Duality, Bridgeland wall-crossing and flips of secant varieties
aa r X i v : . [ m a t h . AG ] A ug DUALITY, BRIDGELAND WALL-CROSSING AND FLIPS OF SECANTVARIETIES
CRISTIAN MARTINEZ
Abstract.
Let v d ( P ) ⊂ |O P ( d ) | denote the d -uple Veronese surface. After studying some generalaspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describea sequence of flips of the secant varieties of v d ( P ) by embedding the blow-up bl v d ( P ) |O P ( d ) | intoa suitable moduli space of Bridgeland semistable objects on P . Contents
1. Introduction 12. Preliminaries 33. Wall and Chamber Structure 74. Duality 115. Bridgeland Walls for 1-Dimensional Plane Sheaves 146. The Embedded Problem: Flips of Secant Varieties. 18Acknowledgments 31References 311.
Introduction
Stability conditions on triangulated categories were introduced by Bridgeland [10], who alsoconstructed the first family of nontrivial examples for K S, H ). By analyzing how these moduli spaces change as the stability condition varies, the authorsare able to describe a sequence of birational transformations of the blow-up of the complete linearseries | H | along S , flipping the secant varieties of S .One of the features of these stability conditions is their “well behaved” wall-crossing. Thisphenomenon was studied in [2] for the topological type v = (1 , , − n ) on P , where it was indicatedthat varying the family of stability conditions introduced in [1] corresponds to running a directedMinimal Model Program (MMP) on the Hilbert scheme of n points (regarded as the moduli spaceof Gieseker semistable sheaves of type v ). The same statement was proven in [9] for any primitivetopological type. The correspondence between wall-crossing for Bridgeland stability conditionsand MMP wall-crossing has been extensively studied by Coskun, Huizenga, and Wolf [13, 14] tocompute the nef and effective cones of moduli spaces of Gieseker semistable plane sheaves.In [24] and [25], Maican constructs cohomological stratifications for the moduli spaces N P ( r, χ )of Gieseker semistable sheaves on P with Hilbert polynomial rm + χ ( r = 5 , N P ( r, χ ), as it was done in [9] forthe moduli spaces N (4 ,
2) and N (5 , Mathematics Subject Classification.
Key words and phrases.
Stability conditions; wall-crossing; moduli of torsion sheaves; secant varieties. ohomological strata and the Bridgeland walls. It was shown in [12] that for the case of N (6 , χ = 0 we can identify all rank-1 walls even when Maican-type stratificationsare unknown. In this case, by restricting the Bridgeland wall-crossing on a suitable subvariety ofa model of N ( d,
0) ( d odd), and following the spirit of [1], we construct a sequence of flips forthe blow-up of the linear series |O ( d − | along the Veronese surface, with the first of these flipscoinciding with the one constructed by Vermeire in [30]. Theorem 33.
Let d ≥ be an integer and let ν d − : P → P ( H ( O ( d − ∨ = P N be ( d − -upleembedding. There exists a sequence of flips bl ν d − ( P ) P N M M · · · M k N ⊃ P N M ′ M ′ · · · (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ (cid:15) (cid:15) ✤✤✤✤✤✤✤✤ / / f / / f (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ where k = ( d − / for d odd, and k = ⌊ ( d − / ⌋ for d even, the exceptional locus of f i is thestrict transform of Sec i ( ν d − ( P )) , and N is the first birational model appearing when running theMMP for N ( d, or N ( d, d/ depending on whether d is odd or even respectively. To obtain this sequence of flips we need to understand the flipping locus for the flips appearingwhen running the MMP for the Gieseker moduli spaces N ( d,
0) for d odd, and N ( d, d/
2) for d even.To do this, we will need a generalization of a result of Maican.In [23], Maican proves that the map F 7→ E xt n − ( F , ω P n ) induces an isomorphism between themoduli spaces N P n ( r, χ ) and N P n ( r, − χ ) of Gieseker semistable sheaves with Hilbert polynomials P = rm + χ and P D = rm − χ respectively. The moduli spaces N X ( r, χ ) were constructed bySimpson [27] for any smooth projective surface via invariant theory and they were proven to beprojective, so one could ask whether Maican’s result extends to any surface. This was proven bySacc`a in her thesis [26]. We recover Sacc`a’s result as a corollary of a more general statement: Theorem 20.
The functor R H om ( · , ω X )[1] induces an isomorphism between the Bridgeland mod-uli spaces M D,tH ( v ) and M − D + K X ,tH ( v D ) provided these moduli spaces exist and Z D,tH ( v ) belongsto the open upper half plane. Notation.
Other than specified we will use the following standard notation: • R ( z ), I ( z ) denote the real and imaginary parts of the complex number z . • D b ( X ) is the bounded derived category of coherent sheaves on X . • K ( X ) denotes the Grothendieck group of the triangulated category D b ( X ). • For
E, F ∈ K ( X ) we define the operator χ ( E, F ) = X i ∈ Z ( − i dim Ext i ( E, F ) . When E = O , we denote χ ( E, F ) by χ ( F ) and refer to it as the Euler characteristic of F . • When X is a smooth projective complex surface, the Hizerburch-Riemann-Roch Theoremcan be written as χ ( F ) = ch ( F ) − ch ( F ) K X ch ( F ) χ ( O ) , here K X is the canonical divisor of X . • Num( X ) denotes the group of cycles A ( X ) up to numerical equivalence, and NS( X ) =Num ( X ) the Ner´on-Severi group of divisors up to numerical equivalence. Also, Num( X ) Q and Num( X ) R denote the tensor products Num( X ) ⊗ Q and Num( X ) ⊗ R , respectively. • We use H i ( · ) to denote the cohomology sheaves of an object in the derived category and H i ( · ) for the cohomology groups of a sheaf. • We use n F to denote the direct sum F ⊕ n . • For a smooth projective surface X , the topological type v ∈ Num( X ) Q of an object E ∈ D b ( X ) is its Chern character vector. • M H ( v ) denotes the moduli space of Gieseker semistable sheaves of topological type v withrespect to the polarization H ∈ Pic( X ). • For a stability condition σ ∈ Stab( X ), M σ ( v ) denotes the moduli space parametrizing S -equivalence classes of σ -semistable objects of topological type v (if such space exists). Forthe stability conditions σ β,ω , M σ β,ω ( v ) is denoted simply by M β,ω ( v ), and by M s,t ( v ) when β = sH and ω = tH . • We refer to an object F fitting into an exact sequence A ֒ → F ։ B as an extension.2. Preliminaries
We assume familiarity with the concept of stability conditions introduced by Bridgeland [10]. Werecall here the relevant theorems and definitions but for a more detailed presentation the unfamiliarreader is encouraged to consult Bridgeland’s original papers [10, 11], or the introduction to the topicby Huybrechts [18].Let X be a smooth projective variety. Definition 1.
A pre-stability condition on X is a pair σ = ( Z, A ) consisting of a linear function Z : K ( X ) → C called the charge and the heart A of a bounded t-structure on D b ( X ) , such that:(a) I ( Z ( E )) ≥ for all E ∈ A and(b) If I ( Z ( E )) = 0 and E = 0 then R ( Z ( E )) < . For every pre-stability condition one can define a slope function µ σ = − R ( Z ) I ( Z )which gives us a notion of (semi)stability: an object E ∈ A is said to be σ -(semi)stable if for anyinclusion A ֒ → E of objects in A one has µ σ ( A )( ≤ ) < µ σ ( E ) . Definition 2.
A pre-stability condition σ = ( Z, A ) is a stability condition if it has the Harder-Narasimhan property: • Every nonzero object E ∈ A admits a finite filtration in A ⊂ E ⊂ E ⊂ · · · ⊂ E n = E uniquely determined by the property that each quotient F i := E i /E i − is σ -semistable and µ σ ( F ) > µ σ ( F ) > · · · > µ σ ( F n − ) . Example . If X = C is a smooth projective curve then ordinary degree and rank of coherentsheaves give a stability condition on A = D b (Coh( C )): Z ( F ) = − deg( F ) + √− F ) . owever, when X is a surface this is not the case. One can still define a Mumford slope (withrespect to some polarization H ): µ H ( E ) = c ( E ) · H rk(E) , but this does not come from any stability condition on Coh( X ) since c ( C p ) = rk( C p ) = 0. Never-theless, it is true that every coherent sheaf E has a filtration E ⊂ · · · ⊂ E n = E such that E is the torsion subsheaf of E and for every i >
0, the factors E i /E i − are semistableof decreasing slopes. (cid:3) Let σ = ( Z, A ) be a stability condition on X . For any nonzero object E ∈ A one can write Z ( E ) = | Z ( E ) | e π √− φ for a unique φ ∈ (0 , E has phase φ . For every φ ∈ (0 ,
1] wedenote by P σ ( φ ) the subcategory consisting of σ -semistable objects of phase φ . Inductively, onecan define P σ ( φ + 1) := P σ ( φ )[1]. For a bounded interval I ⊂ R we denote P σ ( I ) the subcategoryextension-generated by σ -semistable objects of phase in the interval I . For instance, P σ (0 ,
1] = A .One can define semistability in terms of phase just by declaring an object E to be semistableif every subobject has smaller phase. This is equivalent to the definition using slopes since for anobject E ∈ A of phase φ one has µ σ ( E ) = − cot( πφ ) . An easy but important consequence of the definition of stability is
Proposition 3 (Schur’s lemma) . Let σ = ( Z, A ) be a stability condition.(a) If E is σ -stable then Hom(
E, E ) = C .(b) If A, B are different σ -stable objects of the same phase then Hom(
A, B ) = 0 .(c) If A ∈ P σ ( φ ) , B ∈ P σ ( φ ) with φ > φ then Hom(
A, B ) = 0 . Let E ∈ P σ ( φ ). A finite Jordan-H¨older filtration of E is a chain0 = E ⊂ E ⊂ · · · ⊂ E n = E such that the factors E i /E i − are stable of phase φ . In general, finite Jordan-H¨older filtrations donot always exist, and even when they exist, they are not necessarily unique. However, the stablefactors are always unique up to a permutation. Definition 4.
A stability condition is called locally finite if there is some δ > such that eachquasi-abelian category P σ ( φ − δ, φ + δ ) is of finite length. For a locally finite stability condition thecategories P σ ( φ ) have finite length. In particular, every semistable object has a finite Jordan-H¨olderfiltration. Definition 5.
Let σ be a locally finite stability condition. Two objects A, B ∈ P σ ( φ ) are called S -equivalent with respect to σ if they have isomorphic Jordan-H¨older σ -stable factors (up to areordering). Definition 6.
We say that a pre-stability condition σ = ( Z, A ) is numerical if Z : K ( X ) → C factors through the Chern character map ch : K ( X ) → Num ( X ) Q . We continue denoting by Z thecorresponding homomorphism Num ( X ) Q → C . Definition 7.
Fix a norm k k on Num( X ) R . A numerical pre-stability condition σ = ( Z, A ) issaid to satisfy the support property if there exists a constant C > such that for all σ -semistableobjects = E ∈ D b ( X ) , we have k ch ( E ) k ≤ C | Z ( E ) | . t follows from [6, Proposition B4] and [11, Lemma 4.5] that numerical stability conditionssatisfying the support property are locally finite. We denote by Stab( X ) the set of numericalstability conditions satisfying the support property. We can now state Bridgeland’s deformationresult: Theorem 8 ([10, Theorem 1.2]) . There is a natural topology on
Stab( X ) such that the forgetfulmap Z : Stab( X ) → Hom(Num( X ) , C ) , sending σ = ( Z, A ) to Z , is a local homeomorphism. In particular, every connected component of Stab( X ) is a complex manifold. The following proposition due to Bridgeland expresses an important property of Stab( X ): Proposition 9 ([11, Proposition 9.3]) . Let V ⊂ Stab( X ) be a connected component, let K ⊂ V bea compact subset, and let v ∈ Num( X ) Q . Then there is a finite collection { W γ : γ ∈ F } of (real)codimension 1 submanifolds of V (not necessarily closed), such that every connected component C ⊂ K \ [ γ ∈ F W γ has the property that if an object E ∈ D b ( X ) with ch ( E ) = v is semistable for some stabilitycondition in C , then it is semistable for all stability conditions in C . Definition 10.
For a given topological type v ∈ Num( X ) Q , we refer to the submanifolds W γ ofProposition 9 as walls of type v , and to every connected component C as chamber of type v . Proposition 11 ([8, Proposition 2.2.2]) . Given E ∈ D b ( X ) , the set of σ ∈ Stab( X ) for which E is σ -stable is an open subset of Stab( X ) . Further, on the open part of every chamber in the walland chamber decomposition of Stab( X ) , the Harder-Narasimhan filtration of E is constant.Remark . There is a particular instance of Proposition 11 that we will encounter throughoutthe paper. Assume that { σ t = ( Z t , A ) } t ∈ T is a continuous 1-parameter family of stability conditionswith a fixed heart A . Assume further that there is an object E ∈ A that is σ t -stable and σ t -unstable for some t and t . Then the path { σ t } t ∈ T can not be fully contained in a chamber oftype ch ( E ), and there is a value t between t and t that is the intersection of a wall of type ch ( E )with the family { σ t } t ∈ T . By Proposition 11 there is a subobject A ֒ → E in A (the first Harder-Narasimhan factor of E in the nearby chamber where E is unstable) such that µ σ t ( A ) = µ σ t ( E ). Definition 12. [4] . Let S be a scheme of finite type over C . A family of objects in A parametrizedby S is an object E ∈ D b ( X × S ) such that for every closed point s ∈ S we have Li ∗ s ( E ) ∈ A . From now on, X is a smooth projective complex surface. We start by reviewing the examplesof stability conditions constructed by Bridgeland [11] for the K X .Fix a class ω ∈ Amp( X ). One defines, for every s ∈ R , the full subcategories of Coh( X ): • Q s = h E ∈ Coh( X ) : E is µ ω -semistable and µ ω ( E ) > s i , • F s = h E ∈ Coh( X ) : E is µ ω -semistable and µ ω ( E ) ≤ s i ,where for a collection of objects B ⊂ D b ( X ), hBi denotes the smallest subcategory containing B and that is closed under extensions. We follow the convention that a torsion sheaf is semistable ofslope + ∞ .The subcategories Q s , F s are full and ( Q s , F s ) is a torsion pair, i.e., • Hom(
Q, F ) = 0 for all Q ∈ Q s , F ∈ F s . Every coherent sheaf E fits into an exact sequence0 → Q → E → F → Q ∈ Q s , F ∈ F s . This short exact sequence is unique up to isomorphisms ofextensions.By general theory of torsion pairs [16, Proposition 2.1], we know that the extension closure hQ s , F s [1] i is the heart of a bounded t -structure. More precisely, it is the full subcategory A s = { E ∈ D b ( X ) : H − ( E ) ∈ F s , H ( E ) ∈ Q s and H i ( E ) = 0 , i = − , } . Theorem 13 ([11, Proposition 7.1], [1, Corollary 2.1]) . Let β, tω ∈ N ( X ) Q with ω ample. Then, Z β,tω ( E ) = − Z X e − β −√− tω ch ( E ) is the charge of a locally finite stability condition on A β · ω . We denote this stability condition by σ β,tω .Remark . It follows directly from the Hodge index theorem and the Bogomolov-Gieseker in-equality that σ β,tω is a pre-stability condition for every β, ω ∈ NS( X ) R and t >
0. However,producing Harder-Narasimhan filtrations for arbitrary classes is more complicated. If for every β, tω ∈ NS( X ) Q the stability conditions σ β,tω satisfy the support property, then Bridgeland’s defor-mation result guarantees the existence of a lift of Z β,tω ∈ Hom(Num( X ) , C ) to Stab( X ). Becausefor every β, tω ∈ NS( X ) Q the skyscraper sheaves C x are σ β,tω -stable, then for arbitrary real classes β, ω ∈ NS( X ) R and t >
0, there should be a lift of Z β,tω satisfying the same property (within thesame chamber of type ch ( C x ) of nearby rational classes). Then [11, Lemma 10.1(d)] shows thatthe only possible lift of Z β,tω for which all the skyscraper sheaves C x are stable is σ β,tω . The proofthat on an arbitrary smooth projective surface X , the stability conditions σ β,tω satisfy the supportproperty is due to Toda [29, Proposition 3.13].Assume that there is a short exact sequence 0 → A → E → B → A β · ω with ch ( A ) = ch ( E ) = ch ( B ) = 0, then it follows from the definition of Z β,tω that µ σ β,tω ( A ) = µ σ β,tω ( E ) if andonly if(2.1) χ ( A ) ch ( A ) · ω + ch ( A ) · ( K X − β ) ch ( A ) · ω = χ ( E ) ch ( E ) · ω + ch ( E ) · ( K X − β ) ch ( E ) · ω . Because this condition is independent of t , then any such E is not σ β,tω -stable for any value of t . However, it is possible for E to be σ β,tω -semistable for some values of t and σ β,tω -unstable forothers. This motivates the following definition: Definition 14.
An object E ∈ A β · ω of Chern character vector ch ( E ) = (0 , ch , ch ) is said to be σ β,tω -pseudo-stable if E is σ β,tω -semistable, and for any subobject A ֒ → E in A β · ω we have µ σ β,tω ( A ) = µ σ β,tω ( E ) ⇒ ch ( A ) = 0 . Stability conditions on the projective plane.
We now concentrate in the case X = P .In this case (Picard number 1), one can think of ch ( E ) as a vector with numerical entries. Choosing ω = H , the hyperplane class, and β = sH , the central charge takes the form Z s,t ( ch , ch , ch ) = ( − ch + ch s − ch s − t )) + √− ch − ch s ) t. One of the most important results in [2] is the following:
Theorem 15 ([2, Proposition 8.1]) . There are projective coarse moduli spaces M s,t ( v ) classifying S -equivalence classes of families of σ s,t -semistable objects in A s of topological type v . he idea is to identify σ s,t -stability with quiver stability. Let k ∈ Z and consider the extensionclosure A ( k ) = hO ( k − , O ( k − , O ( k ) i . An element of A ( k ) is a complex C n ⊗ O ( k − → C n ⊗ O ( k − → C n ⊗ O ( k )with dimension vector n = ( n , n , n ). Let a be a vector orthogonal to n . An object of dimensionvector n is said to be quiver (semi)stable with respect to a if for any subcomplex in A ( k ) ofdimension vector n ′ one has n ′ · a ( ≥ ) > a have a construction via GIT given in [19]. The change from Chern classes to dimension vectorsis given by the matrix B k := k ( k − − (2 k − k ( k − − (2 k −
2) 2 ( k − k − − (2 k − . Fix a Chern character v so that B k v is a vector of nonnegative entries. It is shown in [2, Proposition7.3] that for every ( s, t ) in the region ( s − ( k − + t < , there exists a choice of an orthogonal vector a s,t such that the moduli spaces of σ s,t -semistableobjects of Chern character v are isomorphic to moduli spaces of semistable objects in A ( k ) ofdimension vector B k v with respect to a s,t . The key point is to observe that each wall and chamberof type v in the ( s, t )-plane intersects at least one of the regions above and so, by Proposition 9,the moduli spaces M s,t ( v ) are projective and may be constructed by GIT.Notice that in general the Gieseker moduli spaces of 1-dimensional plane sheaves are not smooth(although its singular locus has high codimension). Because torsion sheaves are objects of A s thenany short exact sequence of sheaves 0 → A → E → B →
0, with E a torsion sheaf, would alsobe exact in A s . Moreover, it follows from equation (2.1) that a sheaf destabilizing E with respectto Gieseker stability would also destabilize E with respect to Bridgeland stability. Then on anychamber of type v = (0 , ch , ch ) in the ( s, t )-plane one has σ s,t -semistable = σ s,t -pseudo-stable forobjects of topological type v . 3. Wall and Chamber Structure
The results in this section seem to be known to the experts but we decided to include here someproofs for the sake of completeness.In [2] for the case of the stability conditions σ s,t on P , the authors describe what new objectsbecome stable after crossing a wall. Similar reasoning can be applied to study the wall-crossingfor the 1-dimensional family of stability conditions { σ D,tH } t> for fixed D, H with H ample, onan arbitrary smooth projective surface. Following Remark 11.1, assume that a wall at t = t isproduced by a destabilizing sequence 0 → A → E + → B → A and B are stable at the wall and E + is stable above the wall( t > t ). Then the destabilizing sequence is a Jordan-H¨older filtration for the semistable object E + at t . Crossing the wall will produce semistable objects that are S -equivalent to E + at the wall,i.e., the new objects have A and B as stable factors and so they are extensions of the form0 → B → E − → A → . But even more is true, roposition 16. Assume that µ D,t H ( A ) = µ D,t H ( B ) for some objects A, B ∈ A DH and thatthere is ǫ > such that A and B are σ D,tH -stable with µ D,tH ( A ) < µ D,tH ( B ) for t ≤ t < t + ǫ . IfExt ( B, A ) = 0 in A DH then there exists δ > such that every non-split extension → A → E → B → is σ D,tH -stable (or σ D,tH -pseudo-stable when ch ( E ) = 0 ) for all t < t < t + δ .Proof. Assume for the moment that ch ( E ) = 0. Let 0 < δ ≤ ǫ such that there are no walls for E between t and t + δ (this is possible because the walls are locally finite). It is enough to prove thatthere is no stable subobject E ′ ֒ → E , σ D,tH -destabilizing E for t < t < t + δ . If there were such E ′ then at the wall t , E ′ is semistable and µ D,t H ( E ′ ) = µ D,t H ( E ), otherwise it would destabilize E at t since in any case µ D,t H ( E ′ ) ≥ µ D,t H ( E ).Because B is stable at the wall t and µ D,t H ( E ′ ) = µ D,t H ( B ), then the composition E ′ → E → B is either surjective or zero. If it were the zero map, then we would have an inclusion E ′ ֒ → A inwhich case µ ( E ′ ) < µ ( A ) < µ ( E ) above the wall t . Let K be the kernel of E ′ → B , then there isan inclusion K ֒ → A . Since the slopes of K and A are equal at t , then either K = 0 in which casethe sequence A → E → B splits, or K = A and therefore E ′ = E .When ch ( E ) = 0 there is the possibility that E has a subobject E ′ ֒ → E with µ D,tH ( E ′ ) = µ D,tH ( E ) for all t >
0, making it impossible for E to ever be stable. Fortunately, this only happensif ch ( E ′ ) = 0, thus by further assuming ch ( E ′ ) = 0 in the argument above we obtain E σ
D,tH -psudo-stable. (cid:3)
Remark . The conclusion of Proposition 16 also holds below the wall, i.e., if A and B are σ D,tH -stable with µ D,tH ( A ) < µ D,tH ( B ) for t − ǫ < t ≤ t , then there is δ > → A → E → B → σ D,tH -stable (or σ D,tH -pseudo-stable when ch ( E ) = 0)for all t − δ < t < t .Moreover, the more general result holds: Proposition 17.
Let E be an object in A DH which is strictly σ D,t H -semistable for some t > .Assume that E has a Jordan-H¨older filtration at the wall determined by t that becomes the Harder-Narasimhan filtration of E on one of the chambers determined by t , then E is σ D,tH -stable (or σ D,tH -pseudo-stable when ch ( E ) = 0 ) on the other chamber.Proof. Without loss of generality we can assume that E is σ D,tH -unstable for t < t . Assume thatat the wall determined by t , E has a Jordan-H¨older filtration0 = E ⊂ E ⊂ · · · ⊂ E n − ⊂ E n = E such that for t sufficiently near t and above the wall F i = E i /E i − is σ D,tH -stable and the sequence µ D,tH ( F i ) is strictly increasing. Then by applying Proposition 16 to the exact sequences0 → E i − → E i → F i → δ > E i is σ D,tH -stable for all t ∈ ( t , t + δ ). Inparticular E n = E is σ D,tH -stable for every t in this interval (or σ D,tH -pseudo-stable if ch ( E ) =0). (cid:3) roposition 18. Let us assume that the length of a Jordan-H¨older filtration for E (and so of any)is 2 at a wall determined by t > , and that E fits into a diagram B ′ A E BB ′′ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) where A, B ′ and B ′′ are the σ D,t H -stable factors of E . Assume that there is ǫ > such that forall t ∈ ( t , t + ǫ ) A, B ′ , B ′′ are σ D,tH -stable and µ D,tH ( A ) < µ D,tH ( B ′ ) < µ D,tH ( B ′′ ) . Then thereexists δ > such that objects ˜ E that are extensions of the form B ′ A ˜ E ˜ B B ′′ / / / / / / O O O O / / ?(cid:31) O O can not be stable for any t < t < t + δ .Proof. By Proposition 16 there exists δ > → A → E → B → σ D,tH -stable for t < t < t + δ . Taking δ < ǫ we have Hom( B ′ , A ) = 0 since µ D,tH ( A ) <µ D,tH ( B ′ ), and therefore we get an inclusionExt ( B ′′ , A ) ֒ → Ext ( B, A ) . The image of every nonzero element corresponds to a non split extension which is stable by Propo-sition 17. Such extensions admit an injective morphism B ′ ֒ → E that can be visualized in thediagram 0 B ′ A E B A G B ′′ (cid:15) (cid:15) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ ∃ (cid:15) (cid:15) / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / / / / / / / / / (cid:15) (cid:15) tability of E implies µ D,tH ( B ′ ) < µ D,tH ( E ) = µ D,tH ( ˜ E ) and so B ′ destabilizes ˜ E via thecomposition ˜ E ։ ˜ B ։ B ′ . Thus the only other possibility is Ext ( B ′′ , A ) = 0 which gives asurjective map Ext ( B ′ , A ) ։ Ext ( ˜ B, A ) implying that ˜ E is a pullback of an extension of B ′ by A . As before, there is an injective map B ′′ ֒ → ˜ E B ′′ A ˜ E ˜ B A ˜ G B ′ (cid:15) (cid:15) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ ∃ (cid:15) (cid:15) / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / / / / / / / / / (cid:15) (cid:15) Again the stability of E and the composition E ։ B ։ B ′′ imply µ D,tH ( B ′′ ) > µ D,tH ( E ) = µ D,tH ( ˜ E ), and so B ′′ destabilizes ˜ E . (cid:3) The case of the projective plane.
In the case of P and the stability conditions σ s,t , a wallfor a Chern character v is produced when there is an object E with ch ( E ) = v and an inclusion A ֒ → E in some A s such that µ s ,t ( A ) = µ s ,t ( E ) . Using the explicit formula for µ s,t , it is proven in [2] that the walls are nested semicircles in the( s, t )-upper half plane with center on the s -axis. Denote by W ch ( A ) ,ch ( E ) the wall corresponding tothe inclusion A ֒ → E . Lemma 19. [2, Lemma 6.3]
Let E be a coherent sheaf on P which is either a torsion sheafsupported in codimension 1, or a torsion-free sheaf (not necessarily Mumford-semistable) satisfyingthe Bogomolov inequality: ch ( E ) < ch ( E ) r ( E ) and suppose A → E is a map of coherent sheaves which is an inclusion of σ s ,t -semistable objectsof A s of the same slope for some ( s , t ) ∈ W := W ch ( A ) ,ch ( E ) . Then A → E is an inclusion of σ s,t -semi-stable objects of A s of the same slope for every point ( s, t ) ∈ W . Lemma 19 was used in [2] to provided specific bounds on the radius of the walls and via anidentification of σ s,t -stability with quiver stability, it is shown that if E is a Mumford stabletorsion-free sheaf of primitive Chern vector v then there are finitely many isomorphism typesof moduli spaces of σ s,t -stable objects with invariants v , i.e., finitely many walls intersecting theslice { σ s,t } s,t ∈ R ; t> ⊆ Stab( P ).We finish this section by recalling that for a primitive Chern vector v the moduli space M H ( v )of semistable torsion-free plane sheaves of type v is smooth and a Mori dream space (see [17], or
9] for a detailed explanation). It is shown in [2] that above the outermost wall M s,t ( v ) ∼ = M H ( v ),and the argument given in [9] shows that decreasing t corresponds to running a directed minimalmodel program for M H ( v ) so that each M s,t ( v ) is a birational model.Things are slightly different when studying the Gieseker moduli of 1-dimensional sheaves althoughmost of the arguments are the same. By the work of Le Potier [20] we know that these modulispaces are irreducible, locally factorial and their Picard group is free abelian of rank 2. Moreover,a specific set of generators is given, namely: the determinant line bundle and the line bundle givingthe support map. It is not hard then to prove that the Gieseker moduli of 1-dimensional planesheaves of fixed invariants is also a Mori dream space (an argument can be found in [32]).4. Duality
Let X be a smooth projective surface, D, H ∈ N ( X ) R with H ample. Let σ D,tH = ( Z D,tH , A DH )be the stability condition of Theorem 13 and assume that projective coarse moduli spaces for σ D,tH and σ − D + K,tH are known to exist. For example, X can be P [2], P × P or the blow up of P atone point [3], or a K Theorem 20.
The functor
F 7→ F D := R H om ( F , ω X )[1] induces an isomorphism between theBridgeland moduli M D,tH ( ch ( F )) ∼ = M − D + K X ,tH ( ch ( F D )) provided that ch ( F ) is the chern char-acter of an object in A DH of phase in (0 , . This result was proven by Maican [23] for moduli spaces of Gieseker semistable sheaves on P n supported on curves. The theorem above recovers Maican’s for X = P and t ≫
0. For the proofin this context we will need the following
Lemma 21.
Let E be a σ D,tH -(semi)stable object in P σ D,tH (0 , . Then(a) If E is σ D,tH -stable then it is quasi-isomorphic to a two-term complex of vector bundles E − → E .(b) H − ( E ) is torsion-free with Mumford semistable factors of Mumford slope < DH .(c) If A ∈ A DH is an object all of whose σ D,tH -semistable factors belong to P σ D,tH (0 , then A D ∈ A ( − D + K X ) H .(d) E D ∈ A ( − D + K X ) H is σ − D + K X ,tH -(semi)stable.(e) If E, F ∈ A DH are S -equivalent then so are E D , F D ∈ A ( − D + K X ) H .(f ) For any flat family F ∈ D b ( S × X ) of σ D,tH -semistable objects in A DH with fibers ofinvariants ch ( F s ) there is a flat family F D ∈ D b ( S × X ) with fibers of invariants ch ( F sD ) such that Li ∗ s ( F D ) ∼ = ( Li ∗ s F ) D . Proof.
Part (a) is the content of [11, Lemma 10.1(a)]. For (b), notice that E ∈ A DH fits into theshort exact sequence 0 → H − ( E )[1] → E → H ( E ) → , and therefore if µ max ( H − ( E )) = DH then H − ( E )[1] will have a subobject T in A DH of phase 1,which will σ D,tH -destabilize E .Before proving (c), note that for any coherent sheaf F with Mumford semistable factors ofMumford slope < DH (resp. > DH ) we have H i ( F D ) = torsion free sheaf with µ min > − DH + K X H (resp. µ max < − DH + K X H ) if i = − i = 00-dim torsion sheaf or 0 if i = 10 otherwise. ndeed, if F is locally free the statement is clear since F D = H om ( F, O ) ⊗ ω X [1] = F ∗ ⊗ ω X [1].For torsion sheaves the statement follows from [17, Proposition 1.1.6]. If F is torsion-free then F embeds into its double dual and we get an exact sequence0 → F → F ∗∗ → T → , where T is a 0-dimensional torsion sheaf, and the statement follows after taking cohomology on theexact triangle T D → F ∗∗∗ ⊗ ω X [1] → F D → T D [1] . In the general case, F fits into an exact sequence0 → T → F → F → , where T is its torsion subsheaf, and F is torsion-free. Thus the statement follows after takingcohomology on the distinguished triangle F D → F D → T D → F D [1] . Now, assume that E is σ D,tH -stable and let us prove that E D ∈ A ( − D + K X ) H . Taking cohomologyon the exact triangle H ( E ) D → E D → H − ( E )[1] D → H ( E ) D [1]we get the long exact sequence of sheaves0 → H − ( H ( E ) D ) → H − ( E D ) → H − ( H − ( E )[1] D ) →→ H ( H ( E ) D ) → H ( E D ) → H ( H − ( E )[1] D ) → H ( H ( E ) D ) → E D is a two-term complex of vector bundles. But H − ( E )[1] D = H − ( E ) D [ −
1] and so H − ( H − ( E )[1] D ) = 0 . This implies that H − ( E D ) ∈ F ( − D + K X ) H , H ( H ( E ) D ) is the torsion subsheaf of H ( E D ) andbecause H ( H ( E ) D ) is zero-dimensional we have H ( E D ) ∈ Q ( − D + K X ) H .Moreover, this proves that if A ∈ P σ D,tH (0 ,
1) is σ D,tH -stable, then A D is an element of A ( − D + K X ) H .For arbitrary A ∈ P σ D,tH (0 , A is in the extension closure of some σ D,tH -stable objects A , . . . , A k ∈A DH of the same phase and so A D ∈ h A D , . . . , A kD i ⊂ A ( − D + K X ) H . By the same argument we get (c).Assume for the moment that E is σ D,tH -stable. Let us prove that there is no injective map0 → K → E D in A ( − D + K ) H with K having at least one of its Bridgeland semistable factors of phase 1. If so, therewould be an inclusion 0 → A → E D with A ∈ P σ − D + K,tH (1) stable, i.e., A = C x or A = F [1] for some locally free sheaf F with µ H -semistable factors of slope ( − D + K X ) H ([11, Lemma 10.1(b)]). But if E is stable then E D isderived equivalent to a two-term complex of vector bundles implying Hom( C x , E D ) = 0. AlsoHom( F [1] , E D ) = Hom( F, H − ( E D )) = 0in virtue of Schur’s lemma.Let us now prove that E D must be σ − D + K X ,tH -stable. Indeed, if there is a destabilizing sequence0 → A → E D → B → n A ( − D + K X ) H , we can choose B to be σ − D + K X ,tH -stable and by the argument above we know thatall σ − D + K X ,tH -semistable factors of A have phase in (0 , E in A DH since µ D,tH ( · ) = − µ − D + K X ,tH ( · ) D . We conclude that E D is σ − D + K X ,tH -semistable for all σ D,tH -semistable objects E of phase in(0 ,
1) just by dualizing the Jordan-H¨older filtration of E : Let 0 = F ⊂ F ⊂ F ⊂ · · · ⊂ F n = E be a σ D,tH -Jordan-H¨older filtration for E in A DH , then ( E/F n ) D ⊂ ( E/F n − ) D ⊂ · · · ⊂ E D is a σ − D + K X ,tH -Jordan-H¨older filtration for E D in A ( − D + K X ) H with stable factors ( F i /F i − ) D . Thisalso gives part (d).For the last part let F D := R H om ( F , ω S × X/S ) then Li ∗ s ( R H om ( F , ω S × X/S )) ∼ = R H om ( Li ∗ s F , ω X ) ∈ A ( − D + K X ) H . (cid:3) Proof of [ .Theorem 20] Every flat family F ∈ D b ( S × X ) of topological type v gives a morphism π : S → M D,tH ( v ) and a morphism π D : S → M − D + K X ,tH ( v D ) corresponding to the family F D ofthe lemma. Since π D is constant on the fibers of π by part (e) then π D factors through a morphism M D,tH ( v ) → M − D + K X ,tH ( v D ) which sends the closed point representing E to the closed pointrepresenting E D . The symmetry of the situation and the fact that ( ) DD = Id prove that suchmorphism is an isomorphism. (cid:3) Remark . In the special case when X = P and v = (0 , d, − d/
2) duality gives an automorphism( ) D : M − / ,t ( v ) ∼ = M − / ,t ( v ) for all t > Corollary 22.
Let [ C ] ∈ NS( X ) be a curve class and H ∈ Amp( X ) . Then the functor F 7→ E xt ( F , ω X ) induces an isomorphism between the moduli spaces of H -Gieseker semistable sheaves M H (0 , [ C ] , ch ) and M H (0 , [ C ] , K X · C − ch ) .Proof. Take D = K X / µ K X / ,tH ( E ) = ch ( E ) − ch ( E ) · K X + ch ( E )2 ( K X − t H )( ch ( E ) − ch ( E ) K X ) tH . Thus, if ch ( E ) = 0 and ch ( E ) = [ C ] then(4.2) µ K X / ,tH ( E ) = ch ( E ) − C · K X tC · H = χ ( E ) tC · H and therefore a sheaf of those invariants that is σ K X / ,tH -semistable is also H -Gieseker semistable.By [21, Theorem 1.1] we know that the values of t for which there is an inclusion of objects A ֒ → E with µ K X / ,tH ( A ) = µ K X / ,tH ( E ) is bounded above (this also follows by a result of Maciocia [22,Theorem 3.11] when considering the family of stability conditions σ K X / sH,tH ). If E is an objectwith ch ( E ) = 0 that is σ K X / ,tH -semistable for all t ≫ E must be a sheaf since otherwise thenatural inclusion H − ( E )[1] ֒ → E would destabilize E for large values of t . Indeed, if E = H − ( E )is nonzero then µ K X / ,tH ( E [1]) = − ch ( E ) + ch ( E ) · K X − ch ( E )( K X − t H ) − ( ch ( E ) − ch ( E ) tH > χ ( E ) tC · H for t ≫ E is torsion-free.Conversely, let E be a H -Gieseker semistable sheaf with ch ( E ) = (0 , [ C ] , ch ). If A ֒ → E is asubobject in A K X H/ that σ K X / ,tH -destabilizes E for some t then A must be a sheaf, because E s a sheaf, and of positive rank since otherwise it would destabilize E with respect to H -Giesekerstability. Now, since ch ( A ) > t ≫ µ K X / ,tH ( A ) < µ K X / ,tH ( E )and so the inclusion A ֒ → E must produce a wall. Since the walls are bounded above we concludethat above all walls E is σ K X / ,tH -semistable. The coarse moduli spaces M H (0 , [ C ] , ch ) wereconstructed by Simpson [27] via invariant theory, and the conclusion follows from the dualitytheorem. (cid:3) Bridgeland Walls for 1-Dimensional Plane Sheaves
In this section and for the reminder of the paper X = P . As mentioned in Section 2, modulispaces of Gieseker semistable plane sheaves of Hilbert polynomial P ( t ) = ct + χ were studiedby Le Potier in [20], where it is shown that these moduli spaces are irreducible, locally factorial,and smooth at the stable points. For small values of c it is possible to find nice stratifications ofthese moduli spaces by studying their resolutions, see [15] for c = 4 and [24] and [25] for c = 5and 6. Studying a sheaf by studying its possible resolutions is same as replacing such sheaf foran equivalent element in the derived category. Indeed, each strata in the stratifications given byDr´ezet and Maican in [15] and by Maican in [24] and [25] can be interpreted as a set of extensionsin a tilted category [9].Moreover, as in [9], each set of extensions produces a Bridgeland wall, and these are all the wallsfor the directed MMP. The following numerical bound coming from Lemma 19 produces some ofthese sets of extensions for arbitrary value of c even when Maican-type stratifications are unknown.Let E be a sheaf of topological type (0 , c, d ) with c > F be a destabilizing object (whichis necessarily a sheaf), then E and F fit into an exact sequence0 → K → F → E of coherent sheaves. By Lemma 19 we must have K ∈ F s and F ∈ Q s for all s along the wall W ch ( F ) ,ch ( E ) = { ( s, t ) : µ s,t ( F ) = µ s,t ( E ) } = (cid:26) ( s, t ) : A t A s Bs = C (cid:27) , where A = ch ( F ) ch ( E ) − ch ( E ) ch ( F ) ,B = ch ( F ) ch ( E ) − ch ( E ) ch ( F ) ,C = ch ( F ) ch ( E ) − ch ( E ) ch ( F ) . If ch ( F ) = ( r ′ , c ′ , d ′ ), then in our case this wall is a semicircle with center (cid:18) − BA , (cid:19) = (cid:18) ch ( E ) ch ( F ) ch ( F ) ch ( E ) , (cid:19) = (cid:18) dc , (cid:19) , and radius R = s(cid:18) BA (cid:19) + 2 CA = s(cid:18) dc (cid:19) + 2 d ′ r ′ − dc ′ r ′ c . Therefore, we have ch ( K ) ch ( K ) ≤ dc − R and c ′ r ′ ≥ dc + R. Since ch ( K ) = r ′ and ch ( K ) − c ′ + c ≥
0, then combining the inequalities above we get(5.1) R + dc ≤ c ′ r ′ ≤ dc + cr ′ − R, hich immediately produces(5.2) R ≤ c r ′ . The purpose of the next section is to study Veronese surfaces and their secant varieties seeingthese as special loci on a moduli space of Bridgeland semistable objects. To identify which modulispace we should look at, consider the ( d − P for an integer d > ν d − : P → P ( H ( P , O ( d − ∨ ) ∼ = P (Ext ( O ( d ) , O [1])) , where the last isomorphism is given by Serre duality. Thus, if O ( d ) and O [1] are both objects of afull abelian subcategory A ⊂ D b ( P ), then it would be possible to see the ( d − → O [1] → E → O ( d ) → A . The topological type of all such extensions is ch ( E ) = ch ( O [1]) + ch ( O ( d )) = (0 , d, d . When d is odd, we can twist by the line bundle O (( − d − /
2) and obtain the invariants v = ch (( E ( − d − / , d, − d/ d is even, we can twist by O (( − d − /
2) and obtain theinvariants (0 , d, − d ).The Gieseker moduli spaces M H (0 , d, − d/
2) (for d odd) and M H (0 , d, − d ) (for d even) behavesimilarly for our purposes, so we will concentrate in the case when d is odd, and study the casewhen d is even at the end of Section 6, where the differences will be explained. Nevertheless, thetechniques used to study both cases are the same and (except for sections 6.2 and 6.3) every resultfor d odd has a similar statement for d even.Fix the numerical class v = (0 , d, − d ) with d odd. One of the key ingredients in the computationthat follows is the existence of a collapsing wall. The generic element of M H ( v ) corresponds to asheaf E satisfying H ( E ) = 0 (because χ ( E ) = 0). By using the Beilinson spectral sequence (asin [15, Section 2.2] or [25, Proposition 6.1.1]) one can conclude that the general element of M H ( v )has a resolution of the form 0 → d O ( − → d O ( − → E → . Since O ( − O ( −
1) are objects in A − / , then the general element of M H ( v ) fits into anexact sequence 0 → d O ( − → E → d O ( − → A − / . In particular O ( −
1) produces a wall contracting an open set. By Riemann-Roch theradius of a wall produced by a σ s,t -destabilizing subobject F can be written as R = s
94 + 2 ch ( F ) ch ( F ) + 3 ch ( F ) ch ( F ) = s
14 + 2 χ ( F ) ch ( F ) . Thus, the wall W ch ( O ( − ,v has center ( − / ,
0) and radius R = s
14 + 2 χ ( O ( − ch ( O ( − . The complement of such open set is the theta divisor [20] and is the set of semistable sheaves thathave at least one section, i.e., those that have O as a subobject. The corresponding wall W ch ( O ) ,v has radius R = s
14 + 2 χ ( O ) ch ( O ) = 32 . rossing W ch ( O ) ,v corresponds to a divisorial contraction and since M H ( v ) has Picard number 2then there are no walls between W ch ( O ( − ,v and W ch ( O ) ,v . This gives a lower bound for the radiusof walls corresponding to flips: Proposition 23.
Let d > be an odd integer. Let F be a coherent sheaf of positive rank, and let E be a coherent sheaf with ch ( E ) = (0 , d, − d ) . A morphism of sheaves F → E which is an inclusionof objects in the category A − / produces a wall corresponding to a flip if and only if (5.3) 32 < s
14 + 2 χ ( F ) ch ( F ) ≤ d ch ( F ) . Proof.
Since the center of all walls of type v = (0 , d, − d/
2) in the ( s, t )-plane is ( − / , { σ − / ,t } t> intersects every chamber. Because ofthe correspondence between Bridgeland wall-crossing for the topological type v and the MMP forthe moduli spaces M H ( v ) [9, Theorem 1.1], we know that starting with t ≫ t corresponds to run a directed Minimal Model Program on M H ( v ). Moreover, because M H ( v )is a Mori dream space of Picard number 2, then the moduli spaces of σ − / ,t -semistable objectsof topological type v account for all the birational models of M H ( v ). Because there are no wallsbetween W ch ( O ,v ) and W ch ( O ( − ,v ) , then every wall corresponding to a flip has radius larger thanthe radius of W ch ( O ,v ) , i.e., R > /
2. The other inequality is (5.2). (cid:3)
It is useful to know whether the new objects we get after crossing a wall are Bridgeland stableor pseudo-stable, and we can answer this in a very special case:
Proposition 24.
Let v = (0 , d, − d/ and assume that E ∈ A − / is an object with ch ( E ) = v that is Bridgeland semistable with a Jordan-H¨older filtration of length 1 for a stability condition ata wall W of type v . Then E is Bridgeland stable for stability conditions on one of the chambersdetermined by W .Proof. Assume that the Jordan-H¨older filtration of E at W is0 → A → E → B → µ − / ,t ( A ) > µ − / ,t ( B ) above W , then E is Bridgeland pseudo-stable below W byProposition 16. Assume that E is not Bridgeland stable below W , then there should be a subobject E ′ ֒ → E with ch ( E ′ ) = ch ( E ) = 0 such that µ − / ,t ( E ′ ) = µ − / ,t ( E ) ⇔ χ ( E ′ ) ch ( E ′ ) = χ ( E ) ch ( E ) = 0 . Thus, χ ( E ′ ) = χ ( E ) = 0. Moreover, ch ( E ′ ) < ch ( E ) = d since otherwise the quotient E/E ′ in A − / would be mapped to 0 by Z − / ,t . Let K = ker( E ′ ։ B ) and ch ( E ′ ) = d − h , then we havea diagram K E ′ A E B (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:31) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄ (cid:31) (cid:127) / / / / / / If ch ( B ) = r ′ , ch ( B ) = c ′ , and ch ( B ) = δ ′ , then ch ( K ) = ( ch ( K ) , ch ( K ) , ch ( K )) = ( − r ′ , d − h − c ′ , −
32 ( d − h ) − δ ′ ) ,ch ( A ) = ( ch ( A ) , ch ( A ) , ch ( A )) = ( − r ′ , d − c ′ , − d − δ ′ ) . ote that in this case R ( Z − / ,t ( K )) = R ( Z − / ,t ( A )) and I ( Z − / ,t ( K )) = I ( Z − / ,t ( A )) − ht .But A is Bridgeland stable at W and so it is Bridgeland stable for t sufficiently near W , therefore µ − / ,t ( K ) < µ − / ,t ( A )for t above and below W . This implies that − h R ( Z − / ,t ( A )) < W and so h = 0. Thus E is Bridgeland stable. (cid:3) Rank one walls.
Setting ch ( F ) = 1 in (5.3) one finds the set of admissible values for theEuler characteristic of a destabilizing object producing a wall corresponding to a flip: χ ( F ) = 2 , . . . , d − . The possible values for the first Chern class come from solving the inequality (5.1). Assume χ ( F ) = d − − ℓ , for 0 ≤ ℓ ≤ d − −
2, then(5.4) r d − ℓ − ≤ ch ( F ) ≤ −
32 + d − r d − ℓ. It is easy to check that ch ( F ) = ( d − / I Z (( d − /
2) of a 0-dimensional subscheme Z of length ℓ .Now, for a generic 0-dimensional subschemes Z of length ℓ , Gaeta’s theorem states that if wewrite ℓ = r ( r + 1)2 + s, ≤ s ≤ r, then I Z has a free resolution of the form0 → O ( − r − ⊕ ( r − s ) ⊕ O ( − r − ⊕ s → O ( − r ) ⊕ ( r − s +1) → I Z → r ≥ s , or 0 → O ( − r − ⊕ s → O ( − r ) ⊕ ( r − s +1) ⊕ O ( − r − ⊕ (2 s − r ) → I Z → r ≤ s .For 0 ≤ ℓ ≤ d − −
2, a simple computation shows that d − r >
0, and d − r − > r ≤ s . Therefore, for generic 0-dimensional subschemes Z, W ⊂ P of length ℓ we haveHom( O , I W ⊗ L I Z ( d )) = 0. Thus,Ext ( I ∨ W (( − d − /
2) [1] , I Z (( d − / I ∨ W (( − d − / , I Z (( d − / O , I W (( d + 3) / ⊗ L I Z (( d − / O , I W ⊗ L I Z ( d )) = 0 . Since I ∨ W (( − d − /
2) [1] ∈ A − / by Theorem 20 with t ≫
0, then there are nontrivial extensions0 → I Z (( d − / → E → I ∨ W (( − d − /
2) [1] → E ∈ A − / . The generic extension is Bridgeland stable above the wall determinedby I Z (( d − / t > r d − ℓ. This proves that the number of actual rank 1 walls corresponding to flips is d −
98 . otice that the exceptional loci for a rank 1 flip is not irreducible in general. Indeed, theinequality (5.4) has unique solution only when ℓ < d −
12 . However, setting G Wℓ,i := I W (cid:18) d −
32 + i (cid:19) , length( W ) = ℓ + i ( d + i )2 , i ∈ Z we have Proposition 25. If c = d − + i is solution for (5.4) then so is c = d − − i . Generically, thecorresponding destabilizing objects are of the form G Wℓ,i and G Yℓ, − i respectively. Moreover, if E ℓ,k denotes the component containing the locus of sheaves destabilized by an object of the form G Wℓ,k then E ℓ, − k is the image of E ℓ,k by the duality automorphism.Proof. The first part is a trivial computation. For the second one note that a generic destabilizingsequence is of the form 0 → G
Wℓ,i → E → ( G Yℓ, − i ) D → (cid:3) Remark . It follows from [2, Sections 9 & 10] that the outermost wall for the Chern character w = (1 , ( d − / , ( d − / − ℓ ) is produced by the inclusions O (( d − / ֒ → I Z (( d − / , destabilizing twisted ideal sheaves of 0-dimensional subschemes Z ⊂ P of length ℓ supported on aline. This wall is a semicircle with center (cid:0) d − − ℓ, (cid:1) and radius ℓ − . In the outermost chamber,the only Bridgeland semistable (actually stable) objects of Chern character w are the twisted idealsheaves I Z (( d − / ℓ ( P ). Remark . With the notation of Remark 25.1. Notice that the right intercept of a wall W ofradius R for the Chern character v and the s -axis is − + R , and the right intercept of the outermostwall for the Chern character w and the s -axis is d − . Therefore, it follows from Remark 25.1 thatas long as R > d −
1, the only Bridgeland semistable objects of Chern character w along W arethe twisted ideal sheaves I Z (( d − / Z ⊂ P is a 0-dimensional subscheme of length ℓ . Remark . Let W be a wall of radius R for the Chern character v produced by a destabilizingsubobject A → E (here E is a sheaf with ch ( E ) = v ). Because R ≤ d ch ( A ) and d ≥
5, then eachwall of radius
R > d − must be a rank 1 wall. Moreover, since inequality (5.4) has only one solutionwhen R > d − , then the walls of radius R > d − are produced by subobjects A with Chern character ch ( A ) = (cid:18) , d − , ( d − − ℓ (cid:19) , ≤ ℓ < d − . Since A is Bridgeland semistable along W , then by Remark 25.2 we must have A = I Z (( d − / Z ⊂ P of length ℓ .6. The Embedded Problem: Flips of Secant Varieties.
In [30] and [31], Vermeire describes a sequence of flips for the secant varieties of an embedding
X ֒ → P N of an algebraic surface. This sequence of flips is constructed in similar fashion to theflips obtained by Thaddeus [28] when studying variation of GIT for moduli spaces of stable pairson curves. The first of these flips is easy to describe and it is the content of [30, Theorem 4.12]. oughly speaking, if the embedding of X is sufficiently ample such that it can be generated byquadrics with only linear syzygies then there is a flip diagram˜ M bl X ( P N ) M P N P s (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧ π (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ h (cid:15) (cid:15) ✤✤✤✤✤✤✤ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄ ϕ + (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ϕ − / / ϕ where ϕ : P N P s is the rational map given by the forms defining X and ˜ M is the blow-up ofbl X ( P N ) along the strict transform of the secant variety ^ SecX . The diagram restricts to E P ( E ) P ( F )Hilb (X) z z tttttttt π $ $ ❏❏❏❏❏❏❏❏ h $ $ ❏❏❏❏❏❏❏ ϕ + z z ttttttt ϕ − where P ( E ) ∼ = ^ SecX and F = ϕ + ∗ ( N ∗ P ( E ) / bl X ( P N ) ⊗ O P ( E ) ( − d − P , X = ν d − ( P ), such flipsappear naturally when running the MMP for the Gieseker moduli M H (0 , d, − d/
2) for d odd (or M H (0 , d,
0) for d even).From Remark 25.3 we know that the first ( d − / v = (0 , d, − d/
2) ( d odd)correspond to flips and are produced by destabilizing subobjects of the form G Zℓ, = I Z (( d − / , length( Z ) = ℓ, ≤ ℓ < d − . The radius of the wall W ch ( G Zℓ, ) ,v is R ℓ = r d − ℓ. We denote by M ℓ the moduli space of σ − / ,R ℓ -semistable objects of topological type v , and by M ± ℓ the moduli spaces of σ − / ,R ℓ ± ǫ -semistable objects of topological type v for 0 < ǫ ≪
1. Thus, forinstance, M − ℓ = M + ℓ +1 . Also, denote by E ± ℓ the exceptional loci for the contractions π ± ℓ : M ± ℓ → M ℓ . Thus we obtain the exceptional loci for the first flip of M H (0 , d, − d/
2) ( d ≥ E +0 : 0 → O (( d − / → F → O (( − d − /
2) [1] → E − : 0 → O (( − d − /
2) [1] → G • → O (( d − / → , these are obtained from the set-theoretic wall-crossing since the objects O (( d − /
2) and O (( − d − /
2) [1]are σ − / ,t -stable for every value of t , which follows from [2, Proposition 6.2] and Theorem 20. E +019 nd E − are projective spaces, indeed: E +0 ∼ = P (cid:0) Ext ( O (( − d − /
2) [1] , O (( d − / (cid:1) ,E − ∼ = P (cid:0) Ext ( O (( d − / , O (( − d − /
2) [1]) (cid:1) = P (cid:0) Ext ( O (( d − / , O (( − d − / (cid:1) = P ( H ( P , O ( − d ))) ∼ = P ( H ( P , O ( d − ∨ ) . There is a natural P embedded in E − by the complete linear series ν d − : P → E − . The Veronesesurface X := ν d − ( P ) can be described in terms of extensions, it is the set of complexes G • fittinginto a commutative diagram(6.1) I p (( d − / O (( − d − /
2) [1] G • p O (( d − / O (( − d − /
2) [1] G C p w w ♦♦♦♦♦♦♦♦♦♦♦ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / Note that G is unique (up to scalars) since ext ( C p , O (( − d − /
2) [1]) = 1. Thus G • p is the imageunder the pullback homomorphismExt ( C p , O (( − d − /
2) [1]) ֒ → Ext ( O (( d − / , O (( − d − /
2) [1]) . But we know thatExt ( C p , O (( − d − /
2) [1]) ∼ = Ext (( O (( − d − /
2) [1]) D , ( C p ) D )= Ext ( O (( d − / , ( C p ) D ) , so G • p is also the image under the push-forward mapExt ( O (( d − / , ( C p ) D ) ֒ → Ext ( O (( d − / , O (( − d − /
2) [1]) . Applying the functor ( ) D to the pullback diagram above gives us the push-forward diagram C Dp G D O (( d − / O (( − d − /
2) [1] ( G • p ) D O (( d − / I p (( d − / D (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) (cid:15) (cid:15) / / / / (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) / / / / w w ♦♦♦♦♦♦♦♦♦♦ Proposition 26.
The elements of E − are fixed by the duality automorphism.Proof. From the discussion above we know that G • p = ( G • p ) D , and so the duality automorphismwhich restricts to an automorphism ( ) D | E − : E − → E − fixes X . Since every automorphism of E − ∼ = P N is linear, then ( ) D | E − is the identity. (cid:3) he exceptional loci for the second flip are E +1 : 0 → I p (( d − / → F → I ∨ q (( − d − / → p, q ∈ P E − : 0 → I ∨ q (( − d − / → G • → I p (( d − / → p, q ∈ P . This description of E − is given by Proposition 17 since by Remark 25.3 and Theorem 20 we knowthat both I p (( d − /
2) and I ∨ q (( − d − / W ch ( I p (( d − / ,v .The following vanishing theorem will be used repeatedly for the rest of the section: Theorem 27 (Bertram-Ein-Lazarsfeld, [5]) . Assume that X ⊂ P r is (scheme-theoretically) cut outby hypersurfaces of degrees d ≥ d ≥ · · · ≥ d m . Then H i ( P r , I aX ( k )) = 0 for all i ≥ provided that k ≥ ad + d + · · · + d e − r , where e = codim ( X, P r ) . Lemma 28.
The fiber product M +1 × M M − is isomorphic to the common blow-up bl E +1 M +1 =bl E − M − .Proof. A proof of this statement was already given in [9, Section 6.2.1] for the case d = 5, and itgeneralizes for all d (odd) without change. One notices the following vanishingHom( I p (( d − / , I ∨ q (( − d − / ( I ∨ q (( − d − / , I ∨ q (( − d − / ( I p (( d − / , I p (( d − / ( F, I p (( d − / ( I ∨ q (( − d − / , F ) = 0for every p, q ∈ P and F ∈ E +1 . The first is obvious when p = q , for p = q one uses Serre dualityand Bertram-Ein-Lazarsfeld vanishing. The last two are obtained by using Serre duality and thefact that F is Bridgeland stable, which is a consequence of Proposition 24. This allows us to getdiagrams ( I ∨ q (( − d − / , I ∨ q (( − d − / ( F, I p (( d − / ( F, F ) Ext ( F, I ∨ q (( − d − / ( I p (( d − / , I ∨ q (( − d − / (cid:15) (cid:15) (cid:15) (cid:15) / / + + ❲❲❲❲❲❲❲❲❲❲ f / / / / (cid:15) (cid:15) (cid:15) (cid:15) nd ( I p (( d − / , I p (( d − / ( I ∨ q (( − d − / , F ) Ext ( F, F ) Ext ( I p (( d − / , F )Ext ( I p (( d − / , I ∨ q (( − d − / (cid:15) (cid:15) (cid:15) (cid:15) / / + + ❲❲❲❲❲❲❲❲❲❲ f / / / / (cid:15) (cid:15) (cid:15) (cid:15) Then we get an exact sequence0 → ker f → Ext ( F, F ) → Ext ( I p (( d − / , I ∨ q (( − d − / → , and two surjections ker f ։ Ext ( I ∨ q (( − d − / , I ∨ q (( − d − / , ker f ։ Ext ( I p (( d − / , I p (( d − / . Since the compositionsExt ( I ∨ q (( − d − / , I p (( d − / → Ext ( F, I p (( d − / → Ext ( F, F ) , and Ext ( I ∨ q (( − d − / , I p (( d − / → Ext ( I ∨ q (( − d − / , F ) → Ext ( F, F ) , coincide, then ker f fits into an exact sequence 0 C Ext ( I ∨ q (( − d − / , I p (( d − / f Ext ( I ∨ q (( − d − / , I ∨ q (( − d − / ⊕ Ext ( I p (( d − / , I p (( d − / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) Where we take C ∼ = Hom( I p (( d − / , I p (( d − / , or C ∼ = Hom( I ∨ q (( − d − / , I ∨ q (( − d − / . Thus ker f can be identified with the tangent space of E +1 at the point [ F ] and we get an exactsequence 0 → ( T E +1 ) [ F ] → ( T M +1 | E +1 ) [ F ] → Ext ( I p (( d − / , I ∨ q (( − d − / → . and therefore an exact sequence of sheaves0 → T E +1 → T M +1 | E +1 → ( π +1 | E +1 ) ∗ (( π − | E − ) ∗ O E − (1)) → . imilarly one gets 0 → T E − → T M − | E − → ( π − | E − ) ∗ (( π +1 | E +1 ) ∗ O E +1 (1)) → . This proves that we have a fiber square P ( N E +1 /M +1 ) ∼ = P ( N E − /M − ) E − E +1 P × P (cid:15) (cid:15) ✤✤✤✤✤✤✤ / / (cid:15) (cid:15) ✤✤✤✤✤✤✤ π − | E − / / π +1 | E +1 which completes the proof. (cid:3) Proposition 29. (a) E +1 and E − are both projective bundles over P × P .(b) E +1 ∩ E − = X .(c) The closure of E − \ X in M − is isomorphic to the blow-up of E − along X .Proof. For part (a), we only need to verify that ext ( I ∨ q (( − d − / , I p (( d − / ( I p (( d − / , I ∨ q (( − d − / p, q vary, because the rest of the argument follows as in[1, Proposition 4.2]. We haveExt ( I ∨ q (( − d − / , I p (( d − / O , I p ⊗ I q ( d )) , Ext ( I p (( d − / , I ∨ q (( − d − / ( I p (( d − / , I ∨ q (( − d − / ∼ = Hom( O , I p ⊗ I q ( d − ∨ . Note that we can use ordinary tensor instead of derived tensor, because ideal sheaves have a two-term resolution by locally free sheaves. For p = q this follows from standard calculations. For p = q one gets constant dimension because H ( P , I p ( k )) = 0 for k > X ⊂ E +1 ∩ E − since every G • p admits an injectivemap (in A − / ) from I p (( d − / I p (( d − / , O (( d − / I p (( d − / , O (( − d − / E fits into a diagram I p (( d − / O (( − d − /
2) [1] E O (( d − / , (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / then the composition I p (( d − / ֒ → E ։ O (( d − /
2) is the natural inclusion and so E ∼ = G • p .More can be said: since E − is fixed by the duality automorphism, then E +1 intersects E − alonga section over the diagonal ∆ ⊂ P × P . Since the morphism π +1 : M +1 → M collapses the fibers f E +1 , then π | E − : E − → M is a closed immersion. By Lemma 28 we have a diagrambl E +1 M +1 bl X E − M +1 M − E − M (cid:15) (cid:15) ✤✤✤✤✤✤✤ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:15) (cid:15) ✤✤✤✤✤✤✤✤ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ (cid:15) (cid:15) ✤✤✤✤✤✤✤✤ /(cid:15) ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:127) / / π +1 | E − which proves bl X E − ⊂ M − completing the proof of part (c). (cid:3) We now study the third flip for d ≥ < d − , Remark 25.3 with ℓ = 2 andProposition 25 imply that the exceptional loci are: E +2 : 0 → I Z (( d − / → F → I ∨ W (( − d − / → | Z | = | W | = 2 E − : 0 → I ∨ W (( − d − / → G → I Z (( d − / → | Z | = | W | = 2 . Again, Bertram-Ein-Lazarsfeld vanishing exposes E +2 and E − as projective bundles over Hilb ( P ) × Hilb ( P ).Our plan is to study the restriction of the directed MMP for M (0 , d, − d/
2) to E − . It isconvenient to fix some notation. Inductively define Y +1 := E − , Y − i is the closure of the image of Y + i by the rational map M + i M − i and Y + i +1 := Y − i . Then, for instance, Y − = Y +2 = bl X E − . Proposition 30. E +2 intersects Y +2 along the strict transform of the secant variety ^ SecX which isa projective bundle over
Hilb ( P ) .Proof. The computation is very similar to the one we did when computing E +1 ∩ E − . Let Z = p + q where p, q ∈ P and p = q . We have a pullback diagram(6.2) I Z (( d − / O (( − d − /
2) [1] G • Z O (( d − / O (( − d − /
2) [1] G C Z w w ♦♦♦♦♦♦♦♦♦♦♦ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / The difference here is that ext ( C Z , O (( − d − / p and q removing p and q . Thus, the intersection of E +2 \ E − with E − \ X is SecX \ X ,which proves the first claim.A problem arises when considering Z = 2 p , because in this case we have ext ( C Z , O (( − d − / M +2 we already flipped E +1 then not all the complexes G Z obtained this ay are Bridgeland stable. Instead, the complexes G p fitting into a commutative diagram(6.3) I p (( d − / I p (( − d − / G p I p (( d − / I p (( − d − / G C p w w ♦♦♦♦♦♦♦♦♦♦♦ (cid:127) _ (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) / / / / / / are Bridgeland stable. The objects G p form the fiber of ^ SecX over Z = 2 p . (cid:3) Lemma 31.
The fiber product M +2 × M M − is isomorphic to the common blow-up bl E +2 M +2 =bl E − M − .Proof. The proof is similar to the proof of Lemma 28. The right vanishing is again a consequenceof Bertram-Ein-Lazarsfeld vanishing. (cid:3)
This completes Vermeire’s first flip since by restricting the fiber diagram of Lemma 31 one gets E bl ^ SecX (bl X E − ) M +2 × M M − M − ^ SecX bl X ( E − ) M +2 M Hilb ( P ) (cid:31) (cid:127) / / (cid:15) (cid:15) ✤✤✤✤✤✤✤ (cid:31) (cid:127) / / (cid:15) (cid:15) ✤✤✤✤✤✤✤ (cid:15) (cid:15) ✤✤✤✤✤✤✤ / / (cid:15) (cid:15) ✤✤✤✤✤✤✤ , , ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ (cid:31) (cid:127) / / (cid:31) (cid:127) / / / / ?(cid:31) O O ✤✤✤✤✤✤✤ Remark . In [30] it is mentioned that flips of secant varieties are closely related to the geometryof Hilb n (X). By Propositions 25 and 17 and the results of this section, one sees that indeed flips ofsecant varieties of Veronese surfaces are related to the geometry of Hilb n ( P ), and more preciselyto its birational geometry.By using diagrams similar to (6.1) and (6.2) one sees that every rank-1 wall produces a bira-tional transformation of E − whose exceptional locus contains the strict transform of some highersecant variety of X . Indeed, for ℓ < ( d − / E − , corresponding to crossing the wall W ℓ , is the strict transform of Sec ℓ − X .For ℓ ≥ ( d − / E ℓ, intersects E − along the strict transform of Sec ℓ − X . The intersection E ℓ, − i ∩ E ℓ, ∩ E − is the locus in ^ Sec ℓ − X of ( ℓ − ℓ different points, i ( d − i ) / d − C ⊂ P of degree i . Since E − is fixed by theduality automorphism, this completely describes the loci for the restriction to E − of the MMP on M H (0 , d, − d/ The divisorial contraction.
We want to study what happens to our restricted MMP whencrossing the wall W ch ( O ) ,v corresponding to the theta divisor (i.e., the closure of the set of thosesheaves that admit at least one nonzero section). From Lemma 21 it follows that the duality utomorphism preserves the wall-crossing, and so the theta divisor is left invariant by duality.Therefore, it corresponds to extensions of the form0 → N → F → O ( − → N is an object in A − / of Chern character ch ( N ) = (cid:18) , d − , − d (cid:19) = (cid:18) , d − , ( d − − d ( d − (cid:19) . At the wall W ch ( O ) ,v = W ch ( N ) ,v , N must be Bridgeland semistable. The moduli space of Bridgelandsemistable objects at the wall W ch ( N ) ,v of Chern character ch ( N ) is birational to the Hilbert schemeparametrizing (twisted) ideal sheaves I Z ( d −
3) of 0-dimensional subschemes Z ⊂ P of length n = d ( d − / Remark . One can originally think of the dual extensions but this version allows us to computethe intersection with the first flipped locus more effectively.The intersection of the theta divisor with E − corresponds to the extensions F fitting into thepush-forward diagrams O C ( − G O (( d − / O (( − d − / F O (( d − / O ( − (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) z z z z tttttttttttttt / / / / where C ⊂ P is a curve of degree ( d − /
2. Indeed, the middle vertical sequence of arrows is exactand G corresponds to those complexes produced when flipping the locus in Hilb n ( P ) of n pointson a curve of degree ( d − /
2. This intersection is therefore a projective bundle over the Hilbertscheme of plane curves of degree ( d − /
2. An example of this situation was already observed in[9, Section 6] for the case d = 5 where the intersection of the theta divisor with E − was exactlythe strict transform of the secant variety of the Veronese surface in P .Notice that this intersection is not exactly what gets contracted when crossing W ch ( O ) ,v sinceafter several flips we may have replaced some of these objects by new ones. What we know is thatbecause such F is fixed by the duality automorphism then F must have O as a subobject, and sinceHom( O , O ( − G above must have O as a subobject. Therefore crossing W ch ( O ) ,v must introduce objects E fitting into an exact sequence0 → O ( − → E → G → G being S -equivalent to O ⊕G at the wall W ch ( O ) ,v . Notice that ch ( G ) = (0 , d − , − d − / E are all strictly Bridgeland semistable, in fact pseudo-stable when G is pseudo-stable. Remark . Because of the correspondence between Bridgeland wall-crossing and MMP for themoduli spaces M H (0 , d, − d/ W ch ( O ) ,v . emark . After a more detailed analysis one can further prove that if G is a sheaf then it mustfit into an exact sequence 0 → O C ( − → G → O C (( d − / → , and if G is a complex then (using the semistability of G at W ch ( O ) ,v ) that at least it has to fit intoan exact sequence of the form 0 → A → G → A D → A is a semistable object of invariants ch ( A ) = (cid:18) , d − , − ( d − d + 9)8 (cid:19) .6.2. The last birational model.
One could ask what is the moduli space we obtain after thedivisorial contraction and what is the strict transform of E − . The answer to the first questioncomes from the identification with the quiver moduli. Values of ( − / , t ) near W ch ( O ( − ,v are allinside the quiver region ( s − + t < k = −
1. Recall from Theorem 15, that for every ( s, t ) in this region, there is avector a s,t orthogonal to the dimension vector n = B − v = ( − − − − (2( − − − − − − (2( − −
2) 2 ( − − − − − (2( − − d − d/ = dd such that the moduli space of σ s,t -semistable objects of type v = (0 , d, − d/
2) can be identifiedwith the moduli space of complexes C d ⊗ O ( − → C d ⊗ O ( − a s,t . Note that a s,t is of the form ( a, − θ, θ ). Since one naturallyhas the subcomplex C d ⊗ O ( − C d ⊗ O ( − C d ⊗ O ( − / / ?(cid:31) O O then one must have (0 , , d ) · ( a, − θ, θ ) = θ ≥ , d, d ) with respect to a s,t . Moreover, the proof of [2, Proposition 8.1] showsthat above W ch ( O ( − ,v we have θ >
0, at W ch ( O ( − ,v we have θ = 0, and below W ch ( O ( − ,v wehave θ < Remark . In [19], King shows that the GIT quotientHom( W ⊗ O ( − , W ∗ ⊗ O ( − //GL ( W ) × GL ( W ∗ ) , where the action is given by conjugation, is the moduli space of quiver semistable complexes ofdimension vector n = (0 , d, d ), with respect to the orthogonal vector a = ( a, θ, θ ) . Different choicesof the linearization for the GIT quotient correspond to taking θ > θ = 0, or θ < N (3 , d, d ) studied in [15] of morphisms W ⊗ O ( − → W ∗ ⊗ O ( − , that are GIT semistable with respect to the natural action of GL ( W ) × GL ( W ∗ ), where dim W = d .The moduli space at W ch ( O ( − ,v is just a point, and below W ch ( O ( − ,v the moduli space is empty,proving our assertion that W ch ( O ( − ,v was the collapsing wall. n order to understand what is going to be the last birational model of E − , let us take a lookat the simplest but yet interesting M H (0 , , − /
2) studied by Le Potier in [20]. In [20, Th´eor`eme4.4 and Lemme 4.5], Le Potier showed that M H (0 , , − /
2) is the blow up of N (3 , ,
3) at thecomplement of the dense open subset of injective morphisms 3 O ( − ֒ → O ( − GL (3) × GL (3)-orbit, which is the orbit of the skew-symmetric matrix − z xz − y − x y . As a complex, its cohomology is represented in the diagram O ( − O O ( −
2) 3 O ( − (cid:18) r $ $ ❏❏❏❏❏❏ / / − z xz − y − x y ! $ $ $ $ ❏❏❏❏❏❏❏ : : : : ttttttt ,(cid:12) : : ttttttt where the diagonal exact sequences are the Euler sequences.The example above reflects some general features of the general situation. Note that a complex W ⊗ O ( − → W ∗ ⊗ O ( −
1) given by a skew map is fixed by the duality automorphism, since itcorresponds to taking the negative transpose of the corresponding matrix. The general skew mapwill drop rank by 1 everywhere and therefore it must have a kernel and a cokernel that are linebundles. A simple computation of the invariants shows that the kernel should be O (( − d − / O (( d − / → O (( − d − / → [ W ⊗ O ( − → W ∗ ⊗ O ( − → O (( d − / → A − / , giving a point in E − . Conversely, by Proposition 24 all the complexes in E − are stablerather than pseudo-stable. Any stable complex in the last model for E − must be, as E − itself,fixed by the duality automorphism and therefore it must correspond to the orbit of a skew-map W ⊗ O ( − → W ∗ ⊗ O ( − Remark . For d = 5 we have four walls (see [9, Section 6.2] for details): the walls producedby the destabilizing objects O (1) and I p (1), W ch ( O ) ,v , and W ch ( O ( − ,v . At the first two wallsthe Jordan-H¨older filtrations have length 1 and so the strict transform of E − before the divisorialcontraction consists only of stable objects. As above, it can be proven that at the wall W ch ( O ) ,v ,the divisorial contraction produces objects that are S -equivalent to complexes fitting into an exactsequence 0 → O ( − → E → O ⊕ O ℓ ( − ⊕ O ℓ (1) → . In N (3 , ,
5) these correspond to the GL ( W ) × GL ( W ∗ )-orbits of matrices − z x z − y − x y L − L where L is a linear equation defining ℓ . The GL ( W ) × GL ( W ∗ )-orbits of these matrices are strictlysemistable in N (3 , , ow assume that B is a skew-symmetric matrix giving a GL ( W ) × GL ( W ∗ )-stable orbit. If thereare invertible matrices T, S ∈ GL ( W ) such that T BS t is again skew-symmetric then B = ( S − T ) B ( T − S ) t and therefore S = λT for some λ ∈ C ∗ since B is stable and so Hom( B, B ) = C .Since GL ( W ) can be embedded via the diagonal T ( T, T t ) into GL ( W ) × GL ( W ∗ ) then askew-symmetric matrix that is GL ( W ) × GL ( W ∗ )-stable is also GL ( W )-stable for the diagonalaction. Thus we have an injective map { Stable Skew GL ( W ) × GL ( W ∗ ) − orbits } −→ ∧ W ⊗ V //GL ( W )where V = Hom( O ( − , O ( − GL ( W ) on ∧ W ⊗ V is the natural one: GL ( W ) × ∧ W ⊗ V → ∧ W ⊗ V, ( S, B ) SBS t . In the examples above, this map can be extended to the semistable orbits that have a skew rep-resentative. In fact, in a personal communication to the author, Aaron Bertram has made thefollowing
Conjecture 32.
The last birational model of bl X E − is isomorphic to the GIT quotient ∧ W ⊗ V //GL ( W ) . Odd Veronese embeddings.
As mentioned at the beginning of Section 5, one could as wellstudy flips for secant varieties of odd Veronese embeddings by studying the Bridgeland wall crossingfor the Gieseker moduli space M H (0 , d, − d ) for d even, which contains a locus parametrizing curvesof degree d . As before, we want to run the MMP on M H ( u ) for u = (0 , d, − d ).Let E be a Gieseker semistable sheaf with ch ( E ) = u . The center and the radius of a wallproduced by a Bridgeland destabilizing subobject A ֒ → E are respectively C = (cid:18) ch ( E ) ch ( E ) , (cid:19) = ( − , ,R = s(cid:18) ch ( E ) ch ( E ) (cid:19) + 2 ch ( A ) ch ( A ) − ch ( E ) ch ( A ) ch ( A ) ch ( E ) = s ch ( A ) + ch ( A ) ch ( A ) . Thus the category to be considered is A − . Let E be a Gieseker semistable sheaf with ch ( E ) = u ,then χ ( E ) = dim H ( E ) − dim H ( E ) = ch ( E ) + 32 ch ( E ) = − d + 3 d d > . Thus there are nonzero maps
O → E . If K is the kernel in A − of such map, then there is adiagram in A − K O EF (cid:31) (cid:127) / / (cid:31) (cid:31) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄ / / ?(cid:31) O O Because O and E are sheaves so are K and F . Thus K must be a subsheaf of O implying that F has rank 0, and therefore is a subsheaf of E . Now, as mentioned before O is σ s,t -stable for all s, t ( t > µ s,t ( K ) < µ s,t ( O ) < µ s,t ( F ) ≤ µ s,t ( E )unless K is trivial. Since such inequality does not hold for all s, t , then O is a subobject of E in A − . This proves that we have a collapsing wall W ch ( O ) ,u , which has radius R = s ch ( O ) + ch ( O ) ch ( O ) = 1 . rossing this wall collapses (at least) the open set of Gieseker semistable sheaves that are Bridgelandsemistable along W ch ( O ) ,u .There is also a divisorial contraction produced by the tangent sheaf T P ( − M H (0 , d, − d ) / / ❴❴❴ M H (0 , d, − d ) , F 7→ F ⊗ Ω (1) . As in the case when d is odd, the moduli space M H (0 , d, − d ) has a natural divisor Θ (sheaveswith a section), and the image of M H (0 , d, − d ) is not contained in Θ. By pulling back Θ we obtaina divisor Θ ′ consisting of semistable sheaves F such that F ⊗ Ω (1) has a section. SinceHom( O , F ⊗ Ω (1)) = Hom( T P ( − , F )then the divisor Θ ′ is contracted when crossing the wall corresponding to the destabilizing object T P ( − W ch ( T P ( − ,u has radius R = s ch ( T P ( − ch ( T P ( − ch ( T P ( − r − /
22 = r . Remark . To analyze the last birational model one has to use the triad O ( − , Ω (1) , O insteadof O ( − , O ( − , O in the construction of the quiver moduli. This gives a construction of the lastbirational model as the GIT quotientHom( V ⊗ O ( − , W ⊗ O ) //GL ( V ) × GL ( W ) , where V and W are complex vector spaces of dimension d/ < ch ( A ) + 2 ch ( A ) ≤ d R ≤ ch ( A ) + 1 ≤ d − R, where R = p ch ( A ) + 2 ch ( A )is the radius of the corresponding wall.Notice that ch ( A ) = ( d − / d odd case, we would like ch ( A ) = ( d − / A = I Z (cid:18) d − (cid:19) , length( Z ) = ℓ. If this is the only solution to the second inequality, it means that ( d − R ) − R <
1, equivalently wemust have ( d − < ch ( A ) + 2 ch ( A ) . After some elementary computations we obtain0 ≤ ℓ ≤ d − . Remark . The range 0 ≤ ℓ ≤ d − is only to assure that the exceptional locus for the flips isirreducible, and to obtain good wall-crossing since at these walls the strictly semistable objectsbeing destabilized will have Jordan-H¨older filtrations of length 1. he exceptional locus for the first flip corresponds to those sheaves fitting into an exact sequencein A − of the form 0 → O (( d − / → E → O (( − d − / → . The exceptional introduced when crossing this wall is P (Ext ( O (( d − / , O (( − d − / P ( H ( O ( d − ∨ . Propositions 5.2, 5.3, 6.1, Lemma 6.3, Propositions 6.4, 6.5, and Lemma 6.6 hold in this settingwith identical proofs after replacing ( d − / d − / · ) D ⊗ O (1) instead of ( · ) D .We conclude this chapter with the following theorem, which is a corollary of our construction: Theorem 33.
Let d ≥ be an integer and let ν d − : P → P ( H ( O ( d − ∨ = P N be ( d − -upleembedding. There exists a sequence of flips bl ν d − ( P ) P N M M · · · M k N ⊃ P N M ′ M ′ · · · (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ (cid:15) (cid:15) ✤✤✤✤✤✤✤✤ / / f / / f (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ where k = ( d − / for d odd, and k = ⌊ ( d − / ⌋ for d even, the exceptional locus of f i is thestrict transform of Sec i ( ν d − ( P )) , and N is the first birational model appearing when running theMMP for M H (0 , d, − d/ or M H (0 , d, − d ) depending on whether d is odd or even respectively.Remark . As we have seen, this sequence of flips is indeed longer but the exceptional loci afterthe first k flips become more complicated since after this point strictly semistable objects have atleast three Jordan-H¨older factors. Acknowledgments
The present work is part of the author’s Ph.D. thesis at the University of Utah. He would liketo thank his advisor, Professor Aaron Bertram, for his vision, and constant encouragement andpatience throughout the years of graduate school. The author would like to express his gratitude tothe referee(s) for so many valuable comments that have made the present work much more readable.
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Department of Mathematics, University of California, Santa Barbara, South Hall, Room 6607Santa Barbara, CA 93106-3080
E-mail address : [email protected] URL : http://web.math.ucsb.edu/~martinez/http://web.math.ucsb.edu/~martinez/