the pseudoeffective cone of P(Ω 1 S ) of a K3 surface S in some special situations. First we observe that if the Hilbert scheme admits a Lagrangian fibration S [2] → P 2 , i.e., when t ≥ 2 is a square and Equation (1.1) has no solution, we have that the second extremal divisors of Eff(S [2] ) and Nef(S [2] ) agree, and since E is mapped to a lower-dimensional variety, this extremal divisor H − √ tB restricts to an extremal ray of both Eff(E) and Nef(E). In particular, the above bound of Proposition 3.2 is optimal for infinitely many values of d.
We remark, however that Nef(E) need not equal H, L + 2 √ t B in general -this happens for instance for quartic surfaces (see below).
We will now focus our attention on K3 surfaces S of Picard number one and degree d = 2t so that S [2] has two divisorial contractions, namely the Hilbert-Chow and a contraction to a normal variety Y . We denote the extremal divisor of this contraction by D ⊂ S [2] . As pointed out in an earlier section, the image of D via the contraction is a smooth K3 surface T ⊂ Y . Proposition 3.6. Let (S, O S (1) be a K3 surface of degree 2t and Picard number one. Assume that t is not a square and x 2 − 4ty 2 = 5 has no solutions 3 . In the notation above, assume furthermore that the contracted divisor is D P(Ω 1 T ) for T a K3 surface. Then Eff(E) = D| E , H .
Proof. We want to show that D| E is extremal in the pseudoeffective cone, or equivalently that h 0 (E, mD| E ) grows at most quadratically in m. First note that h 0 (S [2] , mD) = 1, since D is an exceptional divisor. From the exact sequence is bounded as a function of m, namely that for m large enough the dimension of this vector space stabilises; this will imply that mD| E can only have one section. For this, we consider the sequence , and this is zero for m 0, since Ω 1 T is not pseudoeffective. Remark 3.7. The above method of proof cannot work in general as Hassett and Macrì explained to us that in degree 16, the second extremal divisor is not even isomorphic to a projective bundle.
A classical case where the assumptions are satisfied is when S [2] admits an involution (which is not induced from an involution of S); this happens for instance when S is a quartic surface (see Section 4.2 for details). In fact, the main theorem of [BCNWS16], gives a complete classification of the degrees for which S [2] admits an involution which acts non-trivially on the cones of divisors. In this case the second extremal effective divisor D of S [2] is simply the image of E under the involution.
More generally, the proposition applies in the case of ambiguous

Examples in low degrees
We work through the first four cases of degree d = 2t Picard number one K3 surfaces, in each case applying and extending the results of the previous section and explain the underlying geometry of the numbers. Note that some values of α e,d , α n,d of low degree K3s were computed in [OP96], although their results are incorrect in the degree 4 case. We correct this below and study the problem for K3 surfaces of Picard number one of degree d ≤ 8. We summarise the results in the following

d = 2, Double covers of P 2
Let S be a generic degree two K3 surface, and let f : S → P 2 be the double cover with ramification divisor R and branch locus a plane sextic curve C. There is an exact sequence Here f restricts to an isomorphism on R, which from the standard cotangent sequence corresponding to the composition R ⊂ S → P 2 implies that Ω 1 . In particular after restricting to R we obtain a negative quotient Ω 1 This gives not only a smooth curve in S which destabilises the cotangent bundle, but also a curve C ⊂ E = P(Ω 1 S ) which is extremal in the cone of curves. Indeed C is exactly the intersection of the P 2 ⊂ S [2] (given by the preimages of S → P 2 ) with E, and so it can be contracted to a point. We come now to movable curves destabilising the cotangent bundle. Proof. To avoid repeating the same argument we refer to the case of the quartic below in Proposition 4.3 for what follows. We saw from the sequence (4.1) that the bundle Ω 1 S (3) is globally generated, and so the divisor D = L + 3H is basepoint-free on P(Ω 1 S ). This implies that there is a smooth surface S ∈ |D| and a smooth movable curve C ∈ |D| S |. Now the intersection number L · D 2 is −6, which means the quotient line bundle Hence Ω 1 S has a negative quotient when restricted to the image of C in S and so it is not nef. We compute that K 2 S = −42 and K S · D = 42 meaning S → S is a smooth blowup along 42 distinct points and the image of C in S is smooth. Note that D 2 · H = 12, so that the images of these curves move in a positive dimensional family in |O S (6)| on S. Since h 0 (S, Ω 1 S (3)) ≥ h 0 (S, f * Ω P 2 (3)) = 8 this family is of dimension at least dim Gr(2, 8) = 12.

d = 4, Quartics in P 3
Let S ⊂ P 3 be a quartic surface of Picard number one. As studied by Beauville in [Bea83], S [2] has an involution i : S [2] → S [2] which maps the reduced length two subscheme p + q ∈ S [2] to the residual degree two subscheme on the intersection S ∩ p,q of S with the line through p and q. When S has Picard number one, this involution must swap the boundaries of the pseudoeffective and nef cones.
There is a natural morphism f : S [2] → Gr(2, 4) taking a 0-cycle of degree two and associating to it the line in P 3 spanned by the two points. Since a general line meets S in four points, this is generically finite of degree 4 2 = 6. In terms

The pseudoeffective cone of P(Ω 1
S ) This has been computed in Theorem 3.8. Since the second boundary of the pseudoeffective cone of S [2] restricted to E is given by ( H − 3 2 B)| E = 3 2 L + 2H we obtain the following. The divisor i(E)| E = 6L + 8H is effective on E. Indeed, this is exactly the surface of bitangents U 0 ⊂ P(Ω 1 S ) (see [Tih80,Proposition 2.3] and [Wel81, Proposition 3.14]). On the other hand, the generator 3L + 4H is not itself effective: taking symmetric powers of the standard cotangent sequence associated to a quartic in P 3 one obtains We can compute the cohomology of the middle term using the sequences where α k = k+n k and β k = k+n−1 k−1 . Hence H 0 (P(Ω 1 S ), 3L + 4H) = 0, since it is sandwiched between H 0 (S, S 3 Ω 1 P 3 | S (4)) and H 1 (S, S 2 Ω 1 P 3 | S ), the first of which is zero (e.g., by Proposition 3.1), whereas the second is zero by Kodaira vanishing.

The Nef cone of P(Ω 1
S ) Going on now to explain Remark 3.4, consider again the surface U 0 = i(E) ∩ E of bitangents of S. The Hilbert-Chow morphism restricts to a finite morphism U 0 → S of degree six (see [Wel81]; counted with multiplicities, there are six bitangent lines through any point in S). In other words, the morphism i(E) → Sym 2 (S), induced by i(H) is finite (since any curve contracted would have to lie in i(E) ∩ E). This implies that the divisor i(H)| E = 4L + 6H is ample on E, and hence α n,4 < 3 2 . Now, from the computation of the pseudoeffective cone above one sees that L + √ 2H is pseudoeffective, but (L + √ 2H) 2 (3L + 4H) < 0 which means that L + √ 2H is not nef. Hence √ 2 < α n,4 < 3 2 .

Families of Destabilising Curves
We will give two families of destabilising curves on S, the second of which has a natural geometric description as the family of ramification divisors from a generic projection S → P 2 . Proof. Observe first that the line bundle D = L + 2H is base-point free on P(Ω 1 S ) by Proposition 3.1. By Bertini, there is a smooth surface S ∈ |D|. Recall that on P(Ω 1 S ), we have L 3 = −24, L 2 · H = 0, L · H 2 = 4. Let = L| S and h = H| S = f * O S (1). This implies that we have the following intersection numbers for and h on S : 2 = −24, · h = 8 and h 2 = 4. It follows that · ( + 2h) = −8. Note that D| S = + 2h is nef, so the restriction of L is not pseudoeffective on S .
Denote by f : S → S the restriction of π to S; S is generically a section of π, so f is birational. In fact, computing the canonical bundle of S gives K S = (K P(Ω 1 S ) + D)| S = 2H − L which gives that K 2 S = −40. In particular f : S → S must be the blow-up of S in 40 distinct points since any exceptional curve is contracted to S so must be a fibre of P(Ω 1 S ) → S. Moreover D · K S = (L + 2H) · (−L + 2H) = 40 so the general divisor in C ∈ |D| S | passes through each exceptional divisor with multiplicity one, implying that its image C = f (C) in S is smooth. We have 16 = D 2 · H = π * D 2 · O S (1) and so it follows that C lies in |O S (4)|. In fact we obtain an 8-dimensional family this way by intersecting two generic divisors in |D|.
The second family was already proven to be destabilising in Proposition 2.3. Using [CMT01, Proposition 2.6] we compute that for p ∈ P 3 generic, the ramification divisor R is a smooth genus 19 curve and it is the normalisation of its image, the branch divisor B. Also, the branch divisor B ⊂ P 2 is irreducible of degree 12 and from [CF11] has only cusps and nodes as singularities. Using the results loc. cit. one computes it has 12 nodes and 24 cusps, and since only the latter contribute to ramification, this implies that deg Ω 1 R/B = 24. The conormal bundle I/I 2 of R ⊂ S is O R (−3) and from the sequence → 0 we see that since the left has degree −36 and the right 24, we obtain that deg Ω 1 f | R = −12. From the sequence Ω 1 S | R → Ω 1 f | R → 0 we thus obtain a negative quotient, which is the desired explicit destabilising quotient.

d = 6, Intersections of cubics and quadrics in P 4
Here Nef(S In this case the second birational model is a divisorial contraction like in the t = 2 case, but S [2] is not ambiguous, so we are not in the case of Theorem 3.8. We will compute the effective cone, in particular proving that α e,6 = 1, using the beautiful geometry of S [2] related to nodal cubic fourfolds. We will also show that like in the d = 4 case, the restriction of the nef cone from S [2] is in the interior of Nef(P(Ω 1 S )), obtaining at least in this case that α n,6 < 4 3 .
Remark 4.4. One could alternatively use cohomological techniques to compute that the divisor given by the restriction 2L + 2H = ( H − 2B)| E is extremal in the pseudoeffective cone. We do not reproduce the proof here as it is a lengthy computation, but it follows from the fact that S is the intersection K ∩ Q ⊂ P 4 of a smooth cubic and quadric, so one can take symmetric products of the Euler sequence restricted to K and the cotangent sequence 0 → O S (−1) → Ω 1 K | S (1) → Ω 1 S (1) → 0 to show that h 0 (S, S m (Ω 1 S (1))) does not have cubic growth in m.
We begin by describing a birational model of S taking two lines 1 , 2 through p and giving the residual line of intersection of Z with the plane spanned by 1 , 2 . Note that this is indeed a morphism (it is not defined precisely when Z contains a 2-plane through p, which we may assume does not happen by genericity). Starting with a line ∈ F \ S and considering the plane spanned by and p, we obtain a residual singular conic in Z whose constituent two lines give a unique point of S [2] . This implies that the map f is birational. Moreover one can prove that F has A 1 singularities along S and is smooth otherwise. Finally S [2] is isomorphic to the blowup Bl S F of F at S (see [Has00, Lemma 6.3.1]), and M is the pullback of the Plücker polarisation. Let D denote the exceptional divisor of f ; the map f restricts to a smooth P 1 -fibration D → S. Geometrically, this means that if one fixes a line through p, then there is a family over P 1 of 2-planes through whose residual conic is two lines 1 + 2 ∈ S [2] passing through p.
Consider now Gr(2, 4) the Grassmannian of lines in P 3 , embedded as a quadric in P 5 and embed Q from above as a hyperplane section of Gr(2, 4).
Lemma 4.5. The exceptional divisor D is isomorphic to P(U ∨ | S ) where U is the universal bundle on Gr(2, 4).
Proof. Note that points of D ⊂ S [2] parameterise pairs of lines through the node p ∈ Z so that the residual line in the 2-plane spanned by the two lines also passes through p. Projecting from p, we see that the images of the three lines thus lie on a line ⊂ Q, which is a trisecant to S.
Let S denote the rank 2 spinor bundle on Q. We have S = U | Q where U is the universal subbundle on Gr(2, 4). Also S ∨ S(1). The projective bundle π : P(S ∨ ) → Q parameterises lines in Q; the fiber over q ∈ Q is the P 1 of lines ⊂ Q through q. We assume that S contains no lines, so for a point s ∈ S ⊂ Q, and a line containing s, the intersection ∩ S consists of s and a length two subscheme ξ ∈ S. This ξ in turn corresponds to two (possibly equal) lines 1 , 2 in Z passing through p, and the residual line of these is exactly the line corresponding to s. We have therefore defined a morphism P(S ∨ ) → D. It is clear that this morphism is bijective, since given 1 , 2 , the line is determined from the secant line through the images of the projections of 1 , 2 in S.
Corollary 4.6. The divisor L + H is extremal in the effective cone, giving α e,6 = 1.
Proof. The line bundle l = O P(U ∨ | S ) (1) is ample since it induces a map P(U ∨ | S ) → P 3 which is finite. Indeed, regarding P 3 as the variety of lines in Q, the preimage of a point [ ] ∈ P 3 consists of the points ∩ S, which is finite since S contains no lines. Note now that −D| D = −K D which equals 2l − g for g the pullback of the hyperplane bundle on S. The curve class l 2 is ample and one computes l 2 · (2l − g) = (σ 2 1 − 2σ 2 ) · σ 2 1 = 0 for σ i the standard Schubert cycles, giving in particular that 2l − g is not pseudoeffective. We will finally sketch how the proof of Proposition 3.6 implies that D = H − 2B is extremal on E, giving that α e,6 = 1. To this aim, we must show that H 1 (D, mD| D − E| D ) = H 1 (P(U ∨ | S ), mD| D − E| D ) vanishes for m large. This group equals H 0 (S, S 2m+a U | S ⊗ (det U | S ) m+b ), for some a, b integers. It follows from the fact that 2l − g is not pseudoeffective as seen above, along with the isomorphism U U ∨ ⊗ det U , that this group is zero for large m.
As far as the nef cone is concerned, we prove the following analogue of the degree four case.
Lemma 4.7. The restriction of the above morphism f : D ∩ E → S is finite. In particular D| E is ample.
Proof. If not there exists a line ⊂ Z through p, and a 1-dimensional family H t of planes intersecting Z in l and a double line 2l t . Consider the projection from p, π : P 5 −{p} → P 4 , which blows down the lines through p to S. The planes H t give a family of lines in P 4 passing through a fixed point q ∈ S and intersecting S with multiplicity two at some other points q t . Since S = K ∩ Q, we see that Q must contain all of these lines. But the lines of Q passing through q are parameterised by a conic and sweep out a 2-dimensional quadric cone Q 0 with vertex at p. This quadric Q 0 spans a H P 3 , and the intersection S ∩ H contains S ∩ Q 0 and thus has a non-reduced divisor as a component. However, this is impossible if Pic(S) = ZH.
In particular, like in the case t = 2, we have that the restriction (2 H − 3B)| E = 3L + 4H is ample on E and one sees that the slope of the nef cone α n,6 is strictly less than 4 3 . On the other hand, we compute that (L + 6 5 H) 2 · (L + H) < 0, so in particular 4 3 < α n,6 < 6 5 . Finally, we can as before use the map φ : P(Ω 1 S ) → Gr(2, 6) to produce an explicit destabilising family of curves on S. Indeed, the curve C ⊂ P(Ω 1 S ) corresponding to lines tangent to S and meeting a general P 2 ⊂ P 4 , has intersection number −6 with L. Pushing this forward to S gives a 6-dimensional family of destabilising curves in |O S (3)|.
The geometry of this fibration is well understood (see [HT00,Proposition 7.1] and [Saw07, §2.1]) and we include some details here for completeness. Consider S as the intersection of three quadrics Q i ⊂ P 5 . Then we have a morphism S [2] → (P 2 ) ∨ P 2 defined by taking a 0-cycle ξ of degree two and giving the base P 1 of the pencil of quadrics (in the net spanned by the Q i ) containing the line in P 5 spanned by ξ. A fibre of this morphism is given by the abelian surface which is the Fano scheme of lines of the threefold which is the base locus of the corresponding pencil. In fact the general fibre will be the Jacobian of a genus two curve: a general pencil of quadrics in P 5 is singular at precisely six points and we take the genus two curve ramified above these six points. In particular, following the description from Theorem 3.3, it is the class of this curve that spans the Mori cone of E, which is also equal to the movable cone of curves of E.

Bott vanishing for K3 surfaces
The following question was raised by Totaro in [Tot20].
Here one does not assume that O S (1) is primitive. As explained in the above reference, by Kodaira vanishing, one need only treat the case p = 1. The problem is essentially solved in [Tot20], but let us show how one can use our above results to easily obtain Bott vanishing for K3 surfaces (S, O S (1)) of Picard number one with O S (1) 2 > 75. Indeed, using the upper bound for the nef slope α n,2t from Proposition 3.2, we obtain that if O S (1) 2 > 75, the divisor D = 3L + H is ample on E = P(Ω 1 S ). Since K E = −2L, one finds that H 1 (S, Ω 1 S ⊗ O S (1)) = H 1 (E, L + H) = H 1 (E, K E + D) = 0 by Kodaira vanishing. In general Bott vanishing can fail; for example because χ(Ω 1 S ⊗ O S (1)) is negative when S has low degree.

Questions
As noted before, we are only able to compute the cones for certain degrees of K3s, leaving several open questions. For instance, we ask for E = P(Ω 1 S ): Question 5.2. What is the second extremal ray of Eff(E) for a degree 2 K3 surface, or more generally when S [2] admits a flop? Question 5.3. What is the second extremal ray of Nef(E) in degree 4 and 6, or more generally when there is a second divisorial contraction S [2] → Z?
Even though the slopes for S [2] are all rational numbers, the situation for P(Ω 1 S ) is less clear: Question 5.4. Let S be a K3 surface with Picard number one. Are the cones Eff(E) and Nef(E) rational polyhedral?