Walls and asymptotics for Bridgeland stability conditions on 3-folds

We consider Bridgeland stability conditions for three-folds conjectured by Bayer-Macr\`i-Toda in the case of Picard rank one. We study the differential geometry of numerical walls, characterizing when they are bounded, discussing possible intersections, and showing that they are essentially regular. Next, we prove that walls within a certain region of the upper half plane that parametrizes geometric stability conditions must always intersect the curve given by the vanishing of the slope function and, for a fixed value of s, have a maximum turning point there. We then use all of these facts to prove that Gieseker semistability is equivalent to asymptotic semistability along a class of paths in the upper half plane, and to show how to find large families of walls. We illustrate how to compute all of the walls and describe the Bridgeland moduli spaces for the Chern character (2,0,-1,0) on complex projective 3-space in a suitable region of the upper half plane.


Introduction
Bridgeland's notion of stability conditions on triangulated categories, introduced in [Bri07] and [Bri08], provides a new set of tools to study moduli spaces of sheaves on smooth projective varieties. Such tools have been successfully applied by many authors first to the study of sheaves on surfaces, see for example [AM16,AB13,BM14a,BM14b,Fey16,Fey17,FL21,MM13,YY14], and more recently on threefolds (especially P 3 ); see for instance [GHS18,MS20,Sch20a,Schi20b]. One way to study moduli spaces of sheaves using Bridgeland stability spaces is to restrict attention to the so-called geometric stability conditions parameterized by (a subset of) the upper half plane H. Once we know that the moduli space of Bridgeland-stable objects is asymptotically given by the Gieseker semistable moduli space along an unbounded path, we can try to locate all the points where the moduli space changes along this path (these isolated points are called walls) and compute the change to the moduli space. Eventually, we might reach a point where the Bridgeland space is empty, and then we can reverse our steps to reconstruct the Gieseker moduli space.
For surfaces, this is a fairly well-understood process. In that case, it is known that the geometric stability space is non-empty, that there only finitely many walls in H away from the β-axis which are nested semicircles centered along the horizontal axis and that Bridgeland stability is asymptotic to (twisted) Gieseker stability. Furthermore, there is an effective algorithm to find all such walls for a given Chern character, and then we can carry out the process above to recover the moduli space of semistable sheaves. One approach to finding walls in this case is to observe that every wall for a given Chern character v intersects a special curve which we will denote by Θ v in this paper, given by the vanishing locus of the slope function ν α,β (v), and we can then restrict our attention to finding walls along Θ v .
The whole process becomes much more complicated for threefolds. We can still use the 2-dimensional construction, but it does not produce full stability conditions, and it is unable to detect sufficient features of the Gieseker moduli spaces since the latter does not coincide, in general, with the asymptotic moduli space.
The first step to improve this was made possible by a number of results guaranteeing the existence of Bridgeland stability conditions on the derived category of sheaves on different types of threefolds, based on the pioneering work of Bayer, Macrì and Toda [BMT14]. Their idea is to start with the surface case and tilt again. This provides a full stability condition, and the family of moduli spaces is considerably more refined than the one provided by the first tilt. Even though there is no general result which shows that their construction works for all smooth threefolds, it is known to work for a wide variety of relevant examples: P 3 [Mac14], smooth quadric threefolds, cf. [Sch14], abelian threefolds, cf. [BMS16,MP16], Fano threefolds with Picard rank 1, cf. [Li19b], more general Fano threefolds, cf. [BMS + 17, Piy17], and smooth quintic threefolds, cf. [Li19a]. More precisely, the geometric stability conditions constructed by Bayer, Macrì and Toda via the generalized Bogomolov-Gieseker inequality proposed in [BMT14] depend on three real parameters (α, β, s) ∈ R + × R × R + . For each of these, we have an abelian category A α,β and a slope function λ α,β,s which allows us to test the stability of objects of A α,β . There are, however, known counterexamples (see [Sch17] and [MS19]) where the generalized Bogomolov-Gieseker inequality fails.
The goal of this paper is to advance on the other two stages of the process outlined above, namely the understanding of the structure of walls and that of asymptotic stability. We only consider the case where X is a smooth projective threefold of Picard rank 1 over an algebraically closed field of characteristic 0. This means that we can view our Chern classes (and their twists) as purely numerical vectors of the form v = (v 0 , v 1 , v 2 , v 3 ).
In order to study walls, we start by providing a uniform way to define the slope functions and their differences in terms of skew-symmetric functions. We go on to consider a number of general properties of numerical λ-walls (defined as the locus where two λ-slopes are equal) in Section 4.1. They are, in general, quartic curves, possibly unbounded and not connected.
In our first main result, we find a simple characterization of those numerical λ-walls which are bounded, and we show that unbounded walls satisfy a version of Bertram's nested wall theorem. Given numerical Chern characters v, u and u , we define δ 01 (u, v) := u 0 v 1 − u 1 v 0 and an equivalence relation u ∼ v u which is essentially that the λ-walls for v corresponding to u and u are the same; see (4.9) for a precise definition. We also remark that when v is a numerical Chern character satisfying the Bogomolov-Gieseker inequality v 2 1 − 2v 0 v 2 ≥ 0 and v 0 0, the curve Θ v allows us to divide the upper half plane H into four regions (see Figure 1 for an example and Section 3 for details).

Main Theorem 1.
Suppose v 0 0 and u v u .
(1) The numerical λ-wall for v corresponding to u is bounded if and only if δ 01 (u, v) 0.
(3) If δ 01 (u, v) 0 and ch ≤2 (u) = ch ≤2 (u ), then the numerical λ-walls for v corresponding to u and u only intersect on Θ v . (4) An unbounded numerical λ-wall for v does not intersect Θ v , and its unbounded connected components are contained in R 0 v .
The different parts of Main Theorem 1 are proved in various results contained in Section 4. The third stage, determining the asymptotics, is not well known. In [BMT14,Section 6], it is shown that the large volume limit as α → ∞ for (α, β) ∈ H gives a polynomial stability condition, but the converse and other directions were not considered. However, unlike the 2-dimensional case, we cannot assume the walls are bounded in all directions. In fact, it is easy to check that as s → 0, the walls are unbounded, and it is theoretically possible that the number of walls is infinite.
This means that the large volume limit is more subtle than for surfaces. We need, therefore, to be more careful about what we mean by asymptotic stability, which we define precisely in Definition 7.1. We use a strong form of such asymptotic stability which effectively includes finiteness of the number of walls for a given object. We also need to be careful to specify the curve along which we are considering the asymptotics. To this end, we introduce the notion of unbounded Θ ± -curve, which is essentially a curve which is asymptotically either to the left or to the right of all Θ-curves; see Definition 5.9.
To help set this up, we also consider what would happen for surfaces, in Section 5. In our case, we look at so-called ν-stability for threefolds, which mimics stability for surfaces given by the first tilt on the category of coherent sheaves, by re-proving results about the large volume limit without the assumption that the walls are bounded.
In order to describe the asymptotics, we need to understand how the stability of an object varies along curves. For ν-stability, it turns out that stable objects can only be destabilized once along inward-moving curves (which cross ν-walls only once). It turns out that this also holds for λ-stability outside the ν-wall. See Theorem 7.8 for the details. Accomplishing such a task requires an understanding of the geometry of λ-walls. For the ν-walls, this was simple because they were circles, and the key property is that they cross the Θ v -curve at their maximum. To understand the similar properties of λ-walls, we need to understand their differential properties in a similar way. We do this is Section 6. Unlike ν-walls, λ-walls need not be regular, and we carefully study when regularity fails. When s = 1/3, it is straightforward to prove that the walls are regular except if they happen to cross a very special point (on Γ u,1/3 and its ν-wall), in which case there is a cusp. For other values of s, it is much harder. It turns out to be easier to study the differential properties of the 2-dimensional wall where we allow s to vary, which we call Σ u,v , regarded as a real algebraic quartic surface in R 3 . We show that Σ u,v is regular except at some exceptional points and for exceptional u and v (see Theorem 6.7 for the details).
Moreover, there is again a special curve, here denoted by Γ v,s , which is defined as the vanishing locus of the slope function λ α,β,s (v). When s = 1/3, a numerical λ-wall for v crosses Γ v,1/3 at its maximum point (just as a ν-wall crosses Θ v at its maximum) and the associated ν-wall at its minimum point. This imposes large constraints on the possible numerical λ-walls when s = 1/3. We also show that for s ≥ 1/3, any wall existing for one value of s must also exist for all s. When s < 1/3, we show that any wall existing for s exists for all value less than that of s.
The key conclusion is the following; see also Theorem 6.17. The proof uses key differential-geometric information about Σ u,v such as its Gauss and mean curvatures.
Main Theorem 2. Suppose a real numerical Chern character v satisfies the Bogomolov-Gieseker inequality and v 0 0. Any connected bounded component of a numerical λ-wall in R − v for some s ≥ 1/3 intersects Γ − v,s . Although the same statement is not true for unbounded walls, we can describe the explicit conditions u must satisfy so that the wall corresponding to u intersects Γ v,s .
We are then finally in position to prove, in Section 7, that strong asymptotic stability is equivalent to Gieseker stability. Our results can be summarized as follows.

Background material and notation
Let X be an irreducible, non-singular projective variety of dimension 3 over an algebraically closed field of characteristic 0 with Pic(X) = Z. Fix an ample generator L of Pic(X), and write = c 1 (L). Our assumptions mean that each object A ∈ D b (X) has a well-defined numerical Chern character ch(A) := (ch 0 (A) 3 , ch 1 (A) · 2 , ch 2 (A) · , ch 3 (A)) ∈ Z × Z × 1 2 Z × 1 6 Z.
Abusing notation, we will simply write ch i (A) for ch i (A) · 3−i . We will refer to an element of R 4 = R ⊗ K num (X) as a real numerical Chern character and an element v of Z × Z × 1 2 Z × 1 6 Z as a Chern character when there is an object A ∈ D b (X) such that v = ch(A). We write the components as v = (v 0 , v 1 , v 2 , v 3 ) corresponding to the Chern characters of objects so that the underlying real Chern character v which is the numerical Chern character ch(A) of an object of D b (X) satisfies v i = ch i (A).
Given Recall that the µ-slope of a coherent sheaf E ∈ Coh(X) is defined as follows: otherwise.
Tilting on (F β , T β ), one obtains an abelian subcategory B β := F β [1], T β of D b (X), which is the heart of a t-structure on D b (X).
For B ∈ D b (X), let H p (B) denote cohomology with respect to Coh(X). Observe that the objects of B β are those B ∈ D b (X) such that: • H p (B) = 0 for p −1, 0; • H −1 (B) ∈ F β ; and • H 0 (B) ∈ T β .
In particular, from the definition of F β , the sheaf H −1 (B) must be torsion-free.
Let H := R + × R, thought of as the upper half plane, with coordinates denoted by (α, β). We will want to consider the slope function as a function of α and β, and to this end it is convenient to define the following function on (α, β) ∈ H: which coincides with the numerator of ν α,β (B) when v = ch(B). To simplify the notation, we define Note that the pair (B β , Z tilt α,β ) is a weak stability condition in D b (X), in the sense of [Tod10, Section 2], for all pairs (α, β) ∈ R + × R. In practical terms, this gives the following.

λ-stability
The next step is to consider the following torsion pair on B β : Tilting on (F α,β , T α,β ), one obtains a new abelian subcategory A α,β := F α,β [1], T α,β of D b (X), which is also the heart of a t-structure on D b (X).
One then introduces a third parameter s > 0 in order to define a family of central charges Z α,β,s : K num (X) → C as follows, for A ∈ A α,β : We could also consider a more general central charge, whose real part is for parameters b ∈ R and a ∈ R + ; see [BMS16,Lemma 8.3] and [Piy17]. However, we will only consider the special case where b = 0, while a = α 2 (s + 1/6).
The generalized Bogomolov-Gieseker inequality can also be used to prove a form of the support property for λ α,β,s -semistability. In the case we are considering where the Picard rank is 1, we can state it as follows.
When we come to do more detailed computations, it will also be useful to have a more uniform notation for the various functions of v ∈ K num (X) introduced above; more precisely, we define the following: By convention, we set ch The reason for setting s = 1/3 will become clearer in Section 4.2, but one technical reason is that the partial derivatives of ch α,β i (v) with respect to α and β behave very well; more precisely, In particular, note that ∆ 10 (α, β) = δ 10 (u, v).
The following is an easy exercise.
Lemma 2.7. Fix real numerical Chern characters u and v.
(2) The partial derivatives of ∆ ij are given by (3) Assume either ch (4) The following are equivalent: (a) There exist α, β and i ∈ {0, 1, 2, 3} such that ch Note that u ∝ v means that u and v are proportional as vectors.
In what follows, we will use M α,β,s (v) to denote the set of λ α,β,s -semistable objects with Chern character v; Piyaratne and Toda proved in [PT19] that M α,β,s (v) has the structure of an algebraic stack, locally of finite type over C. Determining whether M α,β,s (v) also has the structure of a projective variety is an important problem.

The second tilt category
We will now collect some useful facts about the objects in the second tilt category A α,β . Much of the following is well known and is easy to deduce in various ad hoc ways, but we give a novel treatment using higher octahedra which is of independent interest (the idea first appeared in [Bal11]). We will henceforth drop the X from the notation A α,β . Recall that H i denotes cohomology in Coh(X) and So we have three distinguished triangles: (2.12) Because these triangles intersect, we can arrange them into a diagram as follows: Here the squiggly arrows X Y mean X → Y [1]. The diagram is meant to repeat infinitely above and to the right by shifting by [−n] and [n]. Every square commutes, and the triangles along the diagonal are distinguished. Furthermore, each triple of morphisms formed by composing horizontally and then vertically and then looping back via the repeated diagram to the right is distinguished. The additional objects C, C and A[1] are defined as cones on suitable composites. Three of the triangles are given in display (2.12), and the remaining seven are (2.14) The first two tell us what the B β -cohomologies of C and C are. The third tells us that A ∈ Coh(X). The fourth and fifth tell us what the A α,β -cohomologies of C and C are, and the final two tell us what the Coh(X)-cohomologies of C and C are. In particular, we have that There are also five distinguished octahedra which are obtained by removing one row and column from the diagram in display (2.13) (for a more concrete example of an octahedron in this form, see the diagrams (2.22) in the proof of Proposition 2.15). The diagram can also be represented as a 4-dimensional shape given as a truncated 5-simplex with five octahedral and five tetrahedral faces. Another way to express the diagram is that A is filtered in D b (X) by Observe that A i1 ∈ F β , and these must be torsion-free sheaves, while A i0 ∈ T β and, in particular, A 00 ∈ A α,β . In fact, a slightly stronger statement is true.
The following fact will also be useful later on.
Proof. Chasing through the seven triangles listed in display (2.14) with A 11 = A 01 = 0, one concludes that A C A 1 , which is the same as H −1 (A) H −1 β (A) by the isomorphisms in (2.15). The vanishing of A 01 also implies that A 0 A 00 by the sequences in display (2.12); hence H 0 (A) H 0 β (A). We will need to consider short exact sequences in A α,β ; we look at situations where the middle term is in Proof. Apply H B to get a long exact sequence in B β : Note, in particular, that B ∈ B β . Split this via D. Then ν − α,β (D) > 0, and so D ∈ A α,β . Set C = C 0 . Then we have a short exact sequence D → A → C in A α,β together with an injection B → D and a surjection C C also in A α,β , as required. Proof. Apply H B to get a long exact sequence in B β : In particular, C ∈ (1) G ∈ Coh(X) ∩ B β .
Proof. If A = E ∈ A α,β ∩ Coh(X), then the triangles (2.12) and (2.14) imply C = 0, and then A 1 [1] is a sheaf, and so A 1 = 0. This establishes E ∈ B β . Applying cohomology in B β to the triangle D → E → B, where B = E/D in A α,β , we have that D 1 = 0 and have a B β long exact sequence If D 01 0, we set F = Q, and then we repeat the above argument with Q replacing D. So we can assume D 01 = 0. Suppose B 0 = 0. Then Q = E. Now set F = D 00 = D. Then we have case (2) as Otherwise, B 0 0. Note that D → Q in A α,β , and so to find F, we may assume B 1 = 0 (by replacing D by Q in the above). Applying cohomology in Coh(X), we have a long exact sequence If B 00 = 0, then we take F = Q, and we have case (3). Otherwise, we split the sequence via F. From

Duals of semistable sheaves
Given an object A ∈ D b (X), we denote its derived dual by . For a sheaf E, its derived dual E ∨ satisfies H j (E ∨ ) = Ext j+2 (E, O X ) for j = −2, −1, 0, 1 and H j (E ∨ ) = 0 otherwise. If E is torsion-free, then we have the following short exact sequence in Coh(X): Let E be the kernel of the composed epimorphism E * * Q E T E ; it fits into the following short exact sequence in Coh(X): We then have that H j (E ∨ ) = 0 for j −2, −1, 0, and Moreover, Ext 1 (E, O X ) Ext 1 (E , O X ) fits into the short exact sequence Clearly, E * ∈ F β for β > µ + (E * ), while Ext 1 (E, O X ) ∈ T β for every β; it follows that E ∨ [−1] ∈ B β for every β > µ + (E * ). In addition, obtained from dualizing the sequence in display (2.19), we have proved the following.
Proposition 2.13. If E is a torsion-free sheaf and We can also formulate a converse to this construction. Let Coh(X) d denote the category of coherent sheaves on X of dimension at most d. We want to be able to characterize when an object A ∈ D b (X) which has cohomology in three consecutive places A i , A i+1 and A i+2 such that A i is reflexive and A i+j ∈ Coh(X) 2−j for j = 1, 2 is the shift of the dual of a torsion-free sheaf. We will make use of this when we analyze the asymptotics for β 0; we will express them in the form of lifting properties. First we need a technical lemma as an intermediate step. The equivalent statement in dimension 2 is an easy exercise: if A has cohomology in two places with A i locally free and A i+1 ∈ Coh(X) 0 such that any subsheaf S → A i+1 does not lift to A, then A is the shift of the dual of a torsion-free sheaf. Proof. To see where these induced maps come from, consider the triangles Then we have maps The lemma follows immediately from the spectral sequence which converges on the third page to a single entry at (p, q) = (0, −2). Note that E is torsion-free because it is the subsheaf of a reflexive sheaf (namely F * ). We also have that E K, Z E Ext 3 (S, O X ), Q E Ext 2 (G, O X ) and T E Ext 3 (G, O X ).
We can give a more categorical description of the conditions as follows.
Proposition 2.15. Suppose A ∈ D b (X). Then A satisfies the conditions if and only if E := A ∨ is a torsion-free sheaf.
Proof. It is easy to see that the final two conditions are necessary because otherwise we would have maps For the converse, we show that (5) implies Lemma 2.14(7) while (6) implies Lemma 2.14(5) and Lemma 2.14(6) by considering the contrapositives. For the proof, we consider the octahedron G [2].
Observe that the triangle F ∨ [−2] → G ∨ → B ∨ shows that when Lemma 2.14 holds, First suppose K → Ext 3 (S, O X ) fails to surject. Let the quotient be denoted by T ; this is in Coh(X) 0 . Let S be Ext(T , O X ). Then S → S, and the composite with S → B[2] vanishes and so lifts to A, as required. Now suppose item (5) or (6) of Lemma 2.14 fails. Applying cohomology to the left vertical triangle of our octahedron and abbreviating H i (B) to B i , we have Then the cone on B −1 [1] → B is the dual of an object G supported in dimension 1, and there is a map G → G which lifts to G → B. But there are no morphisms G [1] → T [−1], and so this lifts to a non-zero map G [1] → A, as required.
Remark 2.16. We can rephrase item (5) in Proposition 2.15 to say that if K → A is a map from a sheaf in Coh(X) 0 , then the induced map K → S must be zero. Similarly, item (6) becomes the statement that if K[1] → A is a map with K ∈ Coh(X) 1 , then the induced map K → G must be zero. In practice, these are easier to test, but we will make more use of the conditions stated in Proposition 2.15. It is also clear how to extend the statement to higher dimensions.

Refining notions of stability for sheaves
We complete this section by introducing a stability condition in Coh(X) which interpolates between µ-stability and Gieseker stability; it will play a role in some of the proofs below.
The Hilbert polynomial of a coherent sheaf E on X with respect to the polarization L is P E (t) := χ(E ⊗ L ⊗t ) = ch 0 (E)χ(L ⊗t ) + ch 1 (E)x 2 (t) + ch 2 (E)x 1 (t) + ch 3 (E), where x 2 (t) := 1 2 t 2 L 2 + t td 1 (X) · L + td 2 (X) , x 1 (t) := (tL + td 1 (X)) , and the td j (X) denote the Todd classes of X. If F is another coherent free sheaf on X, we define, following [Rud97, Section 2], where δ ij (E, F) := δ ij (ch(E), ch(F)) following the notation introduced in display (2.11). We remark that a coherent sheaf E is Gieseker (semi)stable if and only if every proper, non-trivial subsheaf F → E satisfies Λ(E, F) > (≥) 0 in the lexicographic order. For instance, assume E is torsion-free, and let F → E be a proper subsheaf; letting p E (t) denote the reduced Hilbert polynomial of the sheaf E, we have In what follows, it will be important to consider another notion of stability for torsion-free sheaves, which is equivalent to the notion of stability in the category Coh 3,1 (X) in the sense of [HL10, Definition 1.6.3].
For the sheaf E defined as in display (2.19), we observe that: The first claim follows from the fact that E * = E * . For the second claim, note that ch ≤2 (E ) = ch ≤2 (E) since E /E is 0-dimensional; in addition, any subsheaf F → E will induce a subsheaf F → E such that ch ≤2 (F ) = ch ≤2 (F).
Clearly, one has the following chains of implications: Example 2.18. It is not hard to find explicit examples that show that the reverse implications do not hold in general. Indeed, for X = P 3 , let S be a rank 2 reflexive sheaf given as an extension of an ideal sheaf of a line L ⊂ P 3 by O P 3 ; note that ch(S) = (2, 0, −1, 1). Let C ⊂ P 3 be curve, and consider an epimorphism ϕ : S O C (k); define E ϕ := ker ϕ.
(1) If C is a line not intersecting L and k > 0, then E ϕ is µ ≤2 -semistable but not Gieseker semistable.
(2) If C is a conic not intersecting L and k = 0, then E ϕ is µ ≤2 -stable but not µ-stable.
(3) If C is a line intersecting L in a single point, then E ϕ is Gieseker stable but not µ ≤2 -stable.
Finally, we also recall the following notion of stability for sheaves of dimension 2; compare with [HL10, Definition 1.6.8].
(2) Each factor G k : Proof. Let E 1 be the maximal subsheaf of E of dimension at most 1, so that F := E/T has dimension at least 2. Note that this E 1 might be the zero sheaf. Next, let F = F 0 F 1 · · · ⊂ F n = 0 be the Harder-Narasimhan cofiltration of F as an object in the category Coh 3,1 (X), in the sense of [HL10, Theorem 1.6.7]; each F k is a sheaf of dimension at least 2, either µ-or µ ≤2 -semistable. Composing with E F provides our required cofiltration and hence a filtration in the usual way.

Regions of H
Let v be a real numerical Chern character with v 0 0 satisfying Q tilt (v) ≥ 0. It follows that the curve When v fails the Bogomolov-Gieseker inequality, we occasionally still consider Θ v , but now it is a singlebranch hyperbola cutting the α-axis. The hyperbola Θ v divides H into three regions R − R 0 R + defined as follows: we shall denote these regions by R 0+ v and R 0− v (to include the vertical line) to the left and right, respectively. These regions are illustrated in Figure 1.
Remark 3.1. The various cases which arise in Proposition 2.12 can be rewritten usefully in terms of these regions. In the notation of the statement, we have F ∈ R − ch(F) , and then the cases are: The categories A α,β along the curve Θ v satisfy some strong conditions. These form the basis for Theorem 4.21 (see also [Sch20a, Lemmas 6.2 and 6.4]).

Variation of A α,β along paths
H is a path in the upper half plane with γ(0) = (α, β). Let π i : H → R be the projection maps (so that π 1 picks out the α-value and π 2 the β-value). First assume β(t) := π 2 (γ(t)) is constant and α(t) := π 1 (γ(t)) is monotonic increasing. As before we write A 0 and A 1 for the B β -cohomology of A. Observe that the Harder-Narasimhan factors of A 0 and A 1 with respect to ν α,β are locally constant (they change only on λ-walls corresponding to the factors). Let A − 0 be the Harder-Narasimhan factor of A 0 with ν α,β (A − 0 ) = ν − α,β (A 0 ). Then as we move along γ (upwards), A remains in A α,β until we cross Θ − ch(A − 0 ) (in a Harder-Narasimhan chamber), at which point H 1 A (A) = A − 0 . But since ν + α,β (A 1 ) < 0 and we are traversing the path upwards, for any Harder-Narasimhan factor A 1 of A 1 with ch(A ) 1 = v , we start in the region R 0 v . Hence we can only cross Θ + ch(A 1 ) , but this does not take A out of A γ(t) . The reverse happens if α(t) is monotonic decreasing. In this case, if A + 1 is the Harder-Narasimhan factor corresponding to ν + γ(t) (A 1 ), then as we cross Θ + . This also applies to horizontal paths, but now we need to consider what happens when we cross vertical lines M u = {β = µ(u)} corresponding to u equal to Chern characters of µ-Harder-Narasimhan factors of A 00 , A 10 , A 01 and A 11 . In this case, we see that crossing these lines does not affect A. To see this, consider one of the cases where µ − (A 10 ) = µ(B) and A 10 = ker(A 10 → B) in Coh(X). Then we have an octahedron to the octahedra by inserting a second column and second row to form a higher octahedron (see Section 2.3) The other cases follow similarly.
In conclusion, we have proved the following.

The Γ -curve
The next step is to understand the vanishing locus of λ-slope for a given numerical Chern vector v and to study its relative position with respect to Θ v .
Definition 3.4. For any real numerical Chern character v and s ≥ 0, we define the curve If v 0 0, equation (3.3) is cubic on β; hence Γ v,s has at most three connected components. In fact, there exists an α 0 > 0 such that Γ v,s ∩ {(α, β) ∈ H | α > α 0 } has exactly three connected components. Two of these components are asymptotic to lines α √ 6s + 1 = ±(β − v 1 /v 0 ) and therefore lie in the regions R ± v ; these are the curves Γ ± v,s in Definition 3.4 above. The third component, namely Γ 0 v,s , either coincides with or is asymptotic to the vertical so Γ v,s consists of exactly three components: Γ ± v,s are the two branches of the hyperbola β 2 −(6s+1)α 2 = 6n/m, and Γ 0 v,s coincides with the vertical axis {β = 0}. In the limit s → 0, also assuming that the Bogomolov-Gieseker inequality v 2 1 ≥ 2v 0 v 2 holds for v, the curves Γ ± v,s=0 are co-asymptotic with the two branches of the hyperbola Θ v . It turns out that the position of the different branches of the cubic curve Γ v,s relative to the two branches of the hyperbola Θ v has an important geometric meaning, which is unravelled in the following two statements. Proposition 3.6. Let v be a real numerical Chern character with v 0 0 satisfying the Bogomolov-Gieseker inequality (2.3). If, for each (α, β) ∈ H, there is an object E ∈ A α,β with ch(E) = v, then Γ v,s consists of exactly three connected components: The hypotheses imply that the three asymptotic branches Γ • v,s remain in their respective regions R • v for • = +, −, 0 all the way down to the horizontal axis {α = 0}, and so we have three distinct irreducible components, as given.
Recall that a cubic equation admits three distinct real roots exactly when its discriminant is positive. Regarding equation (3.3) as a cubic in β, an easy computation gives its discriminant as (108 times) (3.5) Setting α = 0 yields (3.4). Finally, we need to know that q α can only change sign once as α varies. To see that, observe that So as a cubic in α 2 , q α has a single point of inflection and so can change sign at most once. Note that, since v 2 1 ≥ 2v 0 v 1 , the point of inflection does not occur for any α > 0.
We also make use of the following. Figure 2. On the other hand, away from Θ v and M v , the curve Γ − v,1/3 is strictly monotonic. This is because ∂ β ch α,β and otherwise this intersection is a single point at which Γ v,s has a minimum turning point, as illustrated in Figure 2.
Proof. This is easiest to see by considering the sign of λ α,β,s (v) for very small α (we can just take α = 0). Observe that for β → −∞, the sign is positive, and the sign changes when we cross either Then the numerical condition is the positivity of the constant term in this cubic Figure 2. This graph contains an example of the situation described in Proposition 3.8 for v = (3, 4, 2, 2/3), which is the Chern character of the tangent bundle of P 3 . We set s = 1/3. The curve We also illustrate Theorem 4.22 here, with the vanishing ν-and λ-walls represented by the curves Ξ u,v (in magenta) and Υ u,v,s (in black), respectively, where u = ch(O P 3 (1)). Finally, Lemma 4.5 is also represented since both of these walls cross Θ − v at the same point.
The following is an easy exercise.

ν-walls
Given a real numerical Chern character v, a curve Ξ u,v ⊂ R + × R is called a numerical ν-wall for v if there is a real numerical Chern character u such that compare with the notation introduced in display (2.11). Note that this coincides with the numerator of the difference ν α,β (u) − ν α,β (v), so its vanishing locus is precisely where it changes sign. Observe that numerical ν-walls are invariant as subsets of the upper half plane under the changes of coordinates u → κv + u, u → ζu and (u, v) → (v, u) for any real constants κ and ζ 0. (1) Ξ u,v is empty.
(2) For all real κ, κv + u satisfies the Bogomolov-Gieseker inequality. ( Proof. The Bogomolov-Gieseker inequality for κv + u as a polynomial in κ is The condition in (3) is the discriminant condition for (4.2) divided by v 2 0 . This shows that (2) and (3) are equivalent. The condition that Ξ u,v = ∅ is that ∆ 21 (0, β) = 0 has at most one solution. Explicitly, this is establishes the equivalence of (4). The structure of numerical ν-walls was described in [Mac14] for the case of projective surfaces, but the same results also hold for projective threefolds; see for instance [Sch20a,Theorem 3.3] and [MS17, Remark 9.2]. Precisely, numerical ν-walls are non-intersecting semicircles centered along the horizontal axis and cross the hyperbola Θ v at their maximum point or the vertical line M v . The semicircles tend to a fixed point (0, C 0 ) on the β-axis as the radius tends to zero.
It then follows that the same property holds for every point of Ξ u,v ; see [Sch20a, Theorem 3.3]. Actual ν-walls are locally finite (that is, any compact subset of the upper half plane intersects only finitely many numerical ν-walls), see [MS17, Lemma 6.23], and there are C max ∈ R and R max > 0 such that every numerical ν-wall is contained in the semicircle centered at (0, C max ) with radius R max . In particular, there is an α > 0 for which there are only finitely many actual ν-walls above the horizontal line {α = α}. However, actual ν-walls may accumulate towards the point Θ v ∩ {α = 0}; see [Mea12] and [YY14] for examples on abelian surfaces.
Another observation is that, since ν-walls are nested, there can be at most two vanishing ν-walls for a given Chern character v, one on each side of the vertical line M v .
The following technical result, which will be useful in Section 8, follows immediately from Lemma 4.1. For a Chern character v, we let for some actual ν-wall given by u}.

Proposition 4.3. Suppose ch(E) = v and ch(F) are two numerical Chern characters which satisfy the Bogomolov-
For example, if a particular Chern character v admits no actual ν-walls, then any Chern character of a sub-object in B β must satisfy those inequalities.
More explicitly, using the notation from display (2.11) (4.5) Numerical λ-walls that are bounded as subsets of H have a simple numerical characterization. Notice that if Υ u,v,s is bounded for some value of the parameter s, then it is bounded for every value of s.
Next, we observe an interesting relation between a numerical λ-wall Υ u,v,s and its associated numerical ν-wall Ξ u,v : if one intersects the hyperbola Θ v , then so does the other.

Lemma 4.5. Let v be a real numerical Chern character satisfying the Bogomolov-Gieseker inequality and
For the second statement, note that Θ v never intersects M v because v is assumed to satisfy the Bogomolov-Gieseker inequality and so ch Proof. Let (α 0 , β 0 ) be the point of intersection, so that ρ v (α 0 , β 0 ) = τ v,s (α 0 , β 0 ) = 0. By the expression in display (4.4), it follows that f u,v,s (α 0 , β 0 ) = 0 for every real numerical Chern character u and every s > 0.
Let us now analyse unbounded numerical λ-walls. Note that if δ 10 (u, v) = 0, then the expression in equation (4.5) reduces to a cubic polynomial in β, with coefficients depending on α 2 , so it always has a real root; in other words, unbounded numerical λ-walls intersect every horizontal line. Furthermore, unbounded numerical λ-walls with different values of the parameter s can look very different; see Figure 3. In particular, an unbounded numerical λ-wall might not be connected, and one of its connected components may be bounded; see Figure 4. The main properties of unbounded walls are described in the following series of lemmas. First, we consider the case s > 1/3.
Proof. If s 1/3, then the equation f u,v,s (α, β) = 0 can be rewritten in the following way: Note that the denominator vanishes at the value β = β given above; therefore, α goes to infinity as β approaches β .
Lemma 4.9. Let s ≤ 1/3. If u and v are numerical Chern characters satisfying δ 10 (u, v) = 0 and δ 20 (u, v) 0, then there exists an α max > 0 (depending on u, v and s) such that Proof. If s < 1/3, then equation (4.7) does admit solutions, which are asymptotically of the form for some β ∈ R given explicitly in Lemma 4.8 above. In other words, there is an α > 0 such that we have Depending on the sign of δ 20 /(δ 21 − δ 30 ), such an equation does admit solutions either for β 0 or for β 0; in both cases, α grows like |β| 3/2 , so Υ u,v,s will lie in the region R 0 v once α is sufficiently large. The same conclusion holds when δ 30 = δ 21 since then f u,v,s (α, β) becomes just a cubic polynomial on β, not depending on α.
Remark 4.10. Note that there are no ν-walls corresponding to unbounded λ-walls, and so by continuity the unbounded component remains in R 0 v as it cannot intersect Θ v . However, an unbounded λ-wall for v may have a bounded connected component contained in R − v , as pictured in Figure 4. We now show that intersection between numerical λ-walls and the curve Γ v,s is independent of s.
In other words, the set of numerical λ-walls for a fixed v ∈ K num (X) only depend on the equivalence class under ∼ v .
Lemma 4.12. Suppose u and v are real numerical Chern characters and v 0 0. Then: It follows that equivalence classes for ∼ v come in three types when v 0 0, according to Table 1 below (x and y are arbitrary rational numbers). In the degenerate case, the numerical λ-wall Υ u,v,s coincides with Θ v .
Proof. We prove (3) and leave (1) and (2) as similar exercises. We simply observe that the equation for the numerical λ-wall Υ u,v,s can be written in the following manner: We conclude that distinct unbounded numerical λ-walls never intersect one another. On the other hand, it is interesting to observe that, in contrast with numerical ν-walls, two distinct bounded numerical λ-walls for Υ u,v,s and Υ u ,v,s can intersect both along Θ v (if ch ≤2 (u) = ch ≤2 (u ), as illustrated in item (3) of Theorem 4.13), and away from it; see Figure 5.
Our normal form for u can also be used to give sufficient conditions for when a numerical λ-wall for v intersects Γ v,s . Proposition 4.14. Let u and v be real numerical Chern characters such that v satisfies the Bogomolov-Gieseker inequality, ch 0 (v) 0 and Proof. Writing u in the canonical form (0, 1, x, y), observe that Γ u,s is given by the hyperbola The conditions in the statement about δ ij are equivalent to the hyperbola crossing the β-axis or not. The hyperbolae are asymptotic to β = x ± 2(s + 1/6)α, while Γ v,s is asymptotic either to M v or to β = µ ± 6(s + 1/6)α. Then the hypotheses guarantee that Γ u,s and Γ v,s intersect as stated.
for some uniquely determined α so long as x is the β-coordinate of some point on Γ v,s . Since the wall Υ u,v,s can only cross Θ v at its intersection with Γ v,s and its asymptotes are in R 0 v , it follows that, for |x| sufficiently large, there is a component of the wall which is bounded and which intersects Γ ± v,s .

Actual, pseudo and vanishing λ-walls
Definition 4.16. An actual λ-wall W u,v,s is the subset of points (α, β) in Υ u,v,s for which there are an object E ∈ A α,β with ch(E) = v and a path γ : This is a rather stronger condition than the usual one, which asks simply that A be properly λ α,β,ssemistable at the point (α, β). Our definition ensures that actual λ-walls are 1-dimensional and avoids degenerate situations like the one described in Remark 2.6. In addition, it allows us to state the following.
Lemma 4.17. An actual λ-wall W u,v,s is a union of segments of arcs within the underlying numerical λ-wall Υ u,v,s whose endpoints lie on another actual λ-wall for u or v, or on Θ u ∩ Θ v , or on the β-axis.
Proof. Note that the wall will have an endpoint if α → 0 along the curve; we take that to be an endpoint by convention. (1) First assume p ∈ W u,v,s is an isolated point; that is, there exists an open neighbourhood B of p on which there are no other actual λ-walls. In contrast with the path γ going through p along which the moduli space M γ(t),s (v) changes as explained above, one can find a path γ : [0, 1] → B such that γ (0) = γ(0) and γ (1) = γ(1) which does not cross any actual λ-wall, implying that M γ(0),s (v) = M γ(1),s (v), and thus providing a contradiction.
(1) In fact, we will see in Section 6 than the wall crosses α = 0 transversely or is singular there.
It follows that W u,v,s is a union of segments of arcs within Υ u,v,s . The same argument shows that if an endpoint (α 0 , β 0 ) of such an arc does not lie on another actual λ-wall for a short exact sequence 0 → A → E → B → 0 in A α,β corresponding to the wall (so ch(E) = v and either ch(A) = u or ch(B) = u), then E, A or B must go out of the category. But if A or B go out of the category, it is because they have a sub-object or quotient object which has a Θ-curve going through that point. But then the λ-slope goes to infinity, and this would contradict the existence of the wall in a small neighbourhood of the point. Consequently, the wall can only end on Θ ch(E) and so also on Θ ch(A) .
In another direction, if an object is strictly destabilized along a curve, then it can only become stable if the curve crosses either an actual λ-wall or a Θ-curve. Note that if an actual λ-wall W u,v crosses Θ u , then it must also cross Θ v .
(1) If A is not λ γ(t),s -semistable for 0 < t < b, then there exist , > 0 and some destabilizing sequence Proof. In either case, we have a long exact sequence for 0 < t < in A γ(t) : This is because as we cross t = 0, the phase changes continuously and the A γ(t) -cohomology of any object in A γ(t) for − < t < 0 in nearby γ(t) is concentrated in positions −1, 0 and/or 1. Split the sequence via B 0 → K → A → Q → C 0 , and note that, for −1 If A is λ γ(t),s -stable beyond t = 0, then we must have equality in one of these. If both are equalities, then the sequence 0 → B → A → C → 0 provides an actual λ-wall at γ(0). If only one is an equality, then γ(0) ∈ Θ u , where u = ch(B) or u = ch(C). But then, λ γ(0),s (A) = +∞ and so γ(0) ∈ Θ ch(A) as well, and so, again, this sequence provides an actual λ-wall.
If our aim is to find all actual λ-walls, then it is easier to first consider a list of necessary numerical conditions in order to reduce the possibilities to a small list of examples (this is exactly what we will do in the example presented in Proposition 8.8 below).
Definition 4.19. By a pseudo λ-wall W u,v,s , we mean the subset of points (α, β) of a numerical λ-wall Υ u,v,s for which: The support property, see Proposition 2.4, implies that an actual λ-wall is also a pseudo λ-wall. We shall see an example in (8.14) in Proposition 8.8 of a pseudo-wall which is not an actual wall but for which there is a destabilizing sequence.
Unlike actual ν-walls, distinct actual λ-walls for a given Chern character v do intersect. This was first noticed by Schmidt in [Sch20a], where he establishes a refinement of Lemma 4.5 providing a relationship between actual ν-and λ-walls for v along Θ − v ; more precisely, he proves the following statement.
We also provide a concrete example of two distinct actual λ-walls for the Chern character v := (2, 0, −1, 0) which intersect away from Θ v , in Section 8.4 below.
Furthermore, an actual λ-wall W u,v,s is called a vanishing λ-wall for v if for each p ∈ W u,v,s there exists a path γ : The meaning of the quartic function q(v) on K num (X) defined in equation (3.4) can now be expressed in the following theorem using the geometry of the Γ -curves we defined in Section 3. Recall that q(E) is the discriminant of the cubic defining Γ whose roots are the β-coordinates of Γ v,s ∩ {α = 0}. Proof. Since E is Gieseker semistable, v satisfies the Bogomolov-Gieseker inequality, and so the curve Θ v divides the plane into three regions, as explained in Section 3 above.
Note that Γ v,s is a smooth cubic curve, and so when q(E) < 0, there is an open neighbourhood of α = 0 such that also q α (E) < 0. It follows that there is some α 0 > 0 such that Γ v,s ∩ {α = α 0 } consists of a single point. Thus, either the asymptotic components Γ − v,s and Γ 0 v,s , or Γ + v,s and Γ 0 v,s , must belong to the same connected component of Γ v,s , which must then cross the hyperbola Θ v .
Fix β and α sufficiently large so that (α, β) lies above any ν-wall for v. Then E is ν α,β -semistable, and Proposition 3.2 guarantees the existence of λ α,β,s -semistable objects in A α,β for (α, β) ∈ H. It follows that there must exist a vanishing ν-wall for v crossing Θ v above the point of intersection But then there is also a vanishing λ-wall through the same point (α,β).
We will see an example of the situation described in Theorem 4.22 in Section 8.1 below, where we study in detail the ideal sheaf of a line in P 3 .

Asymptotic ν α,β -stability
One of the main goals of this paper is to characterize which objects in D b (X) are ν α,β -and λ α,β,ssemistable at infinity, that is, for large values of the parameters α, β. More formally, let γ : [0, ∞) → H be an unbounded path; we consider the following definition. The dual situation is also considered in [Piy17, Proposition 3.2] but from a different perspective.
Definition 5.1. An object A ∈ D b (X) is asymptotically ν α,β -(semi)stable along γ if the following two conditions hold: (i) There is a t 0 > 0 such that A ∈ B γ(t) for every t > t 0 . (ii) There is a t 1 > t 0 such that, for every t > t 1 , every sub-object In this section we characterize asymptotically ν α,β -semistable objects. More precisely, we establish the following results.
Theorem 5.2. Let B ∈ D b (X) be an object with ch 0 (B) 0.
(1) B is asymptotically ν α,β -(semi)stable along a path and only if S := B ∨ [−1] is a µ ≤2 -(semi)stable sheaf such that S * * /S either is empty or has pure dimension 1. . Since we will use Theorem 5.2 and the arguments in its proof in the subsequent sections, we include a full proof here. Our proof is longer than previous ones for two reasons. Firstly, we have a stronger form of asymptotic stability, and secondly we do not assume there are only finitely many ν-walls above a certain horizontal line. Although the latter fact is true, we want to illustrate how it can be avoided, since it is not known for λ-stability. If we do assume the ν-walls are bounded above, then it follows that the t 1 of the definition of asymptotic ν-stability can be chosen uniformly in E, that is, t 1 only depends on ch ≤2 (E).
The study of asymptotic ν α,β -stability is considerably simplified by the following observation.
(2) Suppose E is a µ-semistable sheaf, and suppose there is some (α, β) with β < µ(E) which lies on an actual ν-wall so that there are a ν α,β -semistable object F and a monomorphism F → E in B β . Then this is the only ν-wall for β < µ(E) at which E is destabilized.
Proof For the second item, the idea is that E must be ν α,β -stable above the ν-wall defined by the exact sequence. The local finiteness of the actual ν-walls ensures that E is ν α,β -stable either immediately below or immediately above this wall. But we can show that the latter holds, as follows. First observe that δ 01 (E, F) ≤ 0. This is because if we split the Coh(X) sequence and so µ(F) < µ(K) ≤ µ(E). We use Lemma 2.7(2) to compute the partial derivatives, which satisfy as E is µ-semistable. Consequently, E cannot be destabilized on another wall outside of this ν-wall because it must be stable immediately above any wall and unstable immediately below the wall. But on a path between two adjacent walls, it cannot be both stable and unstable.
To complete the proof, observe that F → E G remains a short exact sequence in B β at all points inside the ν-wall corresponding to F.
We will see that a similar statement holds for β > µ(E). So, for a given E, there is at most one actual ν-wall destabilizing it on each side of the vertical line M v . We can then reduce the proof of Theorem 5.2 to the study of asymptotic ν α,β -stability along horizontal lines. To this end, we define

Asymptotics along
The first part of Theorem 5.2 is proved in two separate lemmas.
Let T → B be a torsion subsheaf of B; if ch 1 (T ) 0, then contradicting asymptotic ν α,β -semistability. If ch 1 (T ) = 0, then for every (α, β) ∈ H, again contradicting asymptotic ν α,β -semistability. So B must be a torsion-free sheaf. Before going on to look at the converse, we can make an interesting deduction from this.
Proof. Suppose otherwise. Then by item (2) in Lemma 5.3, it follows that E is asymptotically ν α,β -semistable, and then by Lemma 5.4 it must be µ ≤2 -semistable, which yields a contradiction.
Now we consider the converse to Lemma 5.4.
This completes the proof of the first part of Theorem 5.2.
Remark 5.7. Recall the definition ofμ-stability from Definition 2.19. We will also need the following version of Lemma 5.6 for torsion sheaves: given any fixed α > 0, if T ∈ Coh(X) 2 isμ-semistable, then there is a β 0 < 0 such that T is ν α,β -semistable for every β < β 0 . The proof of this claim is similar to the proof of Lemma 5.6.
We can now describe an asymptotic Harder-Narasimhan filtration for unstable objects as well.
whose factors G k := B k /B k−1 are asymptotically ν α,β -semistable and satisfy (1) ν γ(t) (B 1 ) = +∞ for every t > 0 if B 1 0; (2) for each k = 2, . . . , n, there is a t k > t 0 such that Proof. The first item of Lemma 5.3 implies that B must be a sheaf, so it admits a filtration as described in Lemma 2.20; this is the filtration we are looking for. Indeed, each factor in the filtration of Lemma 2.20 belongs to B β for β 0, according to Lemma 5.6 and Remark 5.7. The property in item (1) is clear since B 1 ∈ Coh 1 (X). Item (2) is a consequence of the following claim: given E, F ∈ Coh(X) with ch ≤1 (E), ch ≤1 (F) 0, we have ν γ(t) (E) > ν γ(t) (F) for t 0 if and only if Λ 2 (E, F) > 0. This can be explicitly checked for the path γ(t) = (α, −t) using the limits calculated in the proof of Lemma 5.4; the verification for more general paths is similar.

Unbounded Θ * -curves
It is tempting to think that if B ∈ B β (respectively, A ∈ A α,β ) for some α, β, then B ∨ ∈ B −β (respectively, A ∨ ∈ A α,−β ). The problem is that the duals of objects B in F β are not necessarily in T −β because it might be that µ + (B) = β and then µ − (B ∨ ) = −β. The analogous statement holds for objects A in F α,β . However, an asymptotic version of this statement does hold.
First we prove a technical lemma which allows us to turn Proposition 5.8 about the existence of asymptotic Harder-Narasimhan filtrations into a more precise bound on ν − α,β so long as we constrain the unbounded curve we move along. To this end, we introduce the following definition.
In particular, γ is an unbounded Θ − -curve if and only if γ * is an unbounded Θ + -curve.
Remark 5.10. If v 0 = 0 while v 1 0, then the first condition implies the second; indeed, we have ν γ(t) (v) = v 2 /v 1 − β(t), and the first condition implies that this is positive for all t 0.
When v 0 0, the condition on ν γ(t) is satisfied, for example, if there is an > 0 such that for all t 0, So again this is positive for all t 0.
Combining this with Proposition 2.13, we deduce the following.
Proposition 5.13. Consider an unbounded Θ − -curve γ as above. Then the following are equivalent for an object E ∈ D b (X): (1) E ∈ A γ(t) for all t 0.
If E ∈ Coh(X) contains no subsheaf of dimension 0, then there is a t 0 > 0 such that for all t > t 0 , E ∨ ∈ A γ * (t) .
Proof. Lemma 5.12 implies that (2) implies (1) and a similar a similar but simpler argument for µ − (E) shows that (3) implies (2). The idea for the converses is that if , then Lemma 5.12 again shows that ν − γ(t) (E 1 ) becomes positive as t increases, and then E 1 [1] ∈ B β(t) for t sufficiently large and so E ∈ B γ(t) for the same range of t. A similar argument shows (2) implies (3).
Now assume E is a coherent sheaf. When E is torsion-free, the claim follows immediately from Proposition 2.13 and the observation that ν + γ * (t) (E ∨ ) = −ν − γ(t) (E ) → −∞ as t → ∞. Otherwise, let T ⊂ E be its maximal torsion subsheaf. Then the hypothesis implies that T ∨ ∈ A γ * (t) for all t 0, and so dualizing T → E → E/T , we deduce the last part.

Corollary 5.14. Suppose γ is an unbounded Θ − -curve γ and E is a sheaf in
Proof. Observe that 0 → F → E → G → 0 is also a short exact sequence in B β(0) by the first part of Proposition 2.12 and so also in B β(t) for all t > 0 since γ is a Θ − -curve and so remains to the left of any

Asymptotics along Λ + α
We now move to the proof of the second part of Theorem 5.2, starting with a characterization of objects lying in B β for β 0. thus ch 0 (P ) = 0. Conversely, we have that E ∈ F β for β ≥ µ + (E) and P ∈ T β for every β; it follows immediately that B ∈ B β for β ≥ µ + (E).
Proof. For simplicity, set E := H −1 (B) and P := H 0 (B). We start by checking that E is reflexive and dim P ≤ 1.
Indeed, if E is not reflexive, then Q E := E * * /E ∈ T β ⊂ B β for every β; thus Q E is a sub-object of B within B β . But ν α,β (Q E ) = +∞ for every (α, β) ∈ H, so we have a contradiction with the asymptotic ν α,β -semistability of B.
Finally, we provide the converse of the previous lemma, thus concluding the proof of Theorem 5.2. For E ∈ Coh(X), note that Lemma 5.15 implies that E ∨ [−1] ∈ B β for β 0 if and only if E has no subsheaf of dimension at most 1, and the cokernel of the canonical morphism E → E * * has pure dimension 1.
Lemma 5.17. If S is a µ ≤2 -semistable sheaf such that S * * /S is either empty or has pure dimension 1, then there is Proof. The observation in the previous paragraph implies that B := S ∨ [−1] ∈ B β for every β ≥ −µ(S).

The differential geometry of surface walls
Definition 6.1. Let u and v be real numerical Chern characters such that v satisfies the Bogomolov-Gieseker inequality. We define the surface wall Σ u,v ⊂ R + × R × R + to be the vanishing locus of the numerator of the difference of slopes λ α,β,s (u) − λ α,β,s (v) now regarded as a function of all three parameters (α, β, s). Note that for any real φ and ψ 0. Throughout this section, we will be referring to the function f u,v frequently, and it is will be more readable to abbreviate it when the context is clear to f (α, β, s) or just f .
In addition, we will denote by Γ v (without the parameter s in the subscript) the surface {τ v,s (α, β) = 0} ⊂ R × R + . In this notation, Our aim in this section is to explore some of the differential-geometric properties of Σ u,v with a view to understanding finiteness properties of λ-walls. We will assume v is a fixed real numerical Chern character. Note that, by (6.1), we can assume u 0 = v 0 . It turns out to be best to consider the two cases u 0 = 0 = v 0 and u 0 0 v 0 . The former is dealt with in Remark 6.5.

Lemma 6.2. At the points of intersection
In other words, a surface wall Σ u,v does have 2-dimensional components whenever it is not empty.
Proof. Note that each pair of curves divides a small ball around their intersection into four regions; otherwise, the (algebraic) curves must coincide. They cannot coincide except possibly for the Theta curves, but then Σ u,v = Θ u × R. Then for a small arc around the intersection point in two of the regions, the function f u,v (α, β, s) is positive on one curve and negative on the other, and so must vanish at some point in the region for each arc sufficiently close to the intersection point. Combining this with Lemma 4.11, we see that Σ u,v is 2-dimensional in a neighbourhood of the point.
We now look more carefully at the differential geometry of the surface wall Σ u,v . The normal vector is given by the gradient of f (α, β, s). Lemma 6.3.
Note that when s = 0, the surface is unbounded along α = ±β.
If v 0 = 0 = u 0 , then ∆ i0 = 0 identically for all i. Then there are no ν-walls. The numerical λ-walls are nested ellipses, much as ν-walls are for the truncated Chern characters. It follows that Σ u,v is always regular and horizontal exactly on Γ v,s . The variation in s is just a vertical scaling by It is interesting to consider the regularity of numerical walls, and we will use this extensively in our analysis of the asymptotics. The regularity of a general wall for arbitrary s is complicated and hard to describe, but the situation is simpler if we consider the whole surface Σ u,v and also for the case s = 1/3, which we describe first. Proposition 6.6. When s = 1/3, a numerical wall Υ u,v,1/3 is regular everywhere in the upper half plane except where it intersects Γ u,1/3 and its numerical ν-wall Ξ u,v .
Looking at the second derivatives, we see that the local model for Υ u,v,1/3 at its singular point is (α − α 0 ) 2 + higher order terms = 0, and so the singular point is a cusp. We see this more generally in case (2) in Theorem 6.7 below. (1) It intersects a ν-wall away from Γ v at α = 0, in which case it is locally α 2 − β 2 .
(2) It intersects a ν-wall and Γ v at s = 1/3 for α > 0, in which case it is locally up to scaling. Note that in (1), we also allow the degenerate case where the ν-wall has radius 0. An example of this can be seen for v = (2, 0, −1, 0), illustrated in Figure 11 in Section 8.
We consider case (a). From ∆ 31 = 0, we then have that u ∝ v unless ch α,β Then we have that the conditions are equivalent to This is a triple zero at α = 0 and looks like y = 0 ∪ y 2 = xz for α 0. In fact, since (u) or we are also on a ν-wall. In the latter case, we are in case (b); see below. Otherwise, ∆ 01 = 0. But then we either have the special situation described before, or ∆ 20 = 0 and ∆ 30 = 0, so we have u ∝ v. This is impossible. In the special case above, we have Hf u,v = 0 again, and the same third derivatives are non-zero as before.
On the other hand, if v 1 0, the picture will be the same translated by µ(v) along β. Note that (6.2) will determine u 1 for arbitrary non-zero ch(v) and u 0 . Then (6.3) will determine v 3 , while (6.4) will determine u 2 : one root is also a root of ∆ 02 = 0 and so must be dismissed, and then u 2 is uniquely determined. This gives a 3-parameter family of possible rational examples, but to correspond to actual objects, there are strong diophantine constraints which will depend on the threefold X.  Hence, ∆ 31 (α, β) = 0, and so ∂ β f u,v,s = 0, and the wall is horizontal. In a local chart away from a point where the wall is vertical, we can view α as a function of β. Then the second derivative is given by If the point is on the associated ν-wall Ξ u,v , then since ch β 1 (v) 0, we have and so if also ∆ 32 = 0, then the Υ u,v,s must be horizontal there, unless the point lies on Θ v . The second derivative (if the point is not on Γ v,s ) is 1/α. Note that when the point is also on Γ v,s , the wall is singular by Theorem 6.7(2); otherwise, ∆ 21 and ∆ 30 cannot vanish simultaneously except in the special case where δ 01 = 0. Consequently, there is never a point of inflection on a numerical λ-wall except at Γ v,s ∩ Ξ u,v in this special case.
Remark 6.11. A numerical λ-wall might be horizontal as it crosses Θ v , though it generally is not. In that case, we can use β and s as local coordinates on the surface, and the second fundamental form at that point is II = 1 α dβ 2 . Then the Gauss curvature is zero, and the mean curvature is 1/2α. (1) In other words, if a numerical λ-wall exists for one s 0 ≥ 1/3, then it must also exist for all s ≥ 1/3, whereas if s 0 < 1/3, then it need only exist for s < s 0 . This means that numerical λ-walls can only be "created" for s < 1/3 and as s decreases. We shall see below that walls cannot be "created" in R ± v,s even when s < 1/3, but they can be created in R 0 v,s .
Proof. First observe that we may assume boundedness away from s = 0 because if ∆ 01 = 0, then ∆ 20 0 and so f u,v,s (α, β) = 0 is a cubic in β. Then it must have a solution for all α. But if a wall crosses α = 0 for one value of s, then it crosses it for all s. By Theorem 6.7, the surface is regular except at special points on α = 0 where there are multiple tangent planes which include the s-direction or we are in case (2) of the theorem. But this latter case cannot arise in the present situation. This means that if Σ u,v ∩ {s = s 1 } is empty for some s 1 > 0, then there is some s 0 such that ∂ α f u,v,s 0 = 0 = ∂ β f u,v,s 0 and ∂ s f u,v | s=s 0 0. It follows that α 0 and ∆ 12 (α, β) 0.
Then Σ u,v | s 0 is a union of closed curves, unbounded curves or one or more distinct points. Suppose (α, β) is a point on a closed curve component. Since f is a quadratic function of α 2 , it follows that nearby (α, β) there are two distinct solutions for a fixed β. If the curve does not cross α = 0, then there are four distinct solutions sufficiently close to two points in α > 0. But then there are also four points for α < 0, which is impossible. So the curve must cross α = 0. But this gives a contradiction as α 0. We can also eliminate the possibility of unbounded curves, as follows. Each unbounded curve must have two distinct unbounded branches. Then nearby s 0 , there will be four unbounded branches. But the implicit function defining the λ-wall Υ u,v,s 0 is only asymptotically cubic (see the proof of Proposition 4.4), and so this is impossible.
Consequently, Σ u,v ∩ {s = s 0 } consists only of isolated points. Now consider one of these points, with coordinates (α 0 , β 0 ). By regularity, we can use α and β as local coordinates on Σ u,v at this point. Note that the first fundamental form at (α 0 , β 0 ) is dα 2 +dβ 2 , from the vanishing of −∂ α f u,v /∂ s f u,v and −∂ β f u,v /∂ s f u,v .
The mean curvature is then We observe that numerical λ-walls do admit isolated points; see Figure 7 below. As a consequence of Theorem 6.7 and the proof of Theorem 6.12, we can deduce that 1-dimensional components of numerical λ-walls for a fixed s are regular away from ν-walls. Corollary 6.13. For any s > 0 and any real numerical Chern characters u and v, any connected component of a numerical λ-wall Υ u,v,s in H is regular as a real curve away from Ξ u,v . In particular, if there is no associated ν-wall for the pair u, v, then Υ u,v,s is always regular.
Proof. Away from the ν-wall Ξ u,v (if non-empty) when s = 1/3, Σ u,v is regular by Proposition 6.6 or Theorem 6.7. But by the proof above, the only place where the plane tangent to Σ u,v is parallel to the (α, β)-plane occurs at isolated points and not in a 1-dimensional portion of the surface wall.
We complete this section by returning to the issue of the intersection of numerical λ-walls for v and the curves Θ v and Γ v,s .
Unravelling the equality in (4.8) yields a cubic polynomial equation for β 0 . This means that the intersection Υ u,v,s ∩ Γ v,s consists of at most three points away from Θ v ; in addition, according to Lemma 4.11, the number of intersection points does not depend on the parameter s. Notice that the total number of intersection points of a λ-wall for v with Γ v,s will increase by 1 if q(v) < 0.
Let us now examine one situation in which Υ u,v,s ∩ Γ v,s contains two points.
Lemma 6.14. If a connected component of a numerical λ-wall Υ u,v,s crosses a connected component of Γ v,s twice away from the hyperbola Θ v , then the associated numerical ν-wall Ξ u,v is non-empty.
Proof. By Lemma 4.11, we can let s = 1/3. Then the two intersection points are local maxima of Υ u,v,1/3 . By Corollary 6.13, the component is regular or has a tacnode on a ν-wall. If it is regular, then it must have a minimum between the two maxima. Proposition 6.10 implies that such a minimum must lie on the associated ν-wall. Figure 8 illustrates the typical situation described in Lemma 6.14 and shows that it does arise. It is easy to see that the intersection with α = 0 must happen to the right of Figure 8. This example contains several interesting features. We set s = 1/3, v = (3, 1, 0, −1) and u = (0, 1, −3, 7). First, the curve Γ v,s (in red) intersects the positive branch of the hyperbola Θ v (in blue); this intersection is marked with a black bullet. Second, the numerical λ-wall Υ u,v,s (in black) crosses the curve Γ v,s four times (marked with green bullets), twice along Γ − v,s . Finally, we can see that the associated numerical ν-wall Ξ u,v (in magenta) cuts Γ − v,s (red bullet) between the two intersection points of Υ u,v,s ∩ Γ − v,s , illustrating the phenomenon described in Lemma 6.14. Both Υ u,v,s and Ξ u,v cut Θ − v at the same point.
The situation in Figure 8 also demonstrates another phenomenon in which a numerical λ-wall intersects horizontal lines four times. When this happens, it must be that f (α, β) has three turning points along this horizontal line. But note that ∂ β f = −∆ 31 when s = 1/3. On the other hand, ∂ α ∆ 31 = 0, and so the solutions of ∆ 31 = 0 are vertical lines. But these intersect the wall at its horizontal turning points (the green points in Figure 8). The middle one also intersects Υ u,v,1/3 again at a minimum or on Θ v , which must therefore also be on Ξ u,v . For a connected component, we could already deduce this because such a component must have at least one minimum, but the same will follow even when the geometry of the wall does not require there to be a minimum. This more precise reasoning allows us to refine Lemma 6.14 as follows.
Lemma 6.15. If a numerical λ-wall Υ u,v,s intersects Γ − v,s twice, then Υ u,v,s must intersect Θ v and Ξ u,v intersects Γ − v,s in between the intersection points. Proof. Note that Lemma 6.14 shows that if the wall is a connected component, then it must intersect Ξ u,v . Since the solutions of ∆ 31 = 0 are vertical lines, there must be minima above or below the maxima (the green dots on Γ − v,1/3 in Figure 8). Since the wall cannot cross Γ − v,s for a third time because it would have to double back on itself and there can only be at most two solutions of ∆ 32 = 0 along any vertical line, it must cross Ξ u,v a third time at a point which is not a minimum, which must therefore be on Θ v . Note that an alternative picture to Figure 8 has the wall cross α = 0 twice to the left of Γ − v,s . In that case, it only crosses Ξ u,v once, and then Ξ u,v does not intersect Γ − u,v between the two intersection points. Now we assume the wall is two nested components. First we observe that this hypothesis is independent of s. To see that, note that if a component were created at (α, β), then there would be six distinct solutions of f (α, β) = 0 along β = β, which is impossible. The two bounded components must have maxima at s = 1/3 which must intersect Γ − v,1/3 . Then there are three vertical line components of ∆ 31 = 0 which intersect Γ − v,1/3 at the maxima of the wall at distinct β-values. Since the inner bounded component must have a maximum (on Γ − v,1/3 ), the middle component of ∆ 31 = 0 intersects the wall at that point and so intersects the outer component away from Γ − v,1/3 . But this point cannot be a point of inflection (see Remark 6.11) and so must be a minimum, which must therefore be on a ν-wall. If the wall is otherwise in the exterior of the ν-wall, then that point must be at the maximum of the ν-wall and so is on Θ v . Otherwise, it must cross the ν-wall again, which is not a turning point, and so must also cross Θ v . In fact this minimum of Υ u,v,1/3 in the outer component must have another maximum, which must be on Γ v (see Figure 9 for a concrete example of this case).
We can argue similarly to Lemma 6.15 in R 0 v .
Proof. First observe that by Lemma 4.11, the hypothesis and conclusion are independent of s, and so we may set s = 1/3. If Υ u,v,s has a single component, then there must be a minimum between the maxima, and so it crosses Ξ u,v at that point. If there are two components, then the geometry does not require there to be a minimum. But then there are three vertical components of ∆ 31 = 0. The middle one intersects the inner component at its maximum, and then it must intersect the other component at a minimum (by Proposition 6.10) which must be on Ξ u,v .
In the last case, either the outer component is unbounded and the minimum occurs on Θ v , or it is bounded and there is a further maximum which must be on Γ ± v,1/3 , and so the wall again crosses Θ v . The shape is illustrated in Figure 9.
We can now state the main theorem of this section. Proof. Let Υ u,v,s be the component, and suppose for a contradiction that it does not cross Γ − v,s . By Theorem 6.12, this component must exist for all s, and so we may assume s = 1/3. By Theorem 6.7, Υ u,v,1/3 is regular except possibly in case (2) of that theorem, in which case the singular point is on Γ − v,s , as required. Otherwise, Υ u,v,1/3 is regular, and so it must have a maximum. By Proposition 6.10, it must intersect Γ − v,1/3 at this point.
Remark 6.18. In other words, if we want to classify all of the actual λ-walls in the region R − v,s then we only need to locate the ones which cross Γ − v,s . But note that the wall may not be actual as it crosses Γ − v,s .
On the other hand, this fails for s < 1/3 because isolated walls can appear as s decreases. Some will be unbounded as s → 0, but others may persist as isolated closed curves.
Finally, observe that analogous statements also hold for Γ + v,s and the region to the right of Θ + v .

v,s
Similarly to Definition 5.1, we introduced the following definition, where γ : [0, ∞) → H is an unbounded path.
Definition 7.1. An object A ∈ D b (X) is asymptotically λ α,β,s -(semi)stable along γ if the following two conditions hold for a given s > 0: Our first goal is to characterize asymptotically λ α,β,s -semistable objects with numerical Chern character v satisfying v 0 > 0 and the Bogomolov-Gieseker inequality along two families of paths contained in the region R − v , namely the paths Γ − v,s for each s > 0 and Λ − α for each α > 0; see the notation introduced in display (5.1). We then invoke Proposition 5.13 to characterize asymptotically λ α,β,s -semistable objects along the paths Γ + v,s and Λ + α . More precisely, we show that λ α,β,s -semistable objects along Γ + v,s and Λ + α are duals of Gieseker semistable sheaves.
First, we provide a simple consequence of asymptotic λ α,β,s -semistability and the support property along unbounded paths.

Lemma 7.2. Let A ∈ D b (X) be an asymptotically λ α,β,s -semistable object along an unbounded path γ(t). Then
Proof. The support property in Proposition 2.4 implies that Q γ(t) (A) ≥ 0 for all large enough t.
First suppose lim t→∞ β(t) = ±∞; then As the first factor is positive for all t, it follows that Q tilt (A) ≥ 0.

Asymptotics along Γ − v,s
We now turn to asymptotically λ α,β,s -semistable objects with Chern character v along the curve Γ − v,s ; we assume from now on v satisfies v 0 0 and the Bogomolov-Gieseker inequality (2.3).
Since ch β 1 (E) 0 along Γ ± v,s , we have , at least away from the point where Γ ± v,s meets Θ v (it meets it exactly once if q(E) < 0 and not at all otherwise), so we can use β < 0 as a parameter.
Note that a numerical λ-wall Υ u,v,s crosses Γ ± v,s precisely at the zeros of the function λ v,β,s (u). The behaviour of the ν-slope of an object F ∈ D b (X) along Γ ± v,s can be analysed in a similar way. Substituting α 2 from equation (7.1) into the expression for ν α,β (F), we obtain We are now in position to state the main result of this section. The proof will be done in a series of lemmas.
This claim actually follows directly from Proposition 5.13, but we give an alternative hands-on proof.
Proof. If E is asymptotically λ α,β,s -semistable along Γ − v,s , then E is a sheaf by Lemma 7.4 or Proposition 5.13. If E is not torsion-free, let F → E be its maximal torsion subsheaf; if T → F is a subsheaf of dimension at most 1, then T → E is a morphism in A α,β for (α, β) ∈ Γ − v,s and β 0 since T ∈ T α,β for every (α, β). We have that ch 0 (T ) = ch 1 (T ) = 0; then so T would destabilize E. Therefore, we can assume F has pure dimension 2; let T be its maximal µ-semistable subsheaf. Remark 5.7 implies that T ∈ A α,β for (α, β) ∈ Γ − v,s and β 0; thus T → E is a morphism in A α,β in the same range. Since ch 0 (T ) = 0 and ch 1 (T ) 0, we have again contradicting asymptotic λ-semistability. We therefore conclude that E must be torsion-free. If F → E is a proper (torsion-free) subsheaf with δ 01 (F, E) 0, then equation ( We now prove the converse of Lemma 7.5. The difficulty is that the strong definition of asymptotic stability includes showing that a given object has only finitely many walls along Γ − v,s . In fact, we can show that there is at most one, at least outside its actual ν-wall.
Assume E is a Gieseker stable sheaf, and suppose there is an actual λ-wall given by the short exact Consider the case where F and G are sheaves.
Proof. We can compute the same limits as in the proof of Lemma 7.5; we first have for every (α, β) ∈ Γ − v,s for β 0. This enables us to prove that Gieseker stable sheaves can only be destabilized as we move down along Γ − v,s . Lemma 7.7. Suppose E is a Gieseker stable sheaf and F → E is a subsheaf which also corresponds to an actual λ-wall W u,v,s crossing Γ − v,s at a point P beyond any ν-wall for E. If E/G ∈ A α,β ∩ Coh(X), then E must be stable above the point P .
Proof. Parameterize the curve Γ − v,s by γ(t) in the decreasing β-direction with γ(0) = P . By assumption and Proposition 5.13, E ∈ A γ(t) for all t ≥ 0. Suppose E is unstable in A γ(t) for some t > 0. Observe that for t 0, we have that F → E is a monomorphism in A γ(t) . This is because if λ γ(t),s (F) ≥ λ γ(t),s (E) for all t > 0, then Corollary 5.14 and Lemma 7.6 give a contradiction. It follows that at some point Q = γ(t 0 ), for t 0 > 0, we have λ Q,s (F) = λ Q,s (E), and so the numerical λ-wall Υ u,v,s crosses Γ − v,s twice. Then Lemma 6.15 M. Jardim and A. Maciocia tells us that there is a numerical ν-wall Ξ u,v intersecting Γ − v,s between P and Q. But 0 → F → E → G → 0 is a short exact sequence in B β(t) for all t ≥ 0, and so this is an actual ν-wall, contradicting the assumption.
We now show that above an actual ν-wall for E, the λ-wall equivalent of Lemma 5.3(2) holds. Proof. We prove the last part first. Suppose a numerical wall Υ u,v,s intersects Γ − v,s at P above the ν-wall for E. Note that Υ u,v,s is a slice of Σ u,v , and since this is an orientable surface, to show that the E can only be destabilized downwards, it suffices to consider the case s = 1/3. We show that f u,v,1/3 (α, β) is increasing as we cross Υ u,v,1/3 moving down Γ − v,1/3 . This is equivalent to showing ∇f · (−∂ β τ, ∂ α τ) P > 0.
Using (2.10) and Lemma 6.3, we have ∇f = (α∆ 21 − α∆ 30 , −∆ 31 ), and the tangent vector down along But ∆ 21 (P ) > 0 as we are outside the ν-wall (where ∆ 21 vanishes), and ch P v,s again, and then there would be an actual ν-wall between P and Q, which is not permitted by hypothesis. So it must be that F 2 → E ceases to be an injection at some point. Let that point be R. We show that E remains unstable as we cross R. In fact, we will prove a stronger result in the following lemma.
Lemma 7.9. Suppose E is a Gieseker stable sheaf with ch(E) = v, and let γ be a curve segment of Γ − v,s from a point Q = γ(0) to P = γ(1) which is outside an actual ν-wall. Suppose 0 → A → E → B → 0 is a short exact sequence in A P corresponding to an actual λ-wall through the point Q. Then E is λ γ(t),s -unstable for all t ∈ (0, 1]. Proof. By the second statement of the previous lemma, we know that 0 → A → E → B → 0 destabilizes for t ∈ (0, ) for some > 0. We also know from that proof that λ γ(t),s (A) > λ γ(t),s (E) for all t ∈ (0, 1]. In the case where A is not a sheaf, we have that 0 → A → E → B → 0 remains short exact to the end of Γ − v,s as E = A 00 and B = A 01 [2] and Γ − v,s ends on either the β-axis or Θ − v . So we may assume A is a sheaf. Then A remains in the category until its Harder-Narasimhan factor with smallest ν, A − , say, goes out of the category. But then just beyond that point λ(A − ) > 0, and so E → ker A (B → A − [1]) still destabilizes E. We can continue until we have exhausted all of the Harder-Narasimhan factors A of A. Finally, as we approach the last Θ − ch(A ) -curve for the filtration of A, we have λ(A ) < 0. But then there would be a wall corresponding to A destabilizing E above, which is impossible. It follows that A remains in A γ(t) for all t ∈ [0, 1].
Similarly, B 0 remains in A γ(t) until Θ − of a factor, but this cannot happen as E remains in the category. But then B remains in the category by Theorem 3.3. This completes the proof.
We can now complete the proof of Theorem 7.3.
Proof. If E is Gieseker stable, then the first part of Definition 7.1 follows from the fact that E ∈ Coh(X) and Proposition 5.13, and the second now follows from Theorem 7.8. The statement for semistability follows by inducting on the length of the Jordan-Hölder filtration of E.
Remark 7.11. Just as for Proposition 5.8, we can deduce from Lemma 7.10 that any E ∈ A α,β for all (α, β) along an unbounded Θ − -curve γ(t) in R − v has an asymptotic Harder-Harasimhan filtration for λ α β,s -stability. The following will be useful in Section 8.4. Proposition 7.12. Let E be a Gieseker stable sheaf with ch(E) = v, and let W u,v,s be an actual λ-wall in R − v crossing Γ − v,s . Then there is an actual λ-wall which either crosses the β-axis between Γ − v,s and Θ − v or cuts Θ − v . In particular, in the latter case, there is an actual ν-wall for E.
Proof. Let P denote the point where W u,v,s cuts Γ − v,s . By Lemma 4.17, W u,v,s ends either on another actual λ-wall which must remain above the original wall or on α = 0. So we have a piecewise path of actual λ-walls in the region between Γ − v,s and Θ − v . By Theorem 7.8, this path cannot cross Γ − v,s again except at P , but then there is a loop in R − v intersecting Γ − v,s at P outside of which E is unstable. But then E would be unstable on both sides of P , contradicting the fact that W u,v,s is an actual wall. The path cannot be unbounded in this region because unbounded curves are only unbounded in R 0 v . So it must cross either Θ − v or the β-axis in this region.
For the last part, observe that the final segment of the path crossing Θ − v is an actual λ-wall, and so by Theorem 4.21, there is an actual ν-wall crossing Θ − v at the same point.

Asymptotics along
Next, we study objects that are asymptotically λ α,β,s -semistable along a horizontal line {α = α}, or asymptotically λ α,β,s -semistable. The key is to observe that Λ − α is eventually in R L v,s , which is the region to the left of Γ − v,s (see Definition 3.7). Then we can use Theorem 6.17. We establish the following result. As before, the claim follows from Proposition 5.13(1); alternatively, it can also be proved in the same manner as Lemma 7.4 above.
With these formulas at hand, we can establish the if part of Theorem 7.13. First we prove a version of Lemma 7.6 along Λ − α .
provided ch 2 (T ) 0, contradicting the asymptotic λ-semistability. If ch(T ) = (0, 0, 0, e), then T clearly destabilizes E as well. So now assume F has pure dimension 2, and let T be its maximalμ-semistable subsheaf. Remark 5.7 implies that T ∈ A α,β for β 0; thus T → E is a morphism in A α,β in the same range. Since ch 0 (T ) = 0 and ch 1 (T ) 0, then again contradicting the asymptotic λ-semistability. We therefore conclude that E is torsion-free. If E is not Gieseker semistable, let F → E be its maximal destabilizing subsheaf. As in the first part of the proof of Lemma 7.17, we can conclude that F ∈ A α,β for β 0, so F → E is a morphism in A α,β in the same range.
It follows that thus δ 10 (F, E) = 0 because E is asymptotically λ α,β,s -semistable. We then have that thus again δ 20 (F, E) = 0. Finally, we have that thus δ 30 (F, E) = 0, meaning that ch(F) = ch(E), contradicting the fact that F is a proper subsheaf of E. We therefore conclude that E must be Gieseker (semi)stable, as desired.

Asymptotics along general unbounded Θ − -curves
We can now extend our results to any unbounded Θ − -curve (recall Definition 5.9).
Proof. We consider the stable case first, and we deduce the semistable case by inducting on the lengths of the Jordan-Hölder filtrations. Suppose E is Gieseker stable. Then Lemma 7.16 as α varies implies that there is at most a single destabilizing piecewise smooth curve of actual λ-walls from the β-axis, and it cannot be unbounded in R − v . So the piecewise curve must either cross the β-axis again or cross Θ − v . In either case, there is a β such that E is λ β,α,s -stable in {β < β} ∩ R − v and so asymptotically λ-stable along any curve in this region. Conversely, suppose that E is asymptotically λ-stable along γ and not Gieseker stable. Then for all t 0, E is not λ-stable along Λ − α(t) . Let t 0 be the least t ∈ R such that E is λ-semistable in A γ(t) for all t < t 0 . By increasing t 0 if necessary, we may assume that for each t > t 0 , there is some F t such that F t destabilizes E at some β = β 1 < β(t) along Λ − α(t) and such that ν − γ(t) (F t ) > 0 (by the assumption that γ is a Θ − -curve and Lemma 5.12). So there must be a wall W ch(E),ch(F t ),s which crosses the vertical line β = β 1 . But that wall must cross either Λ − α(t) at β < β 1 or (γ) at γ(t ) for some t > t. Since it remains an actual λ-wall, the latter possibility cannot happen by the assumption on t 0 . The former possibility means that there must be another F t which destabilizes at some (α(t), β 2 ) with β 2 < β 1 . Repeating, we have an infinite sequence of such destabilizers, which contradicts Lemma 7.16.
Remark 7.20. We expect that the assumption s ≥ 1/3 is unnecessary. But there are instances of walls which vanish as s increases. These could occur to the left of our piecewise destabilizing wall. For any s, we would conjecture that there are only finitely many of these, and then there would be a Gieseker chamber containing a finite number of small regions where E is λ-unstable.
We can now deduce asymptotic conditions along unbounded curves in R + v . Using the notion of unbounded Θ − -curve γ and of its dual curve γ * proposed in Definition 5.9, we state the following.
Proof. By Proposition 5.13, we know that there is a t 1 such that E ∨ ∈ A γ * (t) for all t > t 1 . If F → E destabilizes E for all t > t 0 , then the proposition also shows that there is some t 2 such that F ∨ and E ∨ are in A γ * (t) for all t > t 2 and also E ∨ → F ∨ surjects in A γ * (t) . But λ γ(t),s (F) = −λ γ * (t),s (F ∨ ).
Putting Theorem 7.19 and Proposition 7.21 together, we immediately deduce the following statement. Note that Proposition 2.15 provides an explicit characterization of objects A ∈ D b (X) that are dual to torsion-free sheaves.

Examples of classifying walls: ideal sheaves and null correlation sheaves
We complete this paper by studying some concrete examples of actual λ-walls and asymptotically λ α,β,s -stable objects for X = P 3 which illustrate many of the results established above.

Ideal sheaves of lines in P 3
Let L ⊂ P 3 be a line, and let I L denote its ideal sheaf; recall that v := ch(I L ) = (1, 0, −1, 1). The curves Θ v and Γ v,s are given by Note that for every s > 0, the curve Γ v,s intersects Θ − v at a single point, call it P s , whose β-coordinate is the real root of the polynomial 3sβ 3 + (2 − 6s)β + 3 = 0; the case s = 1/3 is pictured in Figure 2.
Theorem 4.22 guarantees the existence of vanishing ν-and λ-walls, which are precisely given by the triangle We simplify the notation and use Ξ := Ξ u,v and Υ s := Υ u,v,s , where u = ch(O P 3 (−1)), for the vanishing νand λ-walls just described, respectively.
Let ch ≤2 (A) = (r, −c, d); then r > 0, and the condition that both A and I L /A are in B β gives us −c − βr < 0, and so −βr < c ≤ β(1 − r). For β = − √ 2, we see that c = r is the only solution. Then the first inequality implies r < β/(1 + β) = √ 2/( √ 2 − 1) and so 0 < r ≤ 3. Now choosing β on Θ − v , we have d + βc + r = 0 in order for A to destabilize I L . So d = −r(β + 1) > r( √ 2 − 1). On the other hand, the Bogomolov-Gieseker inequality for A gives r 2 ≥ 2rd and so d ≤ r/2. Then d = r/2 as it must be in Z[1/2]. Figure 9. The vanishing ν-and λ-walls (in purple and black, respectively) for the ideal sheaf of a line. Note that the λ-wall is connected but possesses two irreducible components. Both curves intersect Θ v (in blue) and Γ − v,s (in red).

Torsion-free sheaves with Chern character
(2, 0, −1, 0) Our first step towards the classification of λ α,β,s -semistable objects with Chern character v = (2, 0, −1, 0) on P 3 near the curve Γ − v,s is the classification of µ-semistable torsion-free sheaves with this Chern character. Below, p and L, respectively, denote a point and a line in P 3 . Recall also that a torsion-free sheaf E on P 3 is called a null correlation sheaf if it sits in the exact sequence see [Ein82]. Note that ch(E) = (2, 0, −1, 0). A locally free null correlation sheaf is called a null correlation bundle; non-locally free null correlation sheaves satisfy a sequence of the form Proposition 8.1. Let E be a µ-semistable torsion-free sheaf on X = P 3 with ch(E) = (2, 0, −1, 0).
(1) If E is locally free, then E is a null correlation bundle; in particular, E is µ-stable.
(2) If E is properly torsion-free, then E is strictly µ-semistable, and it is given by one of the following extensions: (a) 0 → I L → E → I p → 0 for p ∈ L with non-trivial extension (in particular, E is a null correlation sheaf, and it is µ ≤2 -stable), (b) 0 → I p → E → I L → 0 for arbitrary p and L (in particular, E is not µ ≤2 -semistable, and it has no global sections), If E is not locally free, then E * * is a µ-semistable rank 2 reflexive sheaf with ch 1 (E * * ) = 0, and either ch 2 (E * * ) = 0 or ch 2 (E * * ) = 1; in both cases, E * * is strictly µ-semistable (cf. [Cha84, Lemma 2.1]), so E is strictly µ-semistable. The first case forces E * * = O ⊕2 P 3 , while, in the second case, E * * must be a properly reflexive sheaf S L given by the sequence note that ch(S L ) = (2, 0, −1, 1). We start by analysing the first case, that is, E * * = O ⊕2 P 3 . We have that ch(Q E ) = (0, 0, 1, 0), where Q E := E * * /E; note that h 0 (E) = 0, 1. Again, two possibilities follow: either Q E has pure dimension 1, in which case Q E = O L (1), or Q E satisfies a sequence of the form where Z is a 0-dimensional sheaf and k − 1 = −h 0 (Z) < 0 because ch 3 (O L (k)) = − ch 3 (Z); since we have h 0 (O L (k)) > 0, we must have k = 0 and thus Z = O p .
The first possibility, namely Q E = O L (1), leads to the sequence (1) → 0 and therefore yields the sheaves described in item (2)(a) since I L is the kernel of the composition In addition, notice that these are the null correlation sheaves. Checking that these sheaves are µ ≥2 -stable is a simple exercise.
The second possibility leads to the sequence 0 → E → O ⊕2 P 3 → OL → 0 and therefore yields the sheaves described in item (2)(c).
Finally, if E * * = S L , then Q E = O p , and the sequence 0 → E → S L → O p → 0 together with the sequence (8.3) lead to the sequence in item (2)(b).
It is important to notice that non-trivial extensions like the ones in items (2)(b) and (2)(c) of Proposition 8.1 do exist. Let us first consider the extension of an ideal sheaf of a line by an ideal sheaf of a point. If p ∈ L, then one can check that Comparing with the exact sequence in display (8.4), we obtain the second part of the statement.
Next we consider the sheaves in item (2)(c) of Proposition 8.1. If E is a null correlation sheaf, then J (E) is a divisor of degree 1 in G. Therefore, for each null correlation sheaf E and L ∈ J (E), there exists an epimorphism E O L (−1) whose kernel is a Gieseker semistable sheaf K with ch(K) = (2, 0, −2, 2).
Such sheaves have been previously considered by Miró-Roig and Trautmann in [MRT94, Section 1.5]; they showed that the family of sheaves K defined by exact sequences of the form where E is a null correlation sheaf and L ∈ J (E), define a locally closed 8-dimensional subscheme of the full moduli space of Gieseker semistable sheaves on P 3 with Chern character equal to (2, 0, −2, 2). In addition, this moduli space, which we will simply denote by Z, is an irreducible projective variety of dimension 13, and every such sheaf satisfies an exact sequence of the form Our next goal in this section is the classification of ν α,β -stable objects for the Chern character v = (−2, 0, 1, 0). First, we show that there are no ν-walls.
Lemma 8.4 has two interesting applications. First we recover the following stronger version of a well-known fact.
Note that S L in the proof of Proposition 8.1 is an example of a µ-semistable reflexive sheaf of Chern character (2, 0, −1, 1). In addition, let E be a null correlation bundle, and let {p 1 , . . . , p k } be distinct points in P 3 ; the kernel of an epimorphism E ⊕ k i=1 O p i provides an example of a µ-stable torsion-free sheaf with Chern character (2, 0, −1, −k) for any k > 0.
Proof. Theorem 5.2 and Proposition 8.1 tell us that the asymptotically ν α,β -stable objects on each side of the α-axis are precisely the ones described above. Since there are no actual ν-walls, such objects are ν α,β -stable everywhere.
In any case, F or G is not λ P -semistable, which is impossible.
We take this inequality as an ansatz to eliminate possibilities rather than a necessary condition for a numerical wall to be actual. where L is a line, C is the union of two lines on a quadric surface in P 3 and H is a hyperplane. Moreover, K is a Gieseker semistable sheaf with ch(K) = (2, 0, −2, 2), as described in display (8.5), and A is quasi-isomorphic to a complex O P 3 (−2) ⊕4 → O P 3 (−1) ⊕5 . The first two are coincident, and only (8.13) is an actual λ-wall below Γ − v,1/3 , while only (8.12) is actual above Γ − v,1/3 . Proof. We assume E is a Gieseker stable torsion-free sheaf. These are classified by Proposition 8.1. We start by looking at an exceptional case not described in Proposition 2.12. This is where F is not a sheaf and the destabilizing sequence is of the form F → E → F 11 [2]. This arises from the presentation of E in (8.6), Viewing the middle map as a 2-step complex A in D b (X), near P we have that A → E → A 11 [2] is short exact in A P . It is easy to see that this is a wall and A 11 = O P 3 (−3), giving (8.13). From the definition of A, it follows that thus where this wall is actual, it is a vanishing wall. The presentation of A shows that it has a vanishing λ-wall given by O(−2) (as a sub-object) which goes through the point P ; denote this wall by W . One can check that W is above (8.13) to the right of Γ − v,1/3 , but it is below (8.13) to the left of Γ − v,1/3 . This shows that (8.13) is actual to the left but not to the right of Γ − v,1/3 . Once we have found the walls for µ-semistable torsion-free sheaves E, we need to also consider new objects which become λ-stable as we cross each wall. Lemma 8.7 and Proposition 2.12 show that it is reasonable to assume our destabilizing objects F and G are in A α,β ∩ B β ∩ Coh(X). Then we have additional constraints to add to (8.7) and (8.8): (8.15) − 2r < x ≤ 0 since E is µ-semistable and F ∈ B β . We also add the ansatz (8.9) and consider various possible cases for the Chern character (r, x, y/2, z/6) of F. First consider the case r = 1.
Finally, we return to the case r = 2. Then G is a torsion sheaf. We have −3 ≤ x ≤ 0. We shall consider these, case by case as before.
We now consider the special case x = 0. The constraints give −6 ≤ y < −2. But y must be even for G to be a sheaf, and so y ≤ −4. When y = −4, we have the same wall as (8.10) but given by the sequence (8.11). When y = −6, we have Q P (F) = −8, and so F cannot be semistable.
It follows from Theorem 6.17 that no other λ-walls in the region R − v exist when s = 1/3. This observation allows us to give a complete chamber decomposition of this region, summarized in the following statement.
The two stability chambers just described are pictured in Figure 11 as follows: the chamber (C1) lies above the green and black curves (which correspond to the walls (8.13) and (8.12), respectively).