Bridgeland Stability on Threefolds -- Some Wall Crossings
BBRIDGELAND STABILITY ON THREEFOLDS - SOME WALLCROSSINGS
BENJAMIN SCHMIDT
Abstract.
Following up on the construction of Bridgeland stability condition on P byMacrì, we compute first examples of wall crossing behaviour. In particular, for Hilbertschemes of curves such as twisted cubics or complete intersections of the same degree, weshow that there are two chambers in the stability manifold where the moduli space is givenby a smooth projective irreducible variety respectively the Hilbert scheme. In the caseof twisted cubics, we compute all walls and moduli spaces on a path between those twochambers. In between slope stability and Bridgeland stability there is the notion of tiltstability that is defined similarly to Bridgeland stability on surfaces. We develop tools touse computations in tilt stability to compute wall crossings in Bridgeland stability. Contents
1. Introduction 12. Very Weak Stability Conditions and the Support Property 43. Constructions and Basic Properties 84. Stability on P Introduction
The introduction of stability condition on triangulated categories by Bridgeland in [Bri07]has revolutionized the study of moduli spaces of sheaves on smooth projective surfaces. Weintroduce techniques that worked on surfaces into the realm of threefolds. As an applicationwe deal with moduli spaces of sheaves on P . It turns out that for certain Chern charactersthere is a chamber in the stability manifold Stab( P ) where the corresponding moduli space issmooth, projective and irreducible. The following theorem applies in particular to completeintersections of the same degree or twisted cubics. Theorem 1.1 (See also Theorem 7.1) . Let v = i ch( O P ( m )) − j ch( O P ( n )) where m, n ∈ Z are integers with n < m and i, j ∈ N are positive integers. Assume that ( v , v , v ) is aprimitive vector. There is a path γ : [0 , → Stab( P ) that satisfies the following properties. Mathematics Subject Classification.
Key words and phrases.
Bridgeland stability conditions, Derived categories, Threefolds, Hilbert Schemesof Curves. a r X i v : . [ m a t h . AG ] S e p At the beginning of the path the semistable objects are exactly slope stable coherentsheaves E with ch( E ) = v . (2) Before the last wall on γ the moduli space is smooth, irreducible and projective. (3) At the end of the path there are no semistable objects, i.e. the moduli space is empty.
As an example we compute all walls on the path of the last Theorem in the case of twistedcubics.
Theorem 1.2 (See also Theorem 7.2) . Let v = (1 , , − ,
5) = ch( I C ) where C ⊂ P is atwisted cubic curve. There is a path γ : [0 , → Stab( P ) such that the moduli spaces for v in its image outside of walls are given in the following order. (1) The empty space M = ∅ . (2) A smooth projective variety M that contains ideal sheaves of twisted cubic curves asan open subset. (3) A space with two components M ∪ M (cid:48) . The space M is a blow up of M in asmooth locus. The exceptional locus parametrizes plane singular cubic curves witha spatial embedded point at a singularity. The second component M (cid:48) is a P -bundleover P × ( P ) ∨ . An open subset in M (cid:48) parametrizes plane cubic curves togetherwith a potentially but not necessarily embedded point that is not scheme theoreticallycontained in the plane. (4) The Hilbert scheme of curves C with ch( I C ) = (1 , , − , . It is given as M ∪ M (cid:48) where M (cid:48) is a blow up of M (cid:48) in a smooth locus. The exceptional locus parametrizesplane cubic curves together with a point scheme theoretically contained in the plane. The Hilbert scheme of twisted cubics has been heavily studied. In [PS85] it was shown thatit has two smooth irreducible components of dimension and intersecting transversallyin a locus of dimension . In [EPS87] it was shown that the closure of the space of twistedcubics in this Hilbert scheme is the blow up of another smooth projective variety in a smoothlocus. This matches exactly the description we obtain using stability.The literature on Hilbert schemes on projective space from a more classical point of viewis vast. It turns out that the geometry of these spaces can be quite badly behaved. Forexample Mumford observed that there is an irreducible component in the Hilbert schemeon P containing smooth curves that is generically non reduced in [Mum62]. However,Hartshorne proved that Hilbert schemes in projective space are at least connected in [Har66].1.1. Ingredients.
Bridgeland’s original work was motivated by Calabi-Yau threefolds andrelated questions in physics. A fundamental issue in the theory of stability conditions onthreefolds is the actual construction of Bridgeland stability conditions. A conjectural way hasbeen proposed in [BMT14] and has been proven for P in [MacE14], for the smooth quadricthreefold in [Sch14] and for abelian threefolds in both [MP13a, MP13b] and [BMS14]. Inorder to do so the notion of tilt stability has been introduced in [BMT14] as an intermediatenotion between classical slope stability and Bridgeland stability on a smooth projectivethreefold X over C . The construction is analogous to Bridgeland stability on surfaces. Theheart is a certain abelian category of two term complexes while the central charge is givenby Z tilt α,β = − H · ch β + α H · ch β + iH · ch β here H ∈ Pic( X ) is ample, α > , β ∈ R and ch β = e − βH · ch is the twisted Chern character.More details on the construction of stability is given in Section 3. Many techniques thatworked in the case of surfaces still apply to tilt stability. Bayer, Macrì and Toda proposethat doing another tilt will lead to a Bridgeland stability condition with central charge Z α,β,s = − ch β +( s + ) α H · ch β + i ( H · ch β − α H · ch B ) where s > . The following theorem connects Bridgeland stability with the simpler notionof tilt stability. It is one of the key ingredients for the two theorems above. Theorem 1.3 (See also Theorem 6.1) . Let v be the Chern character of an object in D b ( X ) such that ( v , v , v ) is primitive. Then there are two paths γ , γ : [0 , → Stab( P ) such thatall moduli spaces of tilt stable objects outside of walls occur as moduli spaces of Bridgelandstable objects along either γ or γ . Notice that the Theorem does not preclude the existence of further chambers along thosepaths. In many cases, for example for twisted cubics as above, there are different exactsequences defining identical walls in tilt stability because the defining objects only differ inthe third Chern character. However, by definition, changes in ch cannot be detected viatilt stability. In Bridgeland stability those identical walls often move apart and give rise tofurther chambers.The computations in tilt stability in this article are very similar in nature to many com-putations about stability of sheaves on surfaces in [ABCH13, BM14, CHW14, LZ13, MM13,Nue14, Woo13, YY14]. Despite the tremendous success in the surface case, the threefold casehas barely been explored. Beyond the issue of constructing Bridgeland stability conditionthere are further problems that have made progress difficult.1.2. Further Questions.
For surfaces, or more generally, tilt stability parametrized bythe ( α, β ) upper half-plane, there is at most one unique vertical wall, while all other wallsare nested inside two piles of non intersecting semicircles. This structure is rather simple.However, in the case of Bridgeland stability on threefolds walls are given by real degree 4equation. Already in the case of twisted cubics we can observe that they intersect in Theorem7.2. Question 1.4.
Given a path γ in the stability manifold and a class v ∈ K num ( X ) is there anumerical criterion that determines all the walls on γ with respect to v ? If not, can we atleast numerically restrict the amount of potential walls on γ in an effective way?We are only able to answer this question for the two paths described in Theorem 6.1. Thegeneral situation seems to be more intricate. If we want to study stability in any meaningfulway beyond tilt stability, we need at least partial answers to this question.Another serious problem is the construction of reasonably behaved moduli spaces ofBridgeland semistable objects. A recent result by Piyaratne and Toda is a major steptowards this. Theorem 1.5 ([PT15]) . Let X be a smooth projective threefold such that the conjectural con-struction of Bridgeland stability from [BMT14] works. Then any moduli space of semistableobjects for such a Bridgeland stability condition is a universally closed algebraic stack offinite type over C . f there are no strictly semistable objects, the moduli space becomes a proper algebraicspace of finite type over C . For certain applications such as birational geometry we wouldlike our moduli spaces to be projective. Question 1.6.
Assume σ ∈ Stab( X ) is a Bridgeland stability condition and v ∈ K num ( X ) .Is the moduli space of σ -stable objects with class v quasi-projective?1.3. Organization of the Article.
In Section 2 we recall the notion of a very weak stabil-ity condition from [BMS14] and [PT15]. All our examples of stability conditions fall underthis notion. Section 3 describes the construction of both tilt stability and Bridgeland sta-bility and establishes some basic properties. In particular, we remark which techniques forBridgeland stability on surfaces work without issues in tilt stability. In Section 4 we dealwith stability of line bundles or powers of line bundles on P by connecting these questionsto moduli of quiver representations. Section 5 deals with computing specific examples in P for tilt stability. Moreover, we discuss how many of those calculations can be handledby computer calculations. In Section 6 we prove our main comparison theorem betweenBridgeland stability and tilt stability. Finally, in Section 7 we use this connection to finishthe computations necessary to establish the two main theorems.1.4. Notation. X smooth projective variety over C , n dim X , H fixed ample divisor on X , I Z/X , I Z ideal sheaf of a closed subscheme Z ⊂ X , D b ( X ) bounded derived category of coherentsheaves on X , ch X ( E ) , ch( E ) Chern character of an object E ∈ D b ( X ) , ch ≤ l,X ( E ) , ch ≤ l ( E ) (ch ,X ( E ) , . . . , ch l,X ( E )) , H · ch X ( E ) , H · ch( E ) ( H n · ch ,X ( E ) , H n − · ch ,X ( E ) , . . . , ch n,X ( E )) for an ample divisor H on X , H · ch ≤ l,X ( E ) , H · ch ≤ l ( E ) ( H n · ch ,X ( E ) , . . . , H n − l · ch l,X ( E )) for an ample divisor H on X , K num ( X ) the numerical Grothendieck group of X , Acknowledgements.
I would like to thank David Anderson, Arend Bayer, Patricio Gal-lardo, César Lozano Huerta and Emanuele Macrì for insightful discussions and comments onthis article. I especially thank my advisor Emanuele Macrì for carefully reading preliminaryversions of this article. Most of this work was done at the Ohio State University whosemathematics department was extraordinarily accommodating after my advisor moved. Inparticular, Thomas Kerler and Roman Nitze helped me a lot with handling the situation.Lastly, I would like to thank Northeastern University at which the finals details of this workwere finished for their hospitality. The research was partially supported by NSF grantsDMS-1160466 and DMS-1523496 (PI Emanuele Macrì) and a presidential fellowship of theOhio State University.2.
Very Weak Stability Conditions and the Support Property
All forms of stability occurring in this article are encompassed by the notion of a veryweak stability condition introduced in Appendix B of [BMS14]. It will allow us to treat ifferent forms of stability uniformly. We will recall this notion more closely to how it wasdefined in [PT15]. Definition 2.1. A heart of a bounded t-structure on D b ( X ) is a full additive subcategory A ⊂ D b ( X ) such that • for integers i > j and A ∈ A [ i ] , B ∈ A [ j ] the vanishing Hom(
A, B ) = 0 holds, • for all E ∈ D there are integers k > . . . > k m and a collection of triangles E (cid:47) (cid:47) E (cid:47) (cid:47) (cid:15) (cid:15) E (cid:47) (cid:47) (cid:15) (cid:15) . . . (cid:47) (cid:47) E m − (cid:47) (cid:47) (cid:15) (cid:15) E m = E (cid:15) (cid:15) A [ k ] (cid:100) (cid:100) A [ k ] (cid:100) (cid:100) A m − [ k m − ] (cid:101) (cid:101) A m [ k m ] (cid:103) (cid:103) where A i ∈ A .The heart of a bounded t-structure is automatically abelian. A proof of this fact and a fullintroduction to the theory of t-structures can be found in [BBD82]. The standard exampleof a heart of a bounded t-structure on D b ( X ) is given by Coh( X ) . While it is generally nottrue that D b ( A ) ∼ = D b ( X ) it is still an intuitive way to partially comprehend this notion. Definition 2.2 ([Bri07]) . A slicing of D b ( X ) is a collection of subcategories P ( φ ) ⊂ D b ( X ) for all φ ∈ R such that • P ( φ )[1] = P ( φ + 1) , • if φ > φ and A ∈ P ( φ ) , B ∈ P ( φ ) then Hom(
A, B ) = 0 , • for all E ∈ D b ( X ) there are φ > . . . > φ m and a collection of triangles E (cid:47) (cid:47) E (cid:47) (cid:47) (cid:15) (cid:15) E (cid:47) (cid:47) (cid:15) (cid:15) . . . (cid:47) (cid:47) E m − (cid:47) (cid:47) (cid:15) (cid:15) E m = E (cid:15) (cid:15) A (cid:99) (cid:99) A (cid:96) (cid:96) A m − (cid:98) (cid:98) A m (cid:101) (cid:101) where A i ∈ P ( φ i ) .For this filtration of an element E ∈ D b ( X ) we write φ − ( E ) := φ m and φ + ( E ) := φ .Moreover, for E ∈ P ( φ ) we call φ ( E ) := φ the phase of E .The last property is called the Harder-Narasimhan filtration . By setting A := P ((0 , to be the extension closure of the subcategories { P ( φ ) : φ ∈ (0 , } one gets the heart of abounded t-structure from a slicing. In both cases of a slicing and the heart of a boundedt-structure it is not particularly difficult to show that the Harder-Narasimhan filtration isunique.Let v : K ( X ) → Γ be a homomorphism where Γ is a finite rank lattice. Fix H to be anample divisor on X . Then v will usually be one of the homomorphisms H · ch ≤ l defined by E (cid:55)→ ( H n · ch ( E ) , . . . , H n − l · ch l ( E )) . for some l ≤ n . Definition 2.3 ([PT15]) . A very weak pre-stability condition on D b ( X ) is a pair σ = ( P, Z ) where P is a slicing of D b ( X ) and Z : Γ → C is a homomorphism such that any non zero ∈ P ( φ ) satisfies Z ( v ( E )) ∈ (cid:40) R > e iπφ for φ ∈ R \ ZR ≥ e iπφ for φ ∈ Z . This definition is short and good for abstract argumentation, but it is not very practicalfor defining concrete examples. As before, the heart of a bounded t-structure can be definedby A := P ((0 , . The usual way to define a very weak pre-stability condition is to insteaddefine the heart of a bounded t-structure A and a central charge Z : Γ → C such that Z ◦ v maps A\{ } to the upper half plane plus the non positive real line { re iπϕ : r ≥ , ϕ ∈ (0 , } .The subcategory P ( φ ) for φ ∈ (0 , consists of all semistable objects such that Z ( v ( E )) ∈ (cid:40) R > e iπφ for φ ∈ R \ ZR ≥ e iπφ for φ ∈ Z . More precisely, we can define a slope function by µ σ := − (cid:60) ( Z ) (cid:61) ( Z ) , where dividing by is interpreted as + ∞ . Then an object E ∈ A is called (semi-)stable if forall monomorphisms A (cid:44) → E in A we have µ σ ( A ) < ( ≤ ) µ σ ( A/E ) . More generally, an element E ∈ D b ( X ) is called (semi-)stable if there is m ∈ Z such that E [ m ] ∈ A is (semi-)stable. Asemistable but not stable object is called strictly semistable . Moreover, one needs to showthat Harder-Narasimhan filtrations exist inside A with respect to the slope function µ σ toactually get a very weak pre-stability condition. We interchangeably use ( A , Z ) and ( P, Z ) to denote the same very weak pre-stability condition.An important tool is the support property. It was introduced in [KS08] for Bridgelandstability conditions, but can be adapted without much trouble to very weak stability condi-tions (see [PT15, Section 2]). We also recommend [BMS14, Appendix A] for a nicely writtentreatment of this notion. Without loss of generality we can assume that Z ( v ( E )) = 0 implies v ( E ) = 0 . If not we replace Γ by a suitable quotient. Definition 2.4.
A very weak pre-stability condition σ = ( A , Z ) satisfies the support property if there is a bilinear form Q on Γ ⊗ R such that(1) all semistable objects E ∈ A satisfy the inequality Q ( v ( E ) , v ( E )) ≥ and(2) all non zero vectors v ∈ Γ ⊗ R with Z ( v ) = 0 satisfy Q ( v, v ) < .A very weak pre-stability condition satisfying the support property is called a very weakstability condition .By abuse of notation we will write Q ( E, F ) instead of Q ( v ( E ) , v ( F )) for E, F ∈ D b ( X ) .We will also use the notation Q ( E ) = Q ( E, E ) .Let Stab vw ( X, v ) be the set of very weak stability conditions on X with respect to v .This set can be given a topology as the coarsest topology such that the maps ( A , Z ) (cid:55)→ Z , ( A , Z ) (cid:55)→ φ + ( E ) and ( A , Z ) (cid:55)→ φ − ( E ) for any E ∈ D b ( X ) are continuous. Lemma 2.5 ([BMS14][Section 8, Lemma A.7 & Proposition A.8]) . Assume that Q hassignature (2 , rk Γ − and U is a path connected open subset of Stab vw ( X, v ) such that all σ ∈ U satisfy the support property with respect to Q . If E ∈ D b ( X ) with Q ( E ) = 0 is σ -stable for some σ ∈ U then it is σ (cid:48) -stable for all σ (cid:48) ∈ U unless it is destabilized by an object F with v ( F ) = 0 . • Let ρ be a ray in C starting at the origin. Then C + = Z − ( ρ ) ∩ { Q ≥ } is a convex cone for any very weak stability condition ( A , Z ) ∈ U . • Moreover, any vector w ∈ C + with Q ( w ) = 0 generates an extremal ray of C + . Only the situation of an actual stability condition is handled in [BMS14]. In that situationthere are no objects F in the heart with v ( F ) = 0 . However, exactly the same arguments gothrough in the case of a very weak stability condition. Definition 2.6. A numerical wall inside Stab vw ( X, v ) (or a subspace of it) with respect toan element w ∈ Γ is a proper non trivial solution set of an equation µ σ ( w ) = µ σ ( u ) for avector u ∈ Γ .A subset of a numerical wall is called an actual wall if for each point of the subset thereis an an exact sequence of semistable objects → F → E → G → in A where v ( E ) = w and µ σ ( F ) = µ σ ( G ) numerically defines the wall.Walls in the space of very weak stability conditions satisfy certain numerical restrictionswith respect to Q . Lemma 2.7.
Let σ = ( A , Z ) be a very weak stability condition satisfying the support propertywith respect to Q (it is actually enough for Q to be negative semi-definite on Ker Z ). (1) Let
F, G ∈ A be semistable objects. If µ ( F ) = µ ( G ) , then Q ( F, G ) ≥ . (2) Assume there is an actual wall defined by an exact sequence → F → E → G → .Then ≤ Q ( F ) + Q ( G ) ≤ Q ( E ) .Proof. We start with the first statement. If Z ( F ) = 0 or Z ( G ) = 0 , then Q ( F, G ) = 0 . Ifnot, there is λ > such that Z ( F − λG ) = 0 . Therefore, we get ≥ Q ( F − λG ) = Q ( F ) + λ Q ( G ) − λQ ( F, G ) . The inequalities Q ( F ) ≥ and Q ( G ) ≥ lead to Q ( F, G ) ≥ . For the second statement wehave Q ( E ) = Q ( F ) + Q ( G ) + 2 Q ( F, G ) ≥ . Since all four terms are positive, the claim follows. (cid:3)
Remark 2.8.
Since Q has to be only negative semi-definite on Ker Z for the Lemma toapply, it is sometimes possible to define Q on a bigger lattice than Γ . For example, we willdefine a very weak stability condition factoring through v = H · ch ≤ , but apply the Lemmafor v = H · ch where everything is still well defined later on.The most well known example of a very weak stability condition is slope stability. Wewill slightly generalize it for notational purposes. Let H be a fixed ample divisor on X .Moreover, pick a real number β . Then the twisted Chern character ch β is defined to be e − βH · ch . In more detail, one has h β = ch , ch β = ch − βH · ch , ch β = ch − βH · ch + β H · ch , ch β = ch − βH · ch + β H · ch − β H · ch . In this case v = H · ch ≤ . The central charge is given by Z slβ ( r, c ) = − ( c − βr ) + ir. The heart of a bounded t-structure in this case is simply
Coh( X ) . The existence of Harder-Narasimhan filtration was first proven for curves in [HN74], but holds in general. Finally thesupport property is satisfied for Q = 0 . We will denote the corresponding slope function by µ β := H · ch β H · ch β = H · ch H · ch − β. Note that the modification by β does not change stability itself but just shifts the value ofthe slope. 3. Constructions and Basic Properties
Tilt Stability.
In [BMT14] the notion of tilt stability has been introduced as an aux-iliary notion in between classical slope stability and Bridgeland stability on threefolds. Wewill recall its construction and prove a few properties. From now on let dim X = 3 .The process of tilting is used to obtain a new heart of a bounded t-structure. For moreinformation on the general theory of tilting we refer to [HRS96]. A torsion pair is defined by T β = { E ∈ Coh( X ) : any quotient E (cid:16) G satisfies µ β ( G ) > } , F β = { E ∈ Coh( X ) : any subsheaf F ⊂ E satisfies µ β ( F ) ≤ } . A new heart of a bounded t-structure is defined as the extension closure
Coh β ( X ) := (cid:104)F β [1] , T β (cid:105) . In this case v = H · ch ≤ . Let α > be a positive real number. The cen-tral charge is given by Z tilt α,β ( r, c, d ) = − ( d − βc + β r ) + α r + i ( c − βr ) The corresponding slope function is ν α,β := H · ch β − α H · ch β H · ch β . Note that in regard to [BMT14] this slope has been modified by switching ω with √ ω . Weprefer this point of view for aesthetical reasons because it will make the walls semicirclesand not just ellipses. Every object in Coh β ( X ) has a Harder-Narasimhan filtration due to[BMT14, Lemma 3.2.4]. The support property is directly linked to the Bogomolov inequality.This inequality was first proven for slope semistable sheaves in [Bog78]. We define the bilinearform by Q tilt (( r, c, d ) , ( R, C, D )) = Cc − Rd − Dr . heorem 3.1 (Bogomolov Inequality for Tilt Stability, [BMT14, Corollary 7.3.2]) . Any ν α,β -semistable object E ∈ Coh β ( X ) satisfies Q tilt ( E ) = ( H · ch β ( E )) − H · ch β )( H · ch β )= ( H · ch ( E )) − H · ch )( H · ch ) ≥ . As a consequence (Coh β , Z tilt α,β ) satisfies the support property with respect to Q tilt . Onsmooth projective surfaces this is already enough to get a Bridgeland stability condition(see [Bri08, AB13]). On threefolds this notion is not able to properly handle geometry thatoccurs in codimension three as we will see. Proposition 3.2 ([BMS14, Appendix B]) . The function R > × R → Stab vw ( X, v ) definedby ( α, β ) (cid:55)→ (Coh β ( X ) , Z tilt α,β ) is continuous. Moreover, walls with respect to a class w ∈ Γ inthe image of this map are locally finite. Numerical walls in tilt stability satisfy Bertram’s Nested Wall Theorem. For surfaces itwas proven in [MacA14].
Theorem 3.3 (Structure Theorem for Walls in Tilt Stability) . Fix a vector ( R, C, D ) ∈ Z × / Z . All numerical walls in the following statements are with respect to ( R, C, D ) . (1) Numerical walls in tilt stability are of the form xα + xβ + yβ + z = 0 for x = Rc − Cr , y = 2( Dr − Rd ) and z = 2( Cd − Dc ) . In particular, they are eithersemicircles with center on the β -axis or vertical rays. (2) If two numerical walls given by ν α,β ( r, c, d ) = ν α,β ( R, C, D ) and ν α,β ( r (cid:48) , c (cid:48) , d (cid:48) ) = ν α,β ( R, C, D ) intersect for any α ≥ and β ∈ R then ( r, c, d ) , ( r (cid:48) , c (cid:48) , d (cid:48) ) and ( R, C, D ) are linearly dependent. In particular, the two walls are completely identical. (3) The curve ν α,β ( R, C, D ) = 0 is given by the hyperbola Rα − Rβ + 2 Cβ − D = 0 . Moreover, this hyperbola intersect all semicircles at their top point. (4) If R (cid:54) = 0 there is exactly one vertical numerical wall given by β = C/R . If R = 0 there is no vertical wall. (5) If a numerical wall has a single point at which it is an actual wall, then all of it isan actual wall.Proof.
Part (1) and (3) are straightforward but lengthy computations only relying on thenumerical data.A wall can also be described as two vectors mapping to the same line under the homomor-phism Z tilt α,β . This homomorphism maps surjectively onto C . Therefore, at most two linearlyindependent vectors can be mapped onto the same line. That proves (2).In order to prove (4), observe that a vertical wall occurs when x = 0 holds. By the aboveformula for x this implies c = CrR in case R (cid:54) = 0 . A direct computation shows that the equation simplifies to β = C/R . If R = 0 and C (cid:54) = 0 , then r = 0 . This implies that the two slopes are the same for all or no α, β ) . If R = C = 0 , then all objects with this Chern character are automatically semistableand there are no walls at all.Let → F → E → G → be an exact sequence of tilt semistable objects in Coh β ( X ) that defines an actual wall. If there is a point on the numerical wall at which this sequencedoes not define a wall anymore, then either F , E or G have to destabilize at another pointalong the numerical wall in between the two points. But that would mean two numericalwalls intersect in contradiction to (2). (cid:3) A generalized Bogomolov inequality involving third Chern characters for tilt semistableobjects with ν α,β = 0 has been conjectured in [BMT14]. In [BMS14] it was shown that theconjecture is equivalent to the following more general inequality that drops the hypothesis ν α,β = 0 . Conjecture 3.4 (BMT Inequality) . Any ν α,β -semistable object E ∈ Coh β ( X ) satisfies α Q tilt ( E ) + 4( H · ch β ( E )) − H · ch β ) ch β ≥ . By using the definition of ch β ( E ) and expanding the expression one can find x, y ∈ R depending on E such that the inequality becomes α Q tilt ( E ) + β Q tilt ( E ) + xβ + y ≥ . This means the solution set is given by the complement of a semi-disc with center on the β -axis or a quadrant to one side of a vertical line. The conjecture is known for P [MacE14],the smooth quadric threefold [Sch14] and all abelian threefolds [BMS14, MP13a, MP13b].Another question that comes up in concrete situations is the question whether a given tiltsemistable object is a sheaf. For a fixed β let c := inf { H · ch β ( E ) > E ∈ Coh β ( X ) } . Lemma 3.5 ([BMT14, Lemma 7.2.1 and 7.2.2]) . An object E ∈ Coh β ( X ) that is ν α,β -semistable for all α (cid:29) is given by one of three possibilities. (1) E = H ( E ) is a pure sheaf supported in dimension greater than or equal to two thatis slope semistable. (2) E = H ( E ) is a sheaf supported in dimension less than or equal to one. (3) H − ( E ) is a torsion free slope semistable sheaf and H ( E ) is supported in dimensionless than or equal to one. Moreover, if µ β ( E ) < then Hom(
F, E ) = 0 for all sheaves F of dimension less than or equal to one.An object F ∈ Coh β ( X ) with H · ch β ∈ { , c } is ν α,β -semistable if and only if it is given byone of the three types above. Notice that part of the second statement follows directly from the first as follows. Anysubobject of F in Coh β ( X ) must have H · ch β = 0 or H · ch β = c . In the second case thecorresponding quotient satisfies H · ch β = 0 . Therefore, in both cases either the quotient orthe subobject have infinite slope. This means there is no wall that could destabilize F forany α > . This type of argument will be used several times in the next sections. Using thesame proof as in the surface case in [Bri08, Proposition 14.1] leads to the following lemma. Lemma 3.6.
Assume E ∈ Coh( X ) is a slope stable sheaf and β < µ ( E ) . Then E is ν α,β -stable for all α (cid:29) . .2. Bridgeland Stability.
We will recall the definition of a Bridgeland stability conditionfrom [Bri07] and show how they can be conjecturally constructed on threefolds based on theBMT-inequality as described in [BMT14]. It is known that the inequality holds on P dueto [MacE14] and we will apply it in a later section to study concrete examples of modulispaces of complexes in this case. Definition 3.7. A Bridgeland (pre-)stability condition on the category D b ( X ) is a very weak(pre-)stability condition ( P, Z ) such that Z ( E ) (cid:54) = 0 for all semistable objects E ∈ D b ( X ) .By Stab(
X, v ) we denote the subspace of Bridgeland stability conditions in Stab vw ( X, v ) .If A = P ((0 , is the corresponding heart, then we could have equivalently defined aBridgeland stability condition by the property Z ( E ) (cid:54) = 0 for all non zero E ∈ A . Note thatin this situation choosing the heart to be P ((0 , instead of P (( φ − , φ ]) for any φ ∈ R isarbitrary and any other choice works just as well. In some very special cases it is possible tochoose φ such that the corresponding heart is equivalent to the category of representationsof a quiver with relations. This will be particularly useful in the case of P . Theorem 3.8 ([Bri07, Section 7]) . The map ( A , Z ) (cid:55)→ Z from Stab(
X, v ) to Hom(Γ , C ) isa local homeomorphism. In particular, Stab(
X, v ) is a complex manifold. In order to have any hope of actually computing wall-crossing behaviour it is necessaryfor walls in Bridgeland stability to be somewhat reasonably behaved. The following resultdue to [Bri08, Section 9] is a major step towards that.
Theorem 3.9.
Walls in Bridgeland stability are locally finite, i.e. for a fixed vector v ∈ Γ there are only finitely many walls in any compact subset of Stab(
X, v ) . An important question is how moduli spaces change set theoretically at walls. In case thedestabilizing subobject and quotient are both stable this has a satisfactory answer due to[BM11, Lemma 5.9]. Note that this proof does not work in the case of very weak stabilityconditions due to the lack of unique factors in the Jordan-Hölder filtration.
Lemma 3.10.
Let σ = ( A , Z ) ∈ Stab( X ) such that there are stable object F, G ∈ A with µ σ ( F ) = µ σ ( G ) . Then there is an open neighborhood U around σ where non trivial extensions → F → E → G → are stable for all σ (cid:48) ∈ U such that φ σ (cid:48) ( F ) < φ σ (cid:48) ( G ) .Proof. Since stability is an open property there is an open neighborhood U of σ in whichboth F and G are stable. The category P ( φ σ ( F )) is of finite length with simple objectscorresponding to stable objects. In fact → F → E → G → is a Jordan-Hölder filtration.By shrinking U if necessary we know that if E is unstable at a point in U , there is a sequence → F (cid:48) → E → G (cid:48) → that becomes a Jordan-Hölder filtration at σ . Since the Jordan-Hölder filtration has unique factors and E is a non trivial extension, we get F = F (cid:48) and G = G (cid:48) . Therefore, there is no destabilizing sequence if φ σ (cid:48) ( F ) < φ σ (cid:48) ( G ) . (cid:3) It turns out that while constructing very weak stability conditions is not very difficult,constructing Bridgeland stability conditions is in general a wide open problem. Note thatfor any smooth projective variety of dimension bigger than or equal to two, there is noBridgeland stability condition factoring through the Chern character for A = Coh( X ) dueto [Tod09, Lemma 2.7]. ilt stability is no Bridgeland stability as can be seen by the fact that skyscraper sheavesare mapped to the origin. In [BMT14] it was conjectured that one has to tilt Coh β ( X ) againas follows in order to construct a Bridgeland stability condition on a threefold. Let T α,β = { E ∈ Coh β ( X ) : any quotient E (cid:16) G satisfies ν α,β ( G ) > } , F α,β = { E ∈ Coh β ( X ) : any subobject F (cid:44) → E satisfies ν α,β ( F ) ≤ } and set A α,β ( X ) := (cid:104)F α,β [1] , T α,β (cid:105) . For any s > they define Z α,β,s := − ch β +( s + ) α H · ch β + i ( H · ch β − α H · ch β ) ,λ ω,B,s := − (cid:60) ( Z α,β,s ) (cid:61) ( Z α,β,s ) . In this case the bilinear form is given by Q α,β,K (( r, c, d, e ) , ( R, C, D, E )) := Q tilt (( r, c, d ) , ( R, C, D ))( Kα + β )+ (3 Er + 3 Re − Cd − Dc ) β − Ce − Ec + 4 Dd. for some K ∈ (1 , s + 1) . Notice that for K = 1 this comes directly from the quadratic formin the BMT-inequality. Theorem 3.11 ([BMT14, Corollary 5.2.4], [BMS14, Lemma 8.8]) . If the BMT inequalityholds, then ( A α,β ( X ) , Z α,β,s ) is a Bridgeland stability condition for all s > . The supportproperty is satisfied with respect to Q α,β,K . Note that as a consequence the BMT inequality holds for all λ α,β,s -stable objects. In[BMS14, Proposition 8.10] it is shown that this implies a continuity result just as in the caseof tilt stability. Proposition 3.12.
The function R > × R × R > → Stab(
X, v ) defined by ( α, β, s ) (cid:55)→ ( A α,β ( X ) , Z α,β,s ) is continuous. In the case of tilt stability we have seen that the limiting stability for α → ∞ is closelyrelated with slope stability. The first step in connecting Bridgeland stability with tilt stabilityis a similar result. For an object E ∈ A α,β ( X ) we denote the cohomology with respect tothe heart Coh β ( X ) by H iβ ( E ) . It is defined by the property that H iβ ( E )[ i ] ∈ Coh β ( X ) is afactor in the Harder-Narasimhan filtration of E . Lemma 3.13 ([BMS14, Lemma 8.9]) . If E ∈ A α,β ( X ) is Z α,β,s -semistable for all s (cid:29) ,then one of the following two conditions holds. (1) E = H β ( E ) is a ν α,β -semistable object. (2) H − β ( E ) is ν α,β -semistable and H β ( E ) is a sheaf supported in dimension . Stability on P In the case of P more can be proven than in the general case. In this section the connectionto stability of quiver representations will be recalled and a stability result about line bundleswill be proven. It was already shown in [BMT14] that a line bundle L is tilt stable if Q tilt ( L ) = 0 . This condition always holds in Picard rank . However, we need a slightlymore refined result that holds in the special case of P . roposition 4.1. Let v = ± ch( O ( n ) ⊕ m ) for integers n, m with m > . Then O ( n ) ⊕ m or ashift of it is the unique tilt semistable and Bridgeland semistable object with Chern character ± v for any α > and β . Moreover, in the case m = 1 the line bundle O ( n ) is stable. For the proof we will need a connection between Bridgeland stability and quiver represen-tations. We will recall exceptional collections after [Bon90].
Definition 4.2. (1) An object E ∈ D b ( X ) is called an exceptional object if Ext l ( E, E ) =0 for all l (cid:54) = 0 and Hom(
E, E ) = C .(2) A sequence E , . . . , E n ∈ D b ( X ) of exceptional objects is a full exceptional collection if Ext l ( E i , E j ) = 0 for all l and i > j and D b ( X ) = (cid:104) E , . . . , E n (cid:105) , i.e., D b ( X ) isgenerated from E , . . . , E n by shifts and extensions.(3) A full exceptional collection E , . . . , E n is called strong if additionally Ext l ( E i , E j ) =0 for all l (cid:54) = 0 and i < j . Theorem 4.3 ([Bon90]) . Let E , . . . , E n be a strong full exceptional collection on D b ( X ) , A := End( (cid:76) E i ) and mod − A be the category of right A -modules of finite rank. Then thefunctor R Hom( A, · ) : D b ( X ) → D b (mod − A ) is an exact equivalence. Under this identification the E i correspond to the indecomposableprojective A -modules. In particular, the category mod − A becomes the heart of a bounded t-structure on D b ( X ) with this identification. In the case of P this heart can be connected to some stabilityconditions. Theorem 4.4 ([MacE14]) . If α < / and β ∈ ( − / , then C := (cid:104)O ( − , T ( − , O [1] , O (1) (cid:105) = P α,β (( φ, φ + 1]) for some φ ∈ (0 , and the Bridgeland stability condition ( P α,β , Z α,β,ε ) for small enough ε > . Moreover, C is the category mod − A for some finite dimensional algebra A comingfrom an exceptional collection as in Theorem 4.3. The four objects generating C correspondto the simple representations.Proof of Proposition 4.1. By using the autoequivalence given by tensoring with O ( − n ) , wecan reduce to the case n = 0 . Then v = ± ( m, , .We start by proving the statement in Bridgeland stability for α = and β = 0 . ByTheorem 4.4 the object O [1] corresponds to a simple representation at this point. Thenany object E in the quiver category with ch( E ) = v corresponds to a representation of theform → → C m → . The statement follows in this case, since there is a unique suchrepresentation and it is semistable.Next, we will extend this to all α , β . Notice that Q α,β,K ( v ) = 0 . By Lemma 2.5 the object O is Bridgeland stable for all α , β . Let E ∈ A α,β ( P ) be Z α,β,s -semistable with ch( E ) = v . ByLemma 2.5, the class v spans an extremal ray of the cone C + = Z − α,β,s ( R ≥ v ) ∩ { Q α,β,K ≥ } .In particular, that means all its Jordan-Hölder factors are scalar multiples of v . If m = 1 ,then v is primitive in the lattice. Therefore, E is actually stable and then E is also stablefor α = and β = 0 , i.e. E is O or a shift of it. Assume m > . Since there are no stableobjects with class v at α = and β = 0 , Lemma 2.5 implies that E is strictly semistable.Therefore, the case m = 1 implies that all the Jordan-Hölder factors are O . he next step is to show semistability of O m in tilt stability. For this, we just need dealwith m = 1 . We have Q tilt ( O ) = 0 . By Lemma 2.5 we know that O is tilt stable everywhereor nowhere unless it is destabilized by an object supported in dimension . In that case β = 0 is a wall. However, that cannot happen since there are no morphism from or to O [1] for any skyscraper sheaf. Since v is primitive, semistability of O is equivalent to stability.For β = 0 and α (cid:29) we know that O is semistable due to Lemma 3.5.Now we will show that any tilt semistable object E with ch( E ) = v has to be O m for α = 1 , β = − . We have ν , − ( E ) = 0 . Therefore, E [1] is in the category A , − ( P ) .The Bridgeland slope is λ , − ,s ( E [1]) = ∞ independently of s . This means E is Bridgelandsemistable and by the previous argument E ∼ = O m .We will use Q tilt ( v ) = 0 and Lemma 2.5 similarly as in the Bridgeland stability case toextend it to all of tilt stability. We start with the case β < . Let E ∈ Coh β ( P ) be a tiltsemistable object with ch( E ) = v . By using Lemma 2.5, the class v spans an extremal ray ofthe cone C + = ( Z tilt α,β ) − ( R ≥ v ) ∩ { Q tilt ≥ } . In particular, that means all its stable factorshave Chern character (1 , , , e ) . The BMT inequality shows e ≤ . But since all the stablefactor add up to v this means e = 0 . Therefore, we reduced to the case m = 1 . In this caseLemma 2.5 does the job as before.If β = 0 , the situation is more involved, since skyscraper sheaves can be stable factors.All stable factor have Chern characters of the form ( − , , , e ) or (0 , , , f ) . In this case f ≥ . Let F be such a stable factor with Chern character ( − , , , e ) . By openness ofstability F is stable in a whole neighborhood that includes points with β < and β > .The BMT-inequality in both cases together implies e = 0 . But then f = 0 follows from thefact that Chern characters are additive. Again we reduced to the case m = 1 . By opennessof stability and the result for β < we are done with this case. The case β > can now behandled in the same way as β < by using Lemma 2.5 again. (cid:3) In the case of tilt stability there is an even stronger statement. If β > n , we do not needto fix ch to get the same conclusion. Proposition 4.5.
Let v = − ch ≤ ( O ( n ) ⊕ m ) for integers n, m with m > . Then O ( n ) ⊕ m [1] is the unique tilt semistable object E with ch ≤ ( E ) = v for any α > and β > n .Proof. The semistability of O ( n ) ⊕ m [1] has already been shown in Proposition 4.1. As in theprevious proof, we can use tensoring by O ( − n ) to reduce to the case n = 0 . This means v = ( m, , .Let E ∈ Coh β ( P ) be a tilt stable object for some α > and β > with ch( E ) =( − m, , , e ) . The BMT-inequality implies e ≤ . Since Q tilt ( E ) = 0 , we can use Lemma2.5 to get that E is tilt stable for all β > . If E is also stable for β = 0 , then using theBMT-inequality for β < implies e = 0 . Assume E becomes strictly semistable at β = 0 . ByLemma 2.5 the class v spans an extremal ray of the cone C + = ( Z tilt α,β ) − ( R ≥ v ) ∩ { Q tilt ≥ } .That means all stable factors must have Chern characters of the form ( − m (cid:48) , , , e (cid:48) ) for some ≤ m (cid:48) ≤ m . If m (cid:48) (cid:54) = 0 then using the BMT-inequality for both β < and β > implies e (cid:48) = 0 . If m (cid:48) = 0 , then e (cid:48) > . However, all the third Chern characters add up to thenon positive number e . This is only possible if e = 0 and no stable factor has m (cid:48) = 0 . ByProposition 4.1 this means E ∼ = O [1] m and since E is stable this is only possible if m = 1 .Let E ∈ Coh β ( P ) be a strictly tilt semistable object for some α > and β > with ch ≤ ( E ) = ( − m, , . Since Q tilt ( E ) = 0 , we can use Lemma 2.5 again to get that all stable actors F have ch ≤ ( F ) = ( − m (cid:48) , , for some m (cid:48) > . By the previous part of the proofthis means m (cid:48) = 1 and F ∼ = O [1] finishes the proof. (cid:3) We finish this section by recalling a basic characterization of ideal sheaves in P k . Lemma 4.6.
Let E ∈ Coh( P k ) be torsion free of rank one and ch ( E ) = 0 . Then either E ∼ = O or there is a subscheme Z ⊂ P k of codimension at least two such that E ∼ = I Z .Proof. We have the inclusion
E (cid:44) → E ∨∨ . The sheaf E ∨∨ is reflexive of rank one, i.e. locallyfree (see [Har80, Chapter 1] for basic properties of reflexive sheaves). Due to ch ( E ) = 0 and rk( E ) = 1 , we get E ∨∨ ∼ = O . Therefore, either E ∼ = O or there is a subscheme Z ⊂ P k such that E ∼ = I Z . If Z is not of codimension at least two, then c ( E ) (cid:54) = 0 . (cid:3) Examples in Tilt Stability
In examples, techniques from the last two sections can be used to determine walls in tiltstability. This is similar to work on surfaces as done in various articles ([ABCH13, BM14,CHW14, LZ13, MM13, Nue14, Woo13, YY14]). We will showcase this for some cases in P .For any v ∈ K num ( X ) we denote the set of tilt semistable objects with Chern Character ± v for some α > and β ∈ R by M tilt α,β ( v ) .5.1. Certain Sheaves.
Let m, n ∈ Z be integers with n < m and i, j ∈ N positive integers.We define a class as v = i ch( O P ( m )) − j ch( O P ( n )) . In this section we study walls forthis class v in tilt stability. Interesting examples of sheaves with this Chern character areideal sheaves of complete intersection of two surfaces of the same degree or ideal sheaves oftwisted cubics. In this generality we will determine the smallest wall in tilt stability on oneside of the vertical wall. Theorem 5.1.
A wall not containing any smaller wall for M tilt α,β ( v ) is given by the equation α + ( β − m + n ) = ( m − n ) . All semistable objects E at the wall are given by extensions ofthe form → O ( m ) ⊕ i → E → O ( n ) ⊕ j [1] → . Moreover, there are no tilt semistable objectsinside this semicircle.Proof. The semicircle defined by Q α,β, ( v ) = 0 coincides with the wall claimed to exist.Therefore, the BMT-inequality implies that no smaller semicircle can be a wall. Moreover,Proposition 4.1 shows that both O ( m ) ⊕ i and O ( n ) ⊕ j [1] are tilt semistable. The equation ν α,vβ ( O ( m )) = ν α,β ( O ( n )) is exactly the equation α + ( β − m + n ) = ( m − n ) . Therefore, weare left to prove the second assertion.Let F be a stable factor of E at the wall. By Lemma 2.7 and Remark 2.8 we get Q α,β, ( F ) =0 at the wall. Since F is stable, it is stable in a whole neighborhood around the wall. But Q α,β, ( F ) will be negative on one side of the wall unless Q α,β, ( F ) = 0 for all α , β . Takingthe limit α → ∞ implies Q tilt ( F ) = 0 .Assume that ch( F ) = ( r, c, d, e ) . Then Q tilt ( F ) = 0 implies c − rd = 0 . If r = 0 , then c = 0 . That cannot happen since the wall would be a vertical line and not a semicircle inthat situation. Thus, we can assume r (cid:54) = 0 . In particular, the equality d = c r holds. Theequation Q α,β, ( F ) = 0 for all ( α, β ) implies e = c r . In particular, the point α = m − n , β = m + n lies on the wall. Since F and E have the same slope at ( α , β ) , a straightforwardbut lengthy computation shows c = mr or c = nr . That means ch( F ) is a multiple of the hern character of either O ( m ) or O ( n ) . Since F was assumed to be stable, Proposition 4.1shows that F has to be one of those line bundles.Since the Chern characters of these two lines bundles are linearly independent we knowthat any decomposition of E into stable factors must contain i times O ( m ) and j times O ( n )[1] . The proof can be finished by the fact that Ext ( O ( m ) , O ( n )[1]) = 0 . (cid:3) In the case of the Chern character of an ideal sheaf of a curve there is also a bound on thebiggest wall.
Proposition 5.2.
Let v = (1 , , − d, e ) be the Chern character of an ideal sheaf of a curve ofdegree d . The biggest wall for M tilt α,β ( v ) and β < is contained inside the semicircle defined by ν α,β ( v ) = ν α,β ( O ( − . The biggest wall in the case β > is contained inside the semicircledefined by ν α,β ( v ) = ν α,β ( O (1)) .Proof. We start by showing there is no wall intersecting β = ± . Let E be tilt semistablefor β = ± and some α with ch( E ) = ± v . Then ch ± ( E ) = 1 holds. If E is strictly tiltsemistable, then there is an exact sequence → F → E → G → of tilt semistable objectswith the same slope. However, either ch ± ( F ) = 0 or ch ± ( G ) = 0 , a contradiction. Thenumerical wall ν α,β ( v ) = ν α,β ( O ( ± contains the point α = 0 , β = ± . The argument isfinished by the fact that numerical walls cannot intersect. (cid:3) Twisted Cubics.
While describing all the walls in general seems to be hard, we canhandle the situation in examples. Let C be a twisted cubic curve in P . We will computeall the walls in tilt stability for β < for the class ch( I C ) . There is a locally free resolution → O ( − ⊕ → O ( − ⊕ → I C → . This leads to ch β ( I C ) = (cid:18) , − β, β − , − β β + 5 (cid:19) . Figure 1.
Walls in tilt stability
Theorem 5.3.
There are two walls for M tilt α,β (1 , , − , for α > and β < . Moreover, thefollowing table lists pairs of tilt semistable objects whose extensions completely describe allstrictly semistable objects at each of the corresponding walls. Let V be a plane in P , P ∈ P and Q ∈ V . + ( β + ) = (cid:0) (cid:1) O ( − ⊕ , O ( − ⊕ α + ( β + ) = (cid:0) (cid:1) I P ( − , O V ( − O ( − , I Q/V ( − The hyperbola ν α,β (1 , , −
3) = 0 is given by the equation β − α = 6 . In order to prove the Theorem we need to put numerical restrictions on potentially desta-bilizing objects. We do this in a series of lemmas.
Lemma 5.4.
Let β ∈ Z and E ∈ Coh β ( P ) be tilt semistable. (1) If ch β ( E ) = (1 , , d, e ) then d − / ∈ Z ≤ . In the case d = − / , we get E ∼ = I L ( β +1) where L is a line plus / − e (possibly embedded) points in P . If d = 1 / , then E ∼ = I Z ( β + 1) for a zero dimensional subscheme Z ⊂ P of length / − e . (2) If ch β ( E ) = (0 , , d, e ) , then d + 1 / ∈ Z and E ∼ = I Z/V ( β + d + 1 / where Z is adimension zero subscheme of length /
24 + d / − e .Proof. Lemma 3.5 implies E to be either a torsion free sheaf or a pure sheaf supported indimension . By tensoring E with O ( − β ) we can reduce to the case β = 0 .In case (i) we have ch( E ⊗ O ( − , , d − / , / − d + e ) . Lemma 4.6 impliesthat E ⊗ O ( − is an ideal sheaf of a subscheme Z ⊂ P . This implies d − / ∈ Z ≤ . If d = 1 / , then Z is zero dimensional of length d − e − / / − e . In case d = − / , thesubscheme Z is a line plus points. The Chern Character of the ideal sheaf of a line is givenby (1 , , − , . Therefore, the number of points is d − e − / / − e .In case (ii) E is supported on a plane V . We will use Lemma 4.6 on V . In order to so, weneed to use the Grothendieck-Riemann-Roch Theorem to compute the Chern character of E on V . The Todd classes of P and P are given by td( P ) = (1 , , and td( P ) = (1 , , , .Therefore, we get i ∗ (cid:18) ch V ( E ) (cid:18) , , (cid:19)(cid:19) = (0 , , d, e ) (cid:18) , , , (cid:19) = (cid:18) , , d + 2 , d + e + 116 (cid:19) where i : V (cid:44) → P is the inclusion. Thus, we have ch V ( E ) = (1 , d + 1 / , d/ e + 1 / and d + 1 / is indeed an integer. Moreover, we can compute ch V ( E ⊗ O ( − d − / , , e − d −
124 ) . Using Lemma 4.6 on V concludes the proof. (cid:3) The next lemma determines the Chern characters of possibly destabilizing objects for β = − . emma 5.5. If an exact sequence → F → E → G → in Coh − ( P ) defines a wall for β = − with ch ≤ ( E ) = (1 , , − then up to interchanging F and G we have ch − ≤ ( F ) =(1 , , ) and ch − ≤ ( G ) = (0 , , − ) .Proof. The argument is completely independent of F being a quotient or a subobject. Wehave ch − ≤ ( E ) = (1 , , − .Let ch − ≤ ( F ) = ( r, c, d ) . By definition of Coh − ( P ) , we have ≤ c ≤ . If c = 0 , then ν α, − ( F ) = ∞ and this is in fact no wall for any α > . If c = 2 , then the same argumentfor the quotient G shows there is no wall. Therefore, c = 1 must hold. We can compute ν α, − ( E ) = − α , ν α, − ( F ) = d − rα . The wall is defined by ν α, − ( E ) = ν α, − ( F ) . This leads to α = 4 d + 22 r − > . (1)The next step is to rule out the cases r ≥ and r ≤ − . If r ≥ , then rk( G ) ≤ − .By exchanging the roles of F and G in the following argument, it is enough to deal withthe situation r ≤ − . In that case we use (1) and the Bogomolov inequality to get thecontradiction rd ≤ , d < − and r ≤ − .Therefore, we know r = 0 or r = 1 . By again interchanging the roles of F and G ifnecessary we only have to handle the case r = 1 . Equation (1) implies d > − . By Lemma5.4 we get d − / ∈ Z ≤ . Therefore, we are left with the case in the claim. (cid:3) Proof of Theorem 5.3.
Since we are only dealing with β < the structure theorem for walls intilt stability implies that all walls intersect the left branch of the hyperbola. In Theorem 5.1we already determined the smallest wall in much more generality. This semicircle intersectsthe β -axis at β = − and β = − . Therefore, all other walls intersecting this branch of thehyperbola have to intersect the ray β = − . By Lemma 5.5 there is at most one wall on thisray. It corresponds to the solution claimed to exist.Let → F → E → G → define a wall in Coh − ( P ) with ch( E ) = (1 , , − , .One can compute ch − ( E ) = (1 , , − , ) . Up to interchanging the roles of F and G wehave ch − ( F ) = (1 , , / , e ) and ch − ( G ) = (0 , , − / , / − e ) . By Lemma 5.4 we get F ∼ = I Z ( − where Z ∈ P is a zero dimensional sheaf of length / − e in P . In particular,the inequality e ≤ / holds. The same lemma also implies that G ∼ = I Z (cid:48) /V ( − where Z (cid:48) isa dimension zero subscheme of length e + 5 / in V . In particular, e ≥ − / . Therefore, thetwo cases e = and e = − remain and correspond exactly to the two sets of objects in theTheorem. (cid:3) Computing Walls Algorithmically.
The computational side in the previous exampleis rather straightforward. In this section we discuss how this problem can be solved by com-puter calculations. The proof of the following Lemma provides useful techniques for actuallydetermining walls. As before X is a smooth projective threefold, H an ample polarizationand for any α > , β ∈ R we have a very weak stability condition (Coh β ( X ) , Z tilt α,β ) . Lemma 5.6.
Let β ∈ Q and v be the Chern character of some object of D b ( X ) . Then thereare only finitely many walls in tilt stability for this fixed β with respect to v . roof. Any wall has to come from an exact sequence → F → E → G → in Coh β ( X ) .Let H · ch β ≤ ( E ) = ( R, C, D ) and H · ch β ≤ ( F ) = ( r, c, d ) . Notice that due to the fact that β ∈ Q the possible values of r , c and d are discrete in R . Therefore, it will be enough tobound those values to get finiteness.By the definition of Coh β ( X ) one has ≤ c ≤ C . If C = 0 , then c = 0 and we are dealingwith the unique vertical wall. Therefore, we may assume C (cid:54) = 0 . Let ∆ := C − RD . TheBogomolov inequality together with Lemma 2.7 implies ≤ c − rd ≤ ∆ . Therefore, we get c ≥ rd ≥ c − ∆2 . Since the possible values of r and d are discrete in R , there are finitely many possible valuesunless r = 0 or d = 0 . If R (cid:54) = 0 and D (cid:54) = 0 , then using the same type of inequality for G instead of E will finish the proof.Assume R = r = 0 . Then the equality ν α,β ( F ) = ν α,β ( E ) holds if and only if Cd − Dc = 0 .In particular, it is independent of ( α, β ) . Therefore, the sequence does not define a wall.Assume D = d = 0 . Then the equality ν α,β ( F ) = ν α,β ( E ) holds if and only if Rc − Cr = 0 .Again this cannot define a wall. (cid:3) Note that together with the structure theorem for walls in tilt stability this lemma impliesthat there is a biggest semicircle on each side of the vertical wall.The proof of the Lemma tells us how to algorithmically solve the problem of determiningall walls on a given vertical line. Assuming that β does not give the unique vertical wall,we have the following inequalities for any exact sequence → F → E → G → defining apotential wall. < H · ch β ( F ) < H · ch β ( E ) , < H · ch β ( G ) < H · ch β ( E ) ,Q tilt ( F, F ) ≥ ,Q tilt ( G, G ) ≥ ,Q tilt ( E, F ) ≥ . Moreover, we need H · ch( F ) and H · ch( G ) to be in the lattice spanned by Chern charactersof objects in D b ( X ) . Finally, the fact that the Chern classes of F and G are integers putsfurther restrictions on the possible values of the Chern characters. The code for a concreteimplementation in [SAGE] can be found at https://people.math.osu.edu/schmidt.707/research.html .We computed the previous example of twisted cubics with it and obtained the same wallsas above. Similar computations for the case of elliptic quartic curves will occur in a futurearticle joint with Patricio Gallardo and César Lozano Huerta.6. Connecting Bridgeland Stability and Tilt Stability
In the example of twisted cubics in the last section, we saw that the biggest wall was definedby two different exact sequences. Their differences were purely determined by differencesin codimension three. It is not very surprising that codimension three geometry cannotbe properly captured by tilt stability, since its definition does not include the third Cherncharacter. It seems difficult to precisely determine how the corresponding sets of stable bjects change at this complicated wall. We will show a general way to handle this issue byusing Bridgeland stability conditions. The problem stems from the fact that Lemma 3.10is in general incorrect in tilt stability. We will see how these multiple walls in tilt stabilityhave to separate in Bridgeland stability in the next section for some examples.Let v = ( v , v , v , v ) be the Chern character of an object in D b ( X ) . For any α > , β ∈ R and s > we denote the set of λ α,β,s -semistable objects with Chern character ± v by M α,β,s ( v ) .Analogous to our notation for twisted Chern characters we write v β = ( v β , v β , v β , v β ) := v · e − βH . We also write P v := { ( α, β ) ∈ R > × R : ν α,β ( v ) > } . The goal of this section is to prove the following theorem. Under some hypotheses, it roughlysays that on one side of the hyperbola { ν α,β ( v ) = 0 } all the chambers and wall crossingsof tilt stability occur in a potentially refined way in Bridgeland stability. In general, thedifference between these wall crossings and the corresponding situation in tilt stability iscomparable to the difference between slope stability and Gieseker stability. Using the theoryof polynomial stability conditions from [Bay09] one can define an analogue of that situationto make this precise. We will not do this as we are not aware of any interesting examples inwhich the difference matters. Theorem 6.1.
Let v be the Chern character of an object in D b ( X ) , α > , β ∈ R and s > such that ν α ,β ( v ) = 0 and H v β > . (1) Assume there is an actual wall in Bridgeland stability for v at ( α , β ) given by → F → E → G → . That means λ α ,β ,s ( F ) = λ α ,β ,s ( G ) and ch( E ) = ± v for semistable E, F, G ∈A α ,β ( X ) . Further assume there is a neighborhood U of ( α , β ) such that the samesequence also defines an actual wall in U ∩ P v , i.e. E, F, G remain semistable in U ∩ P v ∩ { λ α,β,s ( F ) = λ α,β,s ( G ) } . Then E [ − , F [ − , G [ − ∈ Coh β ( X ) are ν α ,β -semistable. In particular, there is an actual wall in tilt stability at ( α , β ) . (2) Assume that all ν α ,β -semistable objects are stable. Then there is a neighborhood U of ( α , β ) such that M α,β,s ( v ) = M tilt α,β ( v ) for all ( α, β ) ∈ U ∩ P v . Moreover, in this case all objects in M α,β,s ( v ) are λ α,β,s -stable. (3) Assume there is a wall in tilt stability intersecting ( α , β ) . If the set of tilt stableobjects is different on the two sides of the wall, then there is at least one actual wallin Bridgeland stability in P v that has ( α , β ) as a limiting point. (4) Assume there is an actual wall in tilt stability for v at ( α , β ) given by → F n → E → G m → such that F, G ∈ Coh β ( X ) are ν α ,β -stable objects, ch( E ) = ± v and ν α ,β ( F ) = ν α ,β ( G ) . Assume further that the set P v ∩ P ch( F ) ∩ P ch( G ) ∩ { λ α,β,s ( F ) = λ α,β,s ( G ) } is non empty. Then there is a neighborhood U of ( α , β ) such that F, G are λ α,β,s -stable for all ( α, β ) ∈ U ∩ P v ∩ { λ α,β,s ( F ) = λ α,β,s ( G ) } . In particular, there is anactual wall in Bridgeland stability restricted to U ∩ P v defined by the same sequence. efore we can prove this theorem, we need three preparatory lemmas. The followinglemma shows how to descend tilt stability on the hyperbola { ν α,β ( v ) = 0 } to Bridgelandstability on one side of the hyperbola. The main issue is that the hyperbola can potentiallybe a wall itself. Lemma 6.2.
Assume E ∈ Coh β ( X ) is a ν α ,β -stable object such that ν α ,β ( E ) = 0 andfix some s > . Then E [1] is λ α ,β ,s -semistable. Moreover, there is a neighborhood U of ( α , β ) such that E is λ α,β,s -stable for all ( α, β ) ∈ U ∩ P ch( E ) .Proof. By definition E [1] ∈ A α ,β ( X ) . Since λ α ,β ,s ( E [1]) = ∞ , the object E [1] is semistableat this point. Let E [1] (cid:16) G be a stable factor in a Jordan-Hölder filtration. There is aneighborhood U of ( α , β ) such that any destabilizing stable quotient of E in U ∩ P ch( E ) isof this form. This can be done since there is a locally finite wall and chamber structure suchthat the Harder-Narasimhan filtration of E is constant in each chamber. Let F be the kernelof this quotient, i.e. there is an exact sequence → F → E [1] → G → in A α ,β ( X ) . Bythe definition of A α ,β ( X ) we must have ν α ,β ( F ) = ν α ,β ( G ) = 0 . The long exact sequencewith respect to Coh β ( X ) leads to → H − β ( F ) → E → H − β ( G ) = G [ − → H β ( F ) → . Due to Lemma 3.13, the object H β ( F ) is supported in dimension . Since E is ν α ,β -stableand G (cid:54) = 0 , we must have H − β ( F ) = 0 . Therefore, F is a sheaf supported in dimension .But that is a contradiction to the fact that we have an exact sequence → F [ − → E → G [ − → in A α,β ( X ) for ( α, β ) ∈ U ∩ P V unless F = 0 . Therefore, E = G [ − is stable. (cid:3) At the hyperbola the Chern character of stable objects usually changes between v and − v .This comes hand in hand with objects leaving the heart while a shift of the object enters theheart. The next lemma deals with the question which shift is at which point in the category. Lemma 6.3.
Let v be the Chern character of an object in D b ( X ) , α > , β ∈ R and s > such that ν α ,β ( v ) = 0 and H v β > . Assume there is a path γ : [0 , → P v with γ (1) = ( α , β ) , γ ([0 , ⊂ P v , E ∈ A γ ( t ) ( X ) is λ γ ( t ) ,s -semistable for all t ∈ [0 , and ch( E ) = ± v . Then E [1] ∈ A α ,β ( X ) .Proof. The map [0 , → R , t (cid:55)→ φ γ ( t ) ,s ( E ) is continuous. Thus, there is m ∈ { , } such that E [ m ] ∈ A α ,β ( X ) is λ α ,β ,s -semistable. Assume m = 0 . Then Lemma 3.13 implies that H − β ( E ) is ν α ,β -semistable and H β ( E ) is a sheaf supported in dimension . This implies H ch β ( E ) ≤ . Therefore, H v β > implies ch( E ) = − v . This leads to (cid:61) Z γ ( t ) ,s ( E ) = −(cid:61) Z γ ( t ) ,s ( v ) < for all t ∈ [0 , in contradiction to E ∈ A α ,β ( X ) . (cid:3) The final lemma restricts the possibilities for semistable objects that leave the heart whilea shift enters the heart.
Lemma 6.4.
Let γ : [0 , → R > × R be a path, γ (1) = ( α , β ) , s > , E ∈ D b ( X ) be anobject such that E ∈ A γ ( t ) ( X ) is λ γ ( t ) ,s -semistable for all t ∈ [0 , and E [1] ∈ A α ,β ( X ) is λ α ,β ,s -semistable. Then E ∈ Coh β ( X ) is ν α ,β -semistable. roof. The continuity of [0 , → R , t (cid:55)→ φ γ ( t ) ,s ( E ) implies (cid:61) Z α ,β ,s ( E ) = 0 . Then Lemma3.13 implies that H − β ( E [1]) is ν α ,β -semistable and H β ( E [1]) is a sheaf supported in di-mension . In particular, there is a non trivial map E [1] → H β ( E [1]) unless H β ( E [1]) = 0 .Since E ∈ A γ ( t ) ( X ) for t ∈ [0 , one obtains φ γ ( t ) ,s ( E [1]) > φ γ ( t ) ,s ( H β ( E [1])) . The semi-stability of E implies H β ( E [1]) = 0 . (cid:3) Together with these three lemmas, we can prove the Theorem.
Proof of Theorem 6.1.
We start by proving (1). Since → F → E → G → also definesa wall in U ∩ P v we know there is m ∈ Z such that E [ m ] , F [ m ] , G [ m ] ∈ A α,β ( X ) for ( α, β ) ∈ U ∩ P v . By Lemma 6.3 this implies m = − and Lemma 6.4 shows E [ − , F [ − and G [ − are all ν α ,β -semistable.This defines a wall in tilt stability unless ν α,β ( F ) = ν α,β ( G ) for all ( α, β ) ∈ R > × R . Butthis is only possible if λ α,β,s ( F ) = λ α,β,s ( G ) is equivalent to ν α,β ( v ) = 0 .We continue by showing part (2). By assumption ( α , β ) does not lie on any wall for v intilt stability. Let U (cid:48) be a neighborhood of ( α , β ) that does not intersect any such wall. Inparticular, this means M tilt α,β ( v ) is constant on U (cid:48) . By part (i) any wall in Bridgeland stabilitythat intersects the hyperbola { ν α,β ( v ) = 0 } and stays an actual wall in some part of P v comes from a wall in tilt stability. Therefore, we can choose a neighborhood U (cid:48)(cid:48) of ( α , β ) such that there is no wall in Bridgeland stability for v in U (cid:48)(cid:48) ∩ P v . We define U := U (cid:48) ∩ U (cid:48)(cid:48) and choose ( α, β ) ∈ U .The inclusion M tilt α,β ( v ) ⊂ M α,β,s ( v ) is a restatement of Lemma 6.2. Let E ∈ M α,β,s ( v ) .There is m ∈ Z such that E [ m ] ∈ A α ,β is a λ α ,β ,s -semistable object. By Lemma 6.3 onegets m = 1 and Lemma 6.4 implies E ∈ Coh β ( X ) is tilt semistable, i.e. E ∈ M tilt α,β ( v ) .Part (3) follows from (2) while (4) is an immediate application of Lemma 6.2. (cid:3) Examples in Bridgeland Stability
In this section the techniques for connecting Bridgeland stability and tilt stability areapplied to the previous examples on P .7.1. Certain Sheaves.
Fix s > . Recall that m, n ∈ Z are integers with n < m and i, j ∈ N are positive integers. There is a class defined by v = i ch( O ( m )) − j ch( O ( n )) .We will show that there is a path close to one branch of the hyperbola {(cid:61) Z α,β,s ( v ) = 0 } where the last wall crossing described in Theorem 5.1 happens in Bridgeland stability. Thefirst moduli space after this wall turns out to be smooth and irreducible. Moreover, at thebeginning of the path stable objects are exactly slope stable sheaves with Chern character v . Theorem 7.1.
Assume that ( v , v , v ) is a primitive vector. There is a path γ : [0 , → R > × R ⊂ Stab( P ) that satisfies the following properties. (1) The last wall on γ is given by λ α,β,s ( O ( m )) = λ α,β,s ( O ( n )) . After the wall there areno semistable objects. Before the wall, the moduli space is smooth, irreducible andprojective. (2) At the beginning of the path the semistable objects are exactly slope stable coherentsheaves E with ch( E ) = v . Moreover, there are no strictly semistable objects. roof. By Theorem 5.1 there is a wall in tilt stability defined by the equation ν α,β ( O ( m )) = ν α,β ( O ( n )) . Moreover, there is no smaller wall. Since ( v , v , v ) is a primitive vector, anymoduli space of ν α,β -semistable objects for v , such that ( α, β ) does not lie on a wall, consistssolely of tilt stable objects. Let Y ⊂ {(cid:61) Z α,β,s ( v ) = 0 } be the branch of the hyperbola thatintersects this wall. Due to Theorem 6.1 we can find a path γ : [0 , → R > × R (cid:44) → Stab( P ) close enough to Y such that all moduli spaces of tilt stable objects that occur on Y outsideof any wall are moduli spaces of Bridgeland stable objects along γ . Moreover, we canassume that γ intersects no wall twice and the last wall crossing is given by λ α,β,s ( O ( m )) = λ α,β,s ( O ( n )) .Part (2) can be proven as follows. By the choice of γ , we have M tilt γ (0) ( v ) = M γ (0) ,s ( v ) . Intilt stability γ (0) is above the largest wall. Therefore, Lemma 3.5 and Lemma 3.6 imply that M tilt γ (0) ( v ) consists of slope stable sheaves E with ch( E ) = v .We will finish the proof of (1) by showing that the first moduli space is a moduli spaceof representations on a Kronecker quiver. Let t ∈ (0 , be such that M γ ( t ) ,s ( v ) is thelast moduli space on γ before the empty space. Let Q be the Kronecker quiver with N = dim Hom( O ( n ) , O ( m )) arrows. ◦ ◦ ... N (cid:52) (cid:52) (cid:42) (cid:42) For any representation V of Q we denote the dimension vector by dim( V ) . If θ : Z ⊕ Z → Z is a homomorphism with θ ( j, i ) = 0 we say that a representation V of Q with dim( V ) = ( j, i ) is θ -(semi)stable if for any subrepresentation W (cid:44) → V the inequality θ ( W ) > ( ≥ )0 holds.Due to [Kin94] there is a projective coarse moduli space K θ that represents stable complexrepresentations with dimension vector ( j, i ) . If there are no strictly semistable representation,then K θ is a fine moduli space. Since we know that the first moduli space consists solelyof extensions of O ( n ) ⊕ j [1] and O ( m ) ⊕ i , we can find θ such that θ -stability and Bridgelandstability at γ ( t ) match. More precisely, there is a bijection between Bridgeland stable objectsat γ ( t ) with Chern character v and θ -stable complex representations with dimension vector ( j, i ) . We denote this specific moduli space of quiver representations by K . Since the quiverhas no relation and i , j have to be coprime, we get that K is a smooth projective variety.We want to construct an isomorphism between K and the moduli space M γ ( t ) ,s ( v ) ofBridgeland stable complexes with Chern character v . In order to do so, we need to makethe above bijection more precise. Let Hom( O ( n ) , O ( m )) = (cid:76) l C ϕ l . There is a functor F : Rep( Q ) → D b ( P ) that sends a representation f l : C j → C i to the two term complex O ( n ) ⊕ j → O ( m ) ⊕ i with map ( s , . . . , s j ) (cid:55)→ (cid:80) l f l ( ϕ l ( s ) , . . . , ϕ l ( s j )) . This functor inducesthe bijection between stable objects mentioned above.Let S be a scheme over C . A representation of Q over S is given by N maps f , . . . , f N : V → W for locally free sheaves V, W ∈ Coh( S ) . The functor above can be generalized tothe relative setting as F S : Rep S ( Q ) → D b ( P × S ) sending f l : V → W to the two termcomplex V (cid:2) O ( n ) → W (cid:2) O ( m ) where the map is given by (cid:80) v ⊗ s (cid:55)→ (cid:80) (cid:80) l f l ( v ) ⊗ ϕ l ( s ) .If E is a family of Bridgeland stable objects at γ ( t ) over S , then we get F ( E s ) = F S ( E ) s for any s ∈ S . That induces a bijective morphism from K to M γ ( t ) ,s ( v ) . We want to showthat this morphism is in fact an isomorphism. In order to so, we will first need to provesmoothness. e have dim M γ ( t ) ,s ( v ) = dim K = jiN − i − j + 1 . For any E ∈ M γ ( t ) ,s ( v ) the Zariskitangent space at E is given by Ext ( E, E ) by standard deformation theory arguments (see[Ina02, Lie06]). We have an exact triangle(2) O ( m ) ⊕ i → E → O ( n ) ⊕ j [1] . Since E is stable we have Hom( O ( n )[1] , E ) = 0 . Applying Hom( O ( n ) , · ) to (2) leads to Hom( O ( n ) , E ) = C Ni − j . The same way we get Hom( O ( m ) , E ) = C i and Ext ( O ( m ) , E ) = 0 .Since E is stable, the equation Hom(
E, E ) = C holds. Applying Hom( · , E ) to (2) leads tothe following long exact sequence. → C → C i → C Nij − j → Ext ( E, E ) → . That means dim Ext ( E, E ) =
N ij − j − i + 1 = dim M γ ( t ) ,s ( v ) , i.e. M γ ( t ) ,s ( v ) is smooth.Since there are no strictly semistable objects, we can use the main result of [PT15] toinfer that M γ ( t ) ,s ( v ) is a smooth proper algebraic space of finite type over C . According to[Knu71, Page 23] there is a fully faithful functor from smooth proper algebraic spaces offinite type over C to complex manifolds. Since any bijective holomorphic map between twocomplex manifolds has a holomorphic inverse we are done. (cid:3) Twisted Cubics.
In the example of twisted cubic curves, we described all walls in tiltstability for β < in Theorem 5.3. We will translate this result into Bridgeland stability viaTheorem 6.1. Figure 2.
Walls in Bridgeland stability
Theorem 7.2.
There is a path γ : [0 , → R > × R ⊂ Stab( P ) that crosses the followingwalls for v = (1 , , − , in the following order. The walls are defined by the two givenobjects having the same slope. Moreover, all strictly semistable objects at each of the wallsare extensions of those two objects. Let V be a plane in P , P ∈ P and Q ∈ V . (1) O ( − , I Q/V ( − (2) I P ( − , O V ( − (3) O ( − ⊕ , O ( − ⊕ The chambers separated by those walls in reverse order have the following moduli spaces. (1)
The empty space M = ∅ . (2) A smooth projective variety M . A space with two components M ∪ M (cid:48) . The space M is a blow up of M in theincidence variety parametrizing a point in a plane in P . The second component M (cid:48) is a P -bundle over the smooth variety P × ( P ) ∨ parametrizing pairs ( I P ( − , O V ( − . The two components intersect transversally in the exceptional locus of theblow up. (4) The Hilbert scheme of curves C with ch( I C ) = (1 , , − , . It is given as M ∪ M (cid:48) where M (cid:48) is a blow up of M (cid:48) in the smooth locus parametrizing objects I Q/V ( − .Proof. Let γ be the path that exists due to Theorem 7.1. The fact that all the walls on thispath occur in this form is a direct consequence of Theorem 6.1 and Theorem 5.3.By Theorem 7.1 we know that M = ∅ , that M is smooth, projective and irreducible andthat the Hilbert scheme occurs at the beginning of the path. The main result in [PS85] isthat this Hilbert scheme has exactly two smooth irreducible components of dimension and that intersect transversally in a locus of dimension . The -dimensional component M contains the space of twisted cubics as an open subset. The -dimensional component M (cid:48) parametrizes plane cubic curves with a potentially but no necessarily embedded point.Moreover, the intersection parametrizes plane singular cubic curves with a spatial embeddedpoint at a singularity. In particular, those curves are not scheme theoretically contained ina plane.Strictly semistable objects at the biggest wall are given by extensions of O ( − , I Q/V ( − .For an ideal sheaf of a curve this can only mean that there is an exact sequence → O ( − → I C → I Q/V ( − . This can only exist if C ⊂ V scheme theoretically. Therefore, the first wall does only modifythe second component. The moduli space of objects I Q/V ( − is the incidence variety ofpoints in the plane inside P × ( P ) ∨ . In particular, it is smooth and of dimension . Astraightforward computation shows Ext ( O ( − , I Q/V ( − C . That means at the firstwall the irreducible locus of extensions Ext ( I Q/V ( − , O ( − C is contracted onto asmooth locus. Moreover, for each sheaf I Q/V ( − the fiber is given by P . This means thecontracted locus is a divisor. By a classical result of Moishezon [Moi67] any proper birationalmorphism f : X → Y between smooth projective varieties such that the contracted locus E is irreducible and the image f ( E ) is smooth is the blow up of Y in f ( E ) . Therefore, to seethat M (cid:48) is the blow up of M (cid:48) we need to show that M (cid:48) is smooth.At the second wall strictly semistable objects are given by extensions of I P ( − and O V ( − . One computes Ext ( I P ( − , O V ( − C for P ∈ V , Ext ( I P ( − , O V ( − for P / ∈ V and Ext ( O V ( − , I P ( − C . The objects I P ( − and O V ( − vary in P respectively ( P ) ∨ that are both fine moduli spaces. Therefore, the component M (cid:48) is a P -bundle over the moduli space of pairs ( O V ( − , I P ( − , i.e. P × ( P ) ∨ . This means M (cid:48) is smooth and projective.We are left to show that M is the blow up of M . We already know that M is the smoothcomponent of the Hilbert scheme containing twisted cubic curves. Moreover, M is smooth byTheorem 7.1. We want to apply the above result of Moishezon again. The exceptional locusof the map from M to M is given by the intersection of the two components in the Hilbertscheme. By [PS85] this is an irreducible divisor in M . Due to Ext ( I P ( − , O V ( − C for P ∈ V the image is as predicted. (cid:3) e believe it should be possible to prove the previous result without referring to [PS85].All the above arguments already show that the result holds set theoretically and one shouldbe able to explicitly construct the universal family by glueing it for the two components. 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