The Picard group of the moduli space of sheaves on a quadric surface
TTHE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRICSURFACE
DMITRII PEDCHENKO
Abstract.
In this paper, we study the Picard group of the moduli space of semistable sheaves on asmooth quadric surface. We polarize the surface by an ample divisor close to the anticanonical class.We focus especially on moduli spaces of sheaves of small discriminant, where we observe new andinteresting behavior. Our method relies on constructing certain resolutions for semistable sheaves andapplying techniques of geometric invariant theory to the resulting families of sheaves.
Contents
1. Introduction 12. Preliminaries 33. Study of the Shatz stratification 144. Group actions and Gaeta-type resolutions 285. The Picard group of the moduli space 336. Bad Chern characters 48References 541.
Introduction
In this paper, we are concerned with the calculation of the Picard group of the moduli space ofsemistable sheaves on the quadric surface P × P .Let Y be a smooth complex projective surface and let H be an ample divisor on Y . Consider themoduli space of sheaves M ( v ) parameterizing S -equivalence classes of H -Gieseker semistable sheaveswith Chern character v on Y . These moduli spaces have been intensively studied over the years, butmany basic questions about their geometry remain open. For that matter, calculating the Picard groupis the first necessary step towards understanding the birational geometry of these spaces.The starting point of our investigations is [Dre88] where Drézet computes the Picard group of themoduli space of semistable sheaves on P . Recall that for a smooth surface Y the total slope and discriminant of a Chern character v ∈ K ( Y ) of rank r are defined by ν = c r , ∆ = 12 ν − ch r . Date : July 24, 2020.2010
Mathematics Subject Classification.
Primary: 14J60, 14C22, 14J26. Secondary: 14D20, 14F05.
Key words and phrases.
Moduli spaces of sheaves, Picard group, quadric surface.During the preparation of this article the author was partially supported by the NSF FRG grant DMS-1664303. a r X i v : . [ m a t h . AG ] J u l D. PEDCHENKO
Let Y = P . By [DLP85], there is a fractal-like curve DLP ( ν ) in the ( ν, ∆) -plane, which we callthe Drézet-Le Potier curve , such that the moduli space M ( v ) is positive dimensional if and only if ∆ ≥ DLP ( ν ) , where v = ( r, ν, ∆) . The Drézet-Le Potier curve is comprised of branches DLP E ( ν ) indexed by all exceptional bundles E on P where for each branch the value DLP E ( ν ) is calculated usingthe numerical invariants of E . Drézet [Dre88] shows that(1) If ∆ > DLP ( ν ) , then Pic ( M ( v )) ∼ = Z ,(2) If ∆ = DLP ( ν ) , then Pic ( M ( v )) ∼ = Z .This way, the Picard number of M ( v ) is determined by the position of v = ( r, ν, ∆) relative to theDLP-curve.Now, let Y = P × P with a generic polarization H close to the anticanonical one. Rudakov [Rud94]constructed a fractal-like surface DLP ( ν ) in the ( ν, ∆) -space, the Drézet-Le Potier surface , such thatagain the moduli space M ( v ) is positive dimensional if and only if ∆ ≥ and ∆ ≥ DLP ( ν ) , where v = ( r, ν, ∆) . This surface is also comprised of branches DLP E ( ν ) indexed by all exceptional bundles E on P × P with r ( E ) < r where for each branch the value DLP E ( ν ) is calculated using the numericalinvariants of E . The main new feature compared with the P case is that now there exist integral Cherncharacters with positive dimensional moduli space M ( v ) which lie on the intersection of two branchesof the DLP-surface.This way, we see that for P × P character v = ( r, ν, ∆) with positive dimensional moduli space M ( v ) can be positioned in three different ways with respect to the DLP-surface: (1) v lies above theDLP-surface, (2) v lies on a single branch of the DLP-surface, and (3) v lies on the intersection of twobranches of the DLP-surface.Our key finding is that contrary to the P case the Picard number ρ of M ( v ) is not determined onlyby the position of v relative to the DLP-surface. The main results of this paper are summarized in thefollowing theorem. Theorem 1.1 (See Theorems 5.1 and 6.3) . Let v = ( r, ν, ∆) ∈ K ( P × P ) be a character with r ≥ and ∆ ≥ . (1) If v (cid:48) = ( r, ν, ∆ − r ) lies above the DLP -surface, then ρ ( M ( v )) = 3 . (2) If v lies on a single branch of the DLP -surface, then ρ ( M ( v )) = 2 or ρ ( M ( v )) = 1 . (3) If v lies on the intersection of two branches of the DLP -surface, then ρ ( M ( v )) = 1 .Furthermore, if v is a primitive character, then Pic ( M ( v )) is a free abelian group of rank ρ . Let us explain the dichotomy in case (2) of the above theorem in greater detail. We split Cherncharacters into two groups, calling the characters in the first group good characters and the charactersin the second group bad characters, see Definition 3.8. For a good character v lying on a single branchof the DLP-surface the Picard number of M ( v ) is equal to , see Theorem 5.1. On the other hand,we construct infinite sequences of bad Chern characters lying on a single branch of the DLP-surface forwhich the Picard number of the moduli space drops to , see Examples 6.1, 6.2. Moreover, we show that ρ ( M ( v )) = 1 for any bad character lying on a single branch of the DLP-surface given by a line bundle ,see Theorem 6.3. When v is a bad character of the smallest rank, the moduli space M ( v ) turns out tobe isomorphic to a projective space, see Example 6.4 and Question 6.5.We emphasize that determining which statement of the above theorem applies to a given character v = ( r, ν, ∆) is a finite computational procedure and therefore can be implemented on a computer:both the computation of DLP ( ν ) and determining whether v is a good or a bad character are finitecomputational procedures. HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 3
The proofs of the above results rely on constructing resolutions for semistable sheaves of a givencharacter v and applying techniques of Mumford’s geometric invariant theory to the resulting familiesof sheaves described by such resolutions. In the case of P the most powerful tool for constructingresolutions of semistable sheaves is the Beilinson-type spectral sequence coming from a choice of a fullexceptional collection (see [CHW17, §5] for a detailed analysis). The main difficulty is that full exceptionalcollections on P × P require four exceptional bundles instead of three in the case of P . As a result, formost characters v writing the associated Beilinson-type spectral sequence no longer gives a resolution ofa semistable sheaf V of character v on P × P as a (co)kernel in a short exact sequence. To circumventthis difficulty we instead use the so-called Gaeta-type resolutions of Coskun and Huizenga constructedin [CH18].We conjecture that for all characters v with positive-dimensional moduli space the Picard number of M ( v ) is fully determined by the relative position of v with respect to the DLP In §2, we recall the preliminary facts and survey the known results con-cerning vector bundles and moduli spaces of sheaves on P × P needed in the rest of the paper.Sections §3-5 form the technical core of the paper. In §3, we study the Shatz stratification of completefamilies of sheaves on P × P . We also prove the irreducibility of families parameterizing sheaves witha fixed Harder-Narasimhan filtration, which we later use to prove the irreducibility of Shatz strata incomplete families of vector bundles admitting a Gaeta-type resolution. In §4, we establish basic factsabout group actions in the context of Gaeta-type resolutions. In §5, we calculate the Picard group of M ( v ) under the assumption that v is a good character.Finally in §6, we study Pic ( M ( v )) for bad characters v that lie on a single branch of the DLP-surfacegiven by a line bundle. Acknowledgments. We would like to thank Jack Huizenga for his support, encouragement and helpfuldiscussions throughout the project. We would also like to express gratitude to K¯ota Yoshioka, Jean-MarcDrézet and Daniel Levine for valuable discussions.2. Preliminaries In this section we recall basic facts, previous results and constructions concerning moduli spaces ofsheaves that will be used in the rest of the paper. We will denote an arbitrary variety or an arbitraryprojective surface by Y , while X will be always reserved for the quadric surface X = P × P .2.1. Chern characters. Given a torsion-free sheaf V on a surface Y and an ample divisor H , the totalslope ν , the H - slope µ H and the discriminant ∆ are defined by ν ( V ) = ch ( V ) ch ( V ) , µ H ( V ) = ch ( V ) · Hch ( V ) , ∆( V ) = 12 ν − ch ( V ) ch . These quantities depend only on the Chern character of V and not on the particular sheaf. Given a Cherncharacter v ∈ K ( Y ) , we define its total slope, H -slope and discriminant by the same formulae. We will D. PEDCHENKO often record Chern characters by the rank, total slope and discriminant. Note that one can recover theChern classes from this data.2.2. Stability. We refer the reader to [Hui17], [HL10] and [LP97] for more detailed discussions. Let Y be a surface and H be an ample divisor on it. A torsion-free coherent sheaf V is called µ H - (semi)stable (or slope (semi)stable) if every proper subsheaf (cid:54) = W (cid:40) V of smaller rank satisfies µ H ( W ) ( ≤ ) µ H ( V ) . Define the H - Hilbert polynomial P H, V ( m ) and the reduced H -Hilbert polynomial p H, V ( m ) of a torsion-free sheaf V by P H, V ( m ) = χ ( V ( mH )) , p H, V ( m ) = P V ( m ) r ( V ) . A torsion-free sheaf V is H - (semi)stable (or Gieseker (semi)stable) if for every proper subsheaf W ⊂ V ,we have p H, W ( m ) ( ≤ ) p H, V ( m ) for m (cid:29) . Slope stability implies Gieseker stability and Gieseker semistability implies slope semistability.Every torsion-free sheaf V admits a Harder-Narasimhan filtration with respect to both µ H − and H -semistability, that is there is a finite filtration V ⊂ V ⊂ V ⊂ ... ⊂ V n = V , such that the quotients W i = V i / V i − are µ H (respectively, H -Gieseker) semistable and µ H ( W i ) > µ H ( W i − ) ( respectively, p H, W i ( m ) > p H, W i − ( m ) for m (cid:29) for ≤ i ≤ n . The Harder-Narasimhan filtration is unique. A semistable sheaf further admits a Jordan-Hölder filtration into stable sheaves. Two semistable sheaves are called S - equivalent if they have thesame associated graded objects with respect to a Jordan-Hölder filtration.Our main object of study will be the moduli space M H ( v ) parameterizing S -equivalence classes of H -Gieseker semistable sheaves of character v on Y . We refer the reader to [HL10, §4.3] for the detailsabout the construction of M H ( v ) and its basic properties.2.3. Choosing the polarization. For our purposes, we would like to work with a locally factorial modulispace. After recalling some definitions and results from [HL10, §4.C], we show that if Y is rational surfaceother than P , then it is always possible to vary the polarization H slightly so that M H ( v ) becomeslocally factorial.Let Y be a smooth projective surface. The intersection pairing defines a bilinear form on N um ( Y ) and the Hodge Index Theorem implies that the extension of this bilinear form to N um R ( Y ) defines theMinkowski metric on N um R (that is the signature of the form is (1 , N ) ). Define the positive cone as K + := { y ∈ N um R ( Y ) | y · y > and y · H > for some ample divisor H } , and note that it contains the positive span of ample divisors as an open subcone Amp ( Y ) . Since wecan think of a polarization given by an ample divisor as a ray R > H ⊂ K + , it is convenient to introduce H as the set of rays in K + . This set becomes a hyperbolic manifold if we make the identification H ∼ = { H ∈ K + | H · H = 1 } . Definition 2.1 ([HL10, Definition 4.C.1]) . Let r ≥ be an integer and ∆ > a real number . A class ξ ∈ N um ( X ) is of type ( r, ∆) if − r ∆ ≤ ξ · ξ < . The wall defined by ξ is the real hypersuface W ξ := { R > H ∈ H | ξ · H = 0 } ⊂ H . Note that the definition of the discriminant ∆ we are using in this paper differs from the definition of discriminant ˜∆ in [HL10]: ˜∆ = 2 r ∆ . That is why some formulas in this subsection differ by a factor r compared to the formulas in[HL10, §4.C]. HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 5 When r ≥ and ∆ > , Lemma 4.C.2 in [HL10] asserts that the set of walls of type ( r, ∆) is locallyfinite in H . It is therefore always possible to choose H to not lie on any wall of type ( r, ∆) by a smallperturbation. Lemma 2.2 ([HL10, Lemma 4.C.3]) . Let H be an ample divisor, let V be a µ H -semistable sheaf of rank r and discriminant ∆ on Y and let V (cid:48) ⊂ V be a subsheaf of rank r (cid:48) , < r (cid:48) < r , with µ H ( V (cid:48) ) = µ H ( V ) .Then ξ := rc ( V (cid:48) ) − r (cid:48) c ( V ) satisfies ξ · H = 0 and − r ≤ ξ · ξ ≤ and ξ · ξ = 0 ⇐⇒ ξ = 0 . From this we can prove that if H does not lie on a wall, then the quotients in a Jordan-Hölder filtrationall have the same numerical invariants. Lemma 2.3. Given a Chern character v = ( r, ν, ∆) ∈ K ( Y ) with r ≥ , choose an ample divisor H not on a wall of type ( r, ∆) . Then for any µ H -semistable sheaf V of Chern character v and a subsheaf V (cid:48) ⊂ V of rank r (cid:48) , < r (cid:48) < r , we have µ H ( V (cid:48) ) = µ H ( V ) ⇐⇒ ν ( V (cid:48) ) = ν ( V ) . Proof. Suppose V (cid:48) ⊂ V with µ H ( V (cid:48) ) = µ H ( V ) , but ν ( V (cid:48) ) (cid:54) = ν ( V ) , or equivalently ξ := rc ( V (cid:48) ) − r (cid:48) c ( V ) (cid:54) = 0 . By Lemma 2.2, we get that ξ satisfies − r ∆ ≤ ξ · ξ < . Since ξ · H = 0 ,we obtain that H lies on a wall W ξ of type ( r, ∆) , contradicting our choice of H . (cid:3) Corollary 2.4. Given a Chern character v = ( r, ν, ∆) ∈ K ( X ) with r ≥ , choose an ample divisor H not on a wall of type ( r, ∆) . Then for any H -semistable sheaf V of Chern character v its Jordan-Hölderfactors gr i ( V ) satisfy ν ( gr i ( V )) = ν and ∆( gr i ( V )) = ∆ . Proof. By the definition of a Jordan-Hölder filtration ⊂ F ⊂ F ⊂ ... ⊂ F l = V , we have µ H ( F i ) = µ H ( V ) . Then by Lemma . we get ν ( F i ) = ν . We apply the "seesaw" property of thetotal slope to the short exact sequence → F i − → F i → gr i ( V ) → to get ν ( gr i ( V )) = ν . The statement about the discriminants then follows from the equality of reduced H -Hilbert polynomials p H, gr i ( V ) = p H, V and Riemann-Roch. (cid:3) Now, let ( Y, H ) be a polarized rational surface with K Y · H < . Drézet [Dré91] calls a point in M H ( v ) a type 2 point if the corresponding S -equivalence class [ V ⊕ ... ⊕ V k ] satisfies ν i (cid:54) = ν j for some ≤ i, j ≤ k. The other points are called type 1 points. Drézet shows in [Dré91, Theorem C] that the moduli space M H ( v ) is not locally factorial at type 2 points.Suppose further that Y is a rational surface other than P . Then there is a morphism Y → P such that the generic fiber is P . Let F be the class of a fiber. Yoshioka [Yos96a] shows that if ( K Y + F ) · H < and ∆( v ) > , then M H ( v ) is locally factorial at points of type 1. In light ofCorollary 2.4, we conclude that under these assumptions M H ( v ) is locally factorial whenever H is noton a wall of type ( r ( v ) , ∆( v )) . D. PEDCHENKO The Donaldson homomorphism. The Donaldson homomorphism will be our main tool for con-structing line bundles on the moduli space. We briefly recall the construction while referring the readerto [HL10, §8.1] and [LP97, §18.2] for full details.Let U /S be a flat family of semistable sheaves of Chern character v on a smooth variety Y parame-terized by a variety S , and let p : S × Y → S and q : S × Y → Y be the two projections. The Donaldsonhomomorphism λ U : K ( Y ) → Pic ( S ) is described as the composition K ( Y ) q ∗ −→ K ( S × Y ) · [ U ] −→ K ( S × Y ) p ! −→ K ( S ) det −→ Pic ( S ) . Functorial properties of λ U are summarized in the following lemma. Lemma 2.5 ([HL10, Lemma 8.1.2.] and [LP97, Lemma 18.2.1]) . Let λ U : K ( Y ) → Pic ( S ) be theDonaldson homomorphism constructed above. (1) If U is an S -flat family and f : S (cid:48) → S a morphism, then for any u ∈ K ( Y ) one has λ f ∗ Y U ( u ) = f ∗ λ U ( u ) . (2) If S is equipped with an action of an algebraic group G and U is a G -linearized family over S ,then λ U factors through the group Pic G ( S ) of isomorphism classes of G -linearized line bundleson S . (3) If → U (cid:48) → U → U (cid:48)(cid:48) → is a short exact sequence of S -flat families of G -linearized coherent,sheaves then λ U ( u ) = λ U (cid:48) ( u ) ⊗ λ U (cid:48)(cid:48) ( u ) in Pic G ( S ) . Using the last property we can construct line bundles on the moduli space of (semi)stable sheaves M H ( v ) . Informally, realize M H ( v ) as a (good) quotient R // G of a subvariety R of a Quot scheme. The G -linearized universal family of quotient sheaves U /R gives a map λ U : K ( X ) → Pic G ( R ) and we wantto descend the G -linearized line bundles in the image along the quotient map R → R // G = M H ( v ) .For this construction to work we, however, need to restrict the domain of λ U . Denote by v ⊥ ⊂ K ( Y ) the complement of v with respect to the Euler pairing χ ( _ · _ ) . We then get the following theorem,which shows that the above construction always produces line bundles on the stable locus M sH ( v ) andis compatible with the universal property of the moduli space M sH ( v ) . Theorem 2.6. [HL10, Theorem 8.1.5] Let v be a class in K ( Y ) . Then there exists a group homomor-phism λ s : v ⊥ → Pic ( M sH ( v )) with the following property:If U is a flat G -linearized family of stable sheaves of class v parameterized by a G -scheme S , and ifthe classifying morphism φ U : S → M sH ( v ) is G -equivariant, then the following diagram commutes : v ⊥ Pic ( M sH ( v )) K ( Y ) Pic G ( S ) . λ s φ ∗U λ U In general, for a polarized variety ( Y, H ) one needs to further restrict the domain of the Donaldsonhomomorphism in order to obtain line bundles on the full locus M H ( v ) (see the rest of [HL10, Theorem8.1.5]). However, when Y is a surface the analysis of the proof of [HL10, Theorem 8.1.5] shows thatfor a polarization which does not lie on a wall of type ( r ( v ) , ∆( v )) we do not need to further shrink thedomain. Proposition 2.7. Let Y be a smooth projective surface. Let v = ( r, ν, ∆) be a class in K ( Y ) and let H be an ample divisor not lying on a wall of type ( r, ∆) . Then there exists a group homomorphism λ : v ⊥ → Pic ( M H ( v )) with the following property: HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 7 If U is a flat G -linearized family of H -semistable sheaves of class v parameterized by a G -scheme S , and if the classifying morphism φ U : S → M H ( v ) is G -equivariant, then the following diagramcommutes: v ⊥ Pic ( M H ( v )) K ( X ) Pic G ( S ) . λ φ ∗U λ U Proof. We follow the notation used in the proof of [HL10, Theorem 8.1.5]. Let R π (cid:16) M H ( v ) be thequotient morphism in the GIT construction of the moduli space, where R is a subvariety of the Quotscheme with a universal family of quotients U . For u ∈ v ⊥ , we would like to descend a GL ( V ) -linearizedline bundle L = λ U ( u ) to M ( v ) along the quotient map π . According to the Descent Lemma [HL10,Theorem 4.2.15], we need to make sure that for any point [ q : H (cid:16) F ] ∈ R in a closed GL ( V ) -orbitthe stabilizer GL ( V ) [ q ] acts trivially on the fiber L | [ q ] of L over the point [ q ] .The orbit of [ q : H (cid:16) F ] ∈ R is closed if and only if F is a polystable sheaf. Thus F ∼ = ⊕ i ( F i ⊗ W i ) with distinct stable Jordan-Hölder factors F i and vector spaces W i . The stabilizer of [ q ] is then isomor-phic to Aut ( F ) ∼ = (cid:81) i GL ( W i ) , and an element ( A , ..., A l ) ∈ (cid:81) i GL ( W i ) acts on the fiber L | [ q ] viamultiplication by(2.7.1) (cid:89) i det( A i ) χ ( u · [ F i ]) . Let v i = [ F i ] and r i = r ( F i ) . According to Corollary 2.4, for H not on a wall of type ( r, ∆) we getthat v i = r i r v for all i , and therefore, the exponents in (2.7.1) all vanish: χ ( u · v i ) = χ ( u · r i r v ) = 0 for u ∈ v ⊥ . It follows that GL ( V ) [ q ] acts trivially on L | [ q ] . (cid:3) When Y is a rational surface other than P , Yoshioka [Yos96a] analyzes the equivariant Picard groupof a subvariety of the Quot scheme parameterizing a certain family of O (0 , -prioritary sheaves (see thenext subsection for a review of prioritary sheaves) and proves the following result as a consequence ofthis analysis. Theorem 2.8 ([Yos96a, Corollary 3.4]) . Let Y be a rational surface other than P and let v = ( r, ν, ∆) ∈ K ( Y ) be a Chern character with ∆ > / . If H is a generic polarization with ( K Y + 2 F ) · H < and if M sH ( v ) is not empty, then the Donaldson homomorphism gives a surjection λ : v ⊥ (cid:16) Pic ( M H ( v )) . Note that Proposition 2.7 ensures that for a generic polarization H the Donaldson homomorphism isdefined as a map λ : v ⊥ → Pic ( M H ( v )) . Therefore, under these assumptions we have a bound on thePicard number of M H ( v ) and the computation of Pic ( M H ( v )) boils down to the computation of thekernel of the Donaldson homomorphism.2.5. Prioritary sheaves. It is often the case that the sheaves in a complete family obtained by consideringvarious resolutions and extensions enjoy an extra cohomological property which, in particular, makes theanalysis of the locus of semistable sheaves in the family much more tractable. D. PEDCHENKO Definition 2.9. Let L be a line bundle on a projective surface Y . A torsion-free sheaf V on Y is called L - prioritary if Ext ( V , V ⊗ L ∨ ) = 0 . Let D be an effective Cartier divisor on a projective surface Y . Denote the stack of torsion-freesheaves on Y and the stack of coherent sheaves on D with fixed numerical invariants by T F Y ( r, c , c ) and Coh D ( r, c · D ) respectively. The next result shows that the restriction of O ( D ) -prioritary sheavesfrom Y to D behaves nicely in families. Lemma 2.10. [Wal93, Lemma 4.] If V is an O ( D ) -prioritary sheaf, then the restriction map T F Y ( r, c , c ) → Coh D ( r, c · D ) is smooth (and therefore open) in a neighborhood of [ V ] . The quadric surface. We specialize some of the above discussion to the case X = P × P .The surface X comes with two natural projections to the P factors. Let F denote the class [ pr ∗ ( pt )] and E denote the class [ pr ∗ ( pt )] . The Picard group of X and the intersection pairing is then given byPic ( X ) = Z E ⊕ Z F, E = F = 0 , E · F = 1 . The canonical class of X is K X = − E − F. A divisor class H = aE + bF is ample if and only if a, b > . For m ∈ Q , we consider the Q -divisor class H m = E + mF. Note that every ample divisor on X is an integer multiple of some H m with m > .For character v = ( r, ν, ∆) = ( r, εE + ϕF, ∆) on X = P × P , the Riemann-Roch Theorem gives χ ( v ) = r ( P ( ν ) − ∆) , where P ( ν ) = ( ε + 1)( ϕ + 1) . Given two sheaves V , W , let ext i ( V , W ) denote dim Ext i ( V , W ) . The Riemann-Roch Theorem says that χ ( V , W ) = (cid:88) i =0 ( − i ext i ( V , W ) = r ( V ) r ( W )( P ( ν ( W ) − ν ( V )) − ∆( V ) − ∆( W )) . Note that on X with an ample divisor H every H -semistable sheaf V of character v is both O (1 , -and O (0 , -prioritary: Ext ( V , V ( − , − Hom ( V , V ( − , − ∨ = 0 , Ext ( V , V (0 , − Hom ( V , V ( − , − ∨ = 0 by Serre duality and semistability. Thus, if we denote the stack of L -prioritary sheaves by P L ( v ) , thenwe have a chain of open substacks M H ( v ) ⊂ P O (1 , ( v ) ⊂ P O (0 , ( v ) . Walter’s Theorem [Wal93] asserts that the stack P O (0 , ( v ) is irreducible and smooth (if nonempty).This implies that the moduli space M H ( v ) is irreducible as well. Furthermore, if r ( v ) ≥ , then thegeneral member of P O (0 , ( v ) is locally free. Additionally, Walter shows that M H ( v ) is unirational.We also have the following useful result of Yoshioka. HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 9 Theorem 2.11 ([Yos95, Proposition 5.1]) . Let v = ( r, ν, ∆) = ( r, c , χ ) ∈ K ( X ) be a primitive H -semistable Chern character.If the polarization H satisfies gcd( r, c · H, χ ) = 1 , then Pic ( M H ( v )) is torsion-free. In this paper, we will be concerned with the calculation of the Picard group of the moduli space M H m ( v ) of H m -semistable sheaves on X when m ∈ Q is sufficiently close to : H m = E + mF, m = 1 + (cid:15), < | (cid:15) | (cid:28) . The reason for doing so is twofold. On the one hand, as we explained above the genericity assumptionmakes M H m ( v ) into a locally factorial variety with a known bound on the Picard number. On theother hand, in the next subsection we recall that when H m is close to the anticanonical class, thereis a complete classification of Chern characters v for which the moduli space M H m ( v ) is nonempty orpositive-dimensional.2.7. Exceptional bundles and existence of semistable sheaves. Let X = P × P polarized by anample divisor H . The question of when M H ( v ) is nonempty was studied by Rudakov in [Rud94] andCoskun and Huizenga in [CH19] (where they studied the existence question for all Hirzebruch surfaces).We follow [CH19] in this subsection. Definition 2.12. A sheaf V on X is(1) simple , if Hom ( V , V ) = C ;(2) rigid , if Ext ( V , V ) = 0 ;(3) exceptional , if it is simple, rigid, and Ext ( V , V ) = 0 ;(4) semiexceptional , if it is a direct sum of copies of an exceptional sheaf.We call a character v ∈ K ( X ) of positive rank potentially exceptional if χ ( v , v ) = 1 , and (semi)exceptional if there is a (semi)exceptional torsion-free sheaf of character v . We also say that char-acter v is H - (semi)stable (resp. µ H - (semi)stable ) if there is an H -(semi)stable (resp., µ H -(semi)stable)sheaf of character v .Any exceptional torsion-free sheaf is locally free and µ − K X -stable by [Gor89] and therefore, remains µ H m -stable for m ∈ Q sufficiently close to by the openness of slope stability in the polarization. Wereproduce a part of [CH19, Lemma 6.7] that further characterizes (potentially) exceptional bundles andcharacters. Lemma 2.13 ([CH19, Lemma 6.7]) . Let v ∈ K ( X ) be a potentially exceptional character of rank r . (1) The rank of v is odd and the discriminant of v is ∆ = 12 − r . (2) The character v is primitive. (3) If m is generic and V is an H m -semistable sheaf of discriminant ∆( V ) < , then V is semiex-ceptional. Heuristically, µ H -stable exceptional bundles give strong bounds on the possible numerical invariantsof µ H -semistable sheaves. In particular, if E is a µ H -stable exceptional bundle and V is a µ H -semistablesheaf with K X · H ≤ µ H ( V ) − µ H ( E ) < , then Hom ( E, V ) = 0 and Ext ( E, V ) = Hom ( V , E ( K X )) ∨ = 0 by semistability and Serre duality. Therefore, χ ( E, V ) ≤ . By the Riemann-Roch Formula, this inequalitycan be viewed as a lower bound on ∆( V ) : ∆( V ) ≥ P ( ν ( V ) − ν ( E )) − ∆( E ) . Likewise, if instead < µ H ( V ) − µ H ( E ) ≤ − K X · H, then the inequality χ ( V , E ) ≤ provides a lower bound ∆( V ) ≥ P ( ν ( E ) − ν ( V )) − ∆( E ) on ∆( V ) .This motivates the following definition. Definition 2.14 ([CH19, Definition 6.13]) . For a µ H -stable exceptional bundle E , define a functionDLP H,E ( ν ) = P ( ν − ν ( E )) − ∆( E ) if K X · H ≤ ( ν − ν ( E )) · H < P ( ν ( E ) − ν ) − ∆( E ) if < ( ν − ν ( E )) · H ≤ − K X · H max { P ( ± ( ν − ν ( E ))) − ∆( E ) } if ( ν − ν ( E )) · H = 0 on the strip of slopes ν = εE + ϕF = ( ε, ϕ ) ∈ Q satisfying | ( ν − ν ( E )) · H | ≤ − K X · H. Let E H be the set of µ H -stable exceptional bundles on X . Further define a functionDLP H,E ( ν ) , where this time ν = ( ε, ϕ ) could be any point in Q . We refer to the above functions as the Drézet-Le-Potier functions , or DLP- functions , for short.One can see the graph of ∆ = DLP Top-down view. Reproducedfrom [CH19, Figure 5]. (b) View from the side Figure 1. The DLP 1) + 1 . This shows that the expected dimension is positive if and only if ∆ ≥ / . Lemma 2.13 (1) implies thatsuch characters are not semiexceptional.Next, we introduce useful terminology describing the position of character v relative to the DLP Let v = ( r, ν, ∆) ∈ K ( X ) be an H m -semistable character with ∆ ≥ and ∆ = DLP Position of v relative to the DLP These resolutions are special resolutions of sheaves on X by direct sumsof line bundles. Their advantage is that they are simple enough to work with and provide unirationalparameterizations of moduli spaces of sheaves. Gaeta-type resolutions were studied in [CH18] for allHirzebruch surfaces F e , but we will only need the case X = F = P × P . Definition 2.18. Let L be a line bundle on X . An L - Gaeta-type resolution of a sheaf V on X is aresolution of V of the form(2.18.1) → L ( − , − α → L ( − , β ⊕ L (0 , − γ ⊕ L δ → V → where α, β, γ, δ are nonnegative integers. We say a sheaf V has a Gaeta-type resolution if it admits an L -Gaeta-type resolution for some line bundle L .The results of [CH18, §4] we will need are summarized in the following statement. Theorem 2.19. If v is a µ H -semistable Chern character with ∆( v ) ≥ , then a general µ H -semistablesheaf V admits a Gaeta-type resolution. More specifically: HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 13 (1) Suppose L is a line bundle such that the inequalities (2.19.1) α = − χ ( v ⊗ L ∨ ( − , − ≥ β = − χ ( v ⊗ L ∨ ( − , ≥ γ = − χ ( v ⊗ L ∨ (0 , − ≥ δ = χ ( v ⊗ L ∨ ) ≥ are satisfied. Then not all of the integers in (2.19.1) are zero and a general µ H -semistable sheaf V admits an L -Gaeta-type resolution with integers in (2.19.1) giving the exponents in (2.18.1) . (2) A line bundle satisfying inequalities (2.19.1) always exists. Gaeta-type resolutions allow us to build complete families of O (1 , -prioritary sheaves. Proposition 2.20. Let α, β, γ, δ be nonnegative integers satisfying r := β + γ + δ − α > . For a line bundle L consider the open subset U ⊂ H := Hom (cid:16) L ( − , − α , L ( − , β ⊕ L (0 , − γ ⊕ L δ (cid:17) parameterizing injective sheaf maps with torsion-free cokernel. For ψ u ∈ U , let V u be the cokernel: → L ( − , − α ψ u −−→ L ( − , β ⊕ L (0 , − γ ⊕ L δ → V u → . If r ≥ , then U is nonempty, codim H ( H \ U ) ≥ , and the family V u /U is a complete family of O (1 , -prioritary sheaves.Proof. Only the statement about the codimension requires a proof as the other statements are provedin [CH18, Theorem 2.10].The statement about the codimension follows from the standard analysis of the incidence correspon-dence Σ := { ( p, ψ ) | ψ | p is not injective } ⊂ X × Hom (cid:16) L ( − , − α , L ( − , β ⊕ L (0 , − γ ⊕ L δ (cid:17) using the fact that H om (cid:16) L ( − , − α , L ( − , β ⊕ L (0 , − γ ⊕ L δ (cid:17) is a globally generated vector bundle. See [LP97, pp. 238-239] and the proof of [DLP85, Theorem 4.7]for details. (cid:3) We finish this subsection by introducing the "dual version" of a Gaeta-type resolution. Specifically,this is a resolution of the form(2.20.1) → V → L (1 , α ⊕ L (0 , β ⊕ L γ → L (1 , δ → . We have the following analogue of Proposition 2.20. Proposition 2.21. Let α, β, γ, δ be nonnegative integers satisfying r := α + β + γ − δ > . For a line bundle L consider the open subset U ⊂ H := Hom (cid:16) L (1 , α ⊕ L (0 , β ⊕ L γ , L (1 , δ (cid:17) parameterizing surjective sheaf maps. For ψ u ∈ U , let V u be the kernel: → V u → L (1 , α ⊕ L (0 , β ⊕ L γ ψ u −−→ L (1 , δ → . If r ≥ , then U is nonempty, codim H ( H \ U ) ≥ , and the family V u /U is a complete family of O (1 , -prioritary vector bundles. Study of the Shatz stratification In this section we use the techniques from [DLP85, §1 & 3], [LP97, Chapter 15] and [HL10, §2.A] todetect strata of unstable sheaves of codimension one in complete families.3.1. Generalities on the Shatz stratifiction. Given a complete family V t /T of torsion-free sheavesof character v , we denote by S H ( P , P , ..., P l ) ⊂ T the Shatz stratum parameterizing sheaves V t with H -Harder-Narasimhan filtration having quotients with H -Hilbert polynomials P , P , ..., P l . If onefurther assumes that T is smooth and for each t ∈ T we have Ext ( V t , V t ) = 0 , then the Shatzstratum S H ( P , P , ..., P l ) is a smooth locally closed subvariety of T with the normal space at point t ∈ S H ( P , P , .., P l ) given by Ext ( V t , V t ) . We refer the reader to [DLP85, §1] for the definitionof Ext , Ext − and the general deformation theory of filtered sheaves. We instead review here thecomputational aspects.For t ∈ S H ( P , P , ..., P l ) equip V t with its H -Harder-Narasimhan filtration with quotientsgr ,t , ..., gr l,t . Then there is a spectral sequence with E -term given by E p,q = (cid:40)(cid:76) i Ext p + q ( gr i,t , gr i − p,t ) if p < if p ≥ which abuts on Ext p + q + ( V t , V t ) in degree p + q . Similarly, there is a spectral sequence with E -term givenby E p,q = (cid:40) if p < (cid:76) i Ext p + q ( gr i,t , gr i − p,t ) if p ≥ which abuts on Ext p + q − ( V t , V t ) in degree p + q .For our purposes, it would be convenient to work with a slightly refined notion of a Shatz stratum.Note that since Pic ( X ) is a discrete algebraic group scheme, for points t within a connected compo-nent of a Shatz stratum S H ( P , P , ..., P l ) the H -Harder-Narasimhan quotients gr ,t , gr ,t , ..., gr l,t of V t not only have the same H -Hilbert polynomials P , P , ..., P l , but also the same numerical invari-ants v , v , ..., v l . Thus, S H ( P , P , ..., P l ) breaks up into a disjoint union of strata S H ( v , v , ..., v l ) ,where each S H ( v , v , ..., v l ) parameterizes sheaves V t with H -Harder-Narasimhan filtration havingquotients with numerical invariants v , v , ..., v l . Later, when we use the notion of Shatz stratumwe will have S H ( v , v , ..., v l ) in mind instead of S H ( P , P , ..., P l ) . The discussion above applies to S H ( v , v , ..., v l ) equally well, and we conclude that when V t /T is a smooth complete family of torsion-free sheaves satisfying Ext ( V t , V t ) = 0 for each t ∈ T the stratum S = S H ( v , v , ..., v l ) is a smoothlocally closed subvariety of T with the normal space at point t described as N S/T | t ∼ = Ext ( V t , V t ) . ∆ i = -strata. Before we proceed with the estimates, let us introduce one more definition. Carefulreading of [LP97, Lemma 18.3.1], [Dre88, Proposition 2.4] and [DLP85, Lemma 4.8] suggests that in the P case the codimension one Shatz strata occur in complete families of O P (1) -prioritary sheaves onlyfor characters v on the DLP P -curve and correspond to sheaves whose first or last Harder-Narasimhanquotient is semiexceptional. The next definition is created ad hoc to capture new codimension one Shatzstrata which did not exist in the P case, but which appear in the P × P case due to the presence ofsemistable Chern characters of discriminant . HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 15 Definition 3.1. Let v = ( r, ν, ∆) be an integral Chern character on X = P × P . Let V t /T be acomplete family of torsion-free sheaves parameterized by a smooth variety T with v ( V t ) = v . We callShatz stratum S ⊂ T a ∆ i = - stratum if S parameterizes sheaves V t with the H -Harder-Narasimhanfiltration of length l = 2 , S = S H ( v , v ) , such that the numerical invariants v = ( r , ν , ∆ ) , v = ( r , ν , ∆ ) of the H -Harder-Narasimhan quotients of V t satisfy the following properties:(1) ∆ , ∆ ≥ with at least one ∆ i = , i = 1 , ,(2) ν − ν = kr r E − kr r F for some integer k with < | k | ≤ r r ,(3) χ ( v , v ) = − .3.3. Codimension of Shatz strata. In this subsection, we present a study of Shatz stratification througha numerical analysis involving Riemann-Roch computations.For the rest of this subsection we adopt the following convention. Consider a family V t /T of sheavesparameterized by a variety T . Suppose V t belongs to a Shatz stratum S H m ( v , v , ..., v l ) ⊂ T with H m -Harder-Narasimhan quotients gr ,t , gr ,t , ..., gr l,t having numerical invariants v , v , ..., v l . To improvereadability we drop the subscript t in gr i,t if any confusion is unlikely. We further write v i = v ( gr i ) = ( r i , ν i , ∆ i ) . We start with a couple of preparatory lemmas. Lemma 3.2. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ . Let (cid:15) ∈ Q be a sufficiently smallnumber (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .Consider a complete family V t /T of O (1 , -prioritary sheaves with v ( V t ) = v parameterized by asmooth variety T . If V t belongs to the Shatz stratum S = S H m ( v , v , ..., v l ) and the inequalities (3.2.1) µ max ,H ( V t ) − µ min ,H ( V t ) ≤ are satisfied for H = O (1 , , O (1 , and O (0 , , then | ( ν − ν i ) · H m | ≤ − K X · H m and codim T,t ( S ) = − (cid:88) i By the above inequalities (6.4.1), for any subsheaf W ⊂ V t and any quotient V t (cid:16) E of V t thedifference of the total slopes ν ( W ) − ν ( E ) lies in a bounded region of the ( ε, ϕ ) -plane of total slopes. Furthermore, inequality (6.4.1) for H = O (1 , implies | ( ν − ν i ) · H | < − K X · H . It follows that since m is close enough to , we have(3.2.2) | ( ν − ν i ) · H m | ≤ − K X · H m . Let gr , gr , ..., gr l be the quotients in the H m -Harder-Narasimhan filtration of V t . Sincecodim T,t ( S ) = dim N S/T | t = ext ( V t , V t ) , we will use the spectral sequences for Ext • + ( V t , V t ) and Ext •− ( V t , V t ) from §3.1 to compute ext ( V t , V t ) .Since Hom ( gr i , gr j ) = 0 for i < j by semistability, we see that Ext ( V t , V t ) = 0 . Likewise, by ourbound (3.2.2) and semistability we haveExt ( gr i , gr j ) ∼ = Hom ( gr j , gr i ⊗ K X ) ∨ = 0 for any i, j, so both Ext ( V t , V t ) = 0 and Ext − ( V t , V t ) = 0 . Therefore, the only nonzero terms in the spectralsequence for Ext p + q + ( V t , V t ) have p + q = 1 . We concludeext ( V t , V t ) = (cid:88) i Let V t /T be a complete family of O (1 , -prioritary sheaves parameterized by a smoothvariety T . Let H be one of line bundles O (1 , , O (1 , or O (0 , . Then the set of points t ∈ T suchthat µ max ,H ( V t ) − µ min ,H ( V t ) > is a closed subset of codimension at least in T .Proof. To show the result for H = O (1 , , one follows the proof of [LP97, Corollary 15.4.4.], replacinga line d on P with a rational curve from the complete linear series |O (1 , | on X = P × P and usingLemma 2.10 together with the O (1 , -prioritariness of sheaves in the family.For H = O (1 , or O (0 , , we recall that O (1 , -prioritariness implies both O (1 , - and O (0 , -prioritariness (see [CH19, Lemma 3.1]). The same argument as above applies in this case too. (cid:3) The next two propositions describe codimension one Shatz strata in complete families of O (1 , -prioritary sheaves. Proposition 3.4. Let v = ( r, ν, ∆) be a Chern character satisfying ∆ > DLP Step 1. We start by making some preliminary reductions. By Lemma 3.3 we can pass to an opensubset of points t ∈ T where(3.4.1) µ max ,H ( V t ) − µ min ,H ( V t ) ≤ , for H = O (1 , , O (1 , and O (0 , .Suppose S := S H m ( v , v , ..., v l ) ⊂ T is a nonempty Shatz stratum of codimension in T . ByLemma 3.2 for t ∈ S we have(3.4.2) | ( ν − ν i ) · H m | ≤ − K X · H m andcodim T,t ( S ) = − (cid:88) i Below, we analyze various possibilities for what the numerical invariants v , v , ..., v l of the H m -Harder-Narasimhan quotients gr , gr , ..., gr l of V t could be and show that they must necessarily satisfyconditions (1)-(3) of Definition 3.1, i.e. S must be a ∆ i = -stratum. Step 2. Suppose that χ ( gr , gr l ) = 0 holds. By the Riemann-Roch Theorem, this gives ∆ + ∆ l = P ( ν l − ν ) . Since m is sufficiently close to , we have that P ( ν l − ν ) ≤ with equality holding only when ν = ν l .If ν (cid:54) = ν l , we get that ∆ + ∆ l = P ( ν l − ν ) < , and therefore ∆ < or ∆ l < .If ν = ν l , then since gr i are the quotients in the Harder-Narasimhan filtration, we must have that ∆ l > ∆ . Since P ( ν l − ν ) = 1 in the case, we get ∆ < .In both of these cases, Lemma 2.13 (3) implies that gr or gr l is semiexceptional and we can followthe argument of [DLP85, Lemma 4.8]. Here we deal with the case where gr is semiexceptional. Theargument for when gr l is semiexceptional is similar. Write gr ∼ = E k with exceptional bundle E . Thenwe get χ ( v , v ) = χ ( E k , V t ) = χ ( gr , V t ) = χ ( gr , gr ) + (cid:88) Now we know χ ( gr , gr l ) = − . If one of gr , gr l is semiexceptional, we arrive to a contra-diction in the same way as above. At this point, we have shown that for points t ∈ S the H m -Harder-Narasimhan quotients of V t satisfy χ ( gr , gr l ) = − and ∆ , ∆ l ≥ .Next, we show that l = 2 . Assume on the contrary that we have l ≥ . Since χ ( gr i , gr j ) = 0 for i < j, ( i, j ) (cid:54) = (1 , l ) , we in particular have χ ( gr , gr i ) = 0 = ⇒ P ( ν i − ν ) − ∆ − ∆ i = 0 for < i < l . Since P ( ν i − ν ) ≤ and we cannot have ν = ν i , ∆ = ∆ i , we get that ∆ i < and gr i is semiexceptional, gr i ∼ = E k . If µ H m ( gr i ) ≥ µ H m ( V t ) , then for a semistable V of character v we have χ ( v i , v ) = χ ( gr i , V ) = k · χ ( E, V ) ≤ , because ∆ ≥ DLP So from now on we use that l = 2 and ∆ , ∆ ≥ . Expanding χ ( gr , gr ) = − by theRiemann-Roch Theorem we get(3.4.3) ∆ + ∆ = P ( ν − ν ) + 1 r r . Below we eliminate various cases for what the values of ∆ i and ν − ν could be. Case 1. Suppose that ∆ , ∆ > . Here we follow the method in [Dre88, Proposition 2.4]. In thiscase, the expected dimension of the moduli spaces M ( v i ) for i = 1 , is exp dim M ( v i ) = r i (2∆ i − 1) + 1 ≥ , which allows us to write(3.4.4) ∆ i ≥ 12 + 12 r i for i = 1 , . Using this estimate in equation (3.4.3), we get P ( ν − ν ) + 1 r r ≥ (cid:18) r + 1 r (cid:19) , which simplifies to − P ( ν − ν ) + 12 (cid:18) r − r (cid:19) ≤ . Since P ( ν − ν ) ≤ , we in fact have − P ( ν − ν ) = 1 r − r = 0 , and r = r , ν = ν . Comparing equations (3.4.3) and (3.4.4), we get ∆ = ∆ . This is a contradiction because gr and gr are quotients in the Harder-Narasimhan filtration.Next we rule out the cases where one of ∆ , ∆ is equal to , but condition (2) of Definition 3.1 doesnot hold. We consider the case when ∆ = , ∆ ≥ ; the case ∆ ≥ , ∆ = is dealt with similarly. Case 2. Assume ∆ = , ∆ ≥ , ν = ν . Since ∆ = , the rank r must be even: r = 2¯ r .Equation (3.4.3) gives(3.4.5) ∆ = 12 + 12¯ r r = ¯ r r + 12¯ r r . If one of ¯ r or r is even, then the right hand side of equation (3.4.5) is an irreducible fraction. Thismeans that after cancelling all the common factors in the numerator and the denominator of(3.4.6) ∆ = c ( gr ) r − r − r · c ( gr ) c ( gr ) r − ( r − · ( c ( gr ) / r the resulting denominator should be equal to r r = r r . This implies that r = kr and character ( r , ν = ν , ) is equal to k · v , so it is an integral Chern character. We can, therefore, write ∆ = ¯ r r + 12¯ r r = 12 + Nr for some integer N . This gives equalities ¯ r r + 1 = ¯ r r + 2 N ¯ r ⇐⇒ N ¯ r = 1 , HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 19 which is impossible.Now assume both ¯ r and r are odd. Then we can cancel in the numerator and the denominatorof (3.4.5) and the result will be an irreducible fraction with denominator ¯ r r . This time, we see fromequation (3.4.6) that ¯ r divides r , so we can write r = ¯ r d for an odd integer d . Since both χ ( gr ) and χ ( gr ) are integers, we have χ ( gr ) = r · (cid:18) P ( ν ) − (cid:19) = 2¯ r P ( ν ) − ¯ r = 2¯ r P ( ν ) − ¯ r ∈ Z ( recall ν = ν ) ,χ ( gr ) = r (cid:18) P ( ν ) − − r r (cid:19) = ¯ r dP ( ν ) − ¯ r d − r = ¯ r d (2¯ r P ( ν ) − ¯ r ) − r == ¯ r d · χ ( gr ) − r ∈ Z . The last expression implies ¯ r = 1 and we get that v = (2 , ν , ) . Now, since r is odd, we canwrite ν = ν = ∗ odd E + ∗ odd F with both coefficients being irreducible fractions. But explicit analysisof the DLP We turn our attention to those cases where ∆ = , ∆ ≥ and ν (cid:54) = ν . We can explicitlywrite ν − ν = (cid:18) a r − a r (cid:19) E + (cid:18) b r − b r (cid:19) F = ar r E + br r F, a, a i , b, b i ∈ Z , so that P ( ν − ν ) = (cid:18) ar r (cid:19) (cid:18) br r (cid:19) = 1 + a + br r + ab ( r r ) . Further cases depend on the values of a and b . Note that by the definition of the Harder-Narasimhanfiltration, we cannot have a > and b > simultaneously. Moreover, since ∆ ≥ , we get an inequality(3.4.7) a + b + 1 r r + ab ( r r ) = P ( ν − ν ) + 1 r r = ∆ + ∆ = 12 + ∆ ≥ , which we use to eliminate certain potential values of a and b . Case 3.1. Assume a < , b < . Rewrite (3.4.7) as(3.4.8) ab ( r r ) ≥ − a − b − r r ⇐⇒ ab ≥ ( − a − b − r r ⇐⇒ a ( b + r r ) ≥ ( − b − r r . If b + r r < , we get a ≤ − b − b + r r r r = − b − − b − r r ( − r r ) ≤ − r r , so that a + r r ≤ . But then µ H ( gr ) − µ H ( gr ) = − a − br r > , contradicting (3.4.1).If b + r r = 0 , then, since r r ≥ , the last inequality in (3.4.8) reads as a ( l + r r ) ≥ ( − b − r r l > , which is a contradiction.If ≤ b + r r ≤ r r − , the last inequality in (3.4.8) reads as a ≥ − b − b + r r r r ≥ , now contradicting the assumption that a < . Case 3.2. Assume a < , b > , | a | > b or a > , b < , | b | > a . We estimate the left hand side in(3.4.7): a + b + 1 r r + ab ( r r ) ≤ ab ( r r ) < , and the necessary condition (3.4.7) does not hold. Case 3.3. Assume a = 0 , b < or a < , b = 0 . Since the calculations are symmetric in a and b , weonly treat a = 0 , b < . The left hand side of (3.4.7) now reads as b + 1 r r . If b ≤ − , then again condition (3.4.7) does not hold.If b = − , we get from (3.4.7) ∆ + ∆ = P ( ν − ν ) + 1 r r = 1 + b + 1 r r = 1 , which gives ∆ = ∆ = . This implies that now both r and r are even, r = 2¯ r , r = 2¯ r , and ν − ν = (cid:18) a r − a r (cid:19) E + (cid:18) b r − b r (cid:19) F = (cid:18) r a − r a r r (cid:19) E + (cid:18) r b − r b r r (cid:19) F, which is never equal to ar r E + br r F = − r r F. Case 3.4. At this point observe that we have ruled out all cases for possible values of a and b , exceptfor the case a < , b > , | a | = b or a > , b < , a = | b | .If | a | = | b | > r r , then inequality (3.4.7) does not hold.Finally, if | a | = | b | ≤ r r , then this case corresponds precisely to conditions (1)-(3) of Definition 3.1.This shows that the nonempty Shatz stratum S of codimension in T must be a ∆ i = -stratum. (cid:3) The proof of Proposition 3.4 can be readily modified to give an analogous statement for characters v = ( r, ν, ∆) , which lie on a single branch of the Drézet-Le Potier surface. Proposition 3.5. Let v = ( r, ν, ∆) be a Chern satisfying ∆ = DLP Proof. Suppose S := S H m ( v , v , ..., v l ) is a Shatz stratum of codimension . Repeating step 1 of theproof of Proposition 3.4, we have for a point point t ∈ S (3.5.1) | ( ν − ν i ) · H m | ≤ − K X · H m andcodim T,t ( S ) = − (cid:88) i Definition 3.6. Let v = ( r, ν, ∆) ∈ K ( X ) be an H m -semistable Chern character, where m = 1 + (cid:15) fora sufficiently small (depending on r ) number (cid:15) ∈ Q , < | (cid:15) | (cid:28) .We call v a bad character if we can find a decomposition v = v + v , where v , v are H m -semistable Chern characters satisfying(1) p H m , v > p H m , v , where p H m , v i is the reduced H m -Hilbert polynomial of v i ,(2) ∆ , ∆ ≥ with at least one ∆ i = , i = 1 , ,(3) ν − ν = kr r E − kr r F, for some integer k with < | k | ≤ r r ,(4) χ ( v , v ) = − . Otherwise, we call an H m -semistable Chern character v a good character. Remark 3.7. Note that a bad character v is always primitive. Indeed, if ∆ = , then χ ( v , v ) = χ ( v , v + v ) = χ ( v , v ) = − . If instead ∆ = , then χ ( v , v ) = − .The point of this notion is that by Definition 3.1 and Propositions 3.4, 3.5 for good H m -semistablecharacters ∆ i = -strata do not appear in smooth complete families of O (1 , -prioritary sheaves. Thisway, for good characters the study of the Shatz stratification yields results that are similar to the P case.On the other hand, when v is a bad Chern character, we get a potentially nonempty divisorial ∆ i = -stratum S H m ( v , v ) in smooth complete families of O (1 , -prioritary sheaves for every decomposition v = v + v as in Definition 3.6.To demonstrate this phenomenon we give an example of a bad Chern character v and a smoothcomplete family of O (1 , -prioritary sheaves of character v for which a ∆ i = -stratum is nonempty. HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 23 Example 3.8. Consider the character v = ( r, ν, ∆) = (4 , − E − F, ) . We have ∆( v ) = 916 = DLP < H m (cid:18) − E − F (cid:19) = DLP H m , O (cid:18) − E − F (cid:19) for m = 1 + ε, < ε (cid:28) , so that v is H m -semistable and the line bundle O is associated to v . One checks that conditions ofDefinition 3.6 are met for the decomposition v = v + v , v = (2 , − F, 12 ) , v = (2 , − E, 12 ) , where the semistability of v , v follows fromDLP < H m (cid:18) − E (cid:19) = DLP < H m (cid:18) − F (cid:19) = 12 . This shows that v is an example of a bad Chern character.The Beilinson-type spectral sequence (see [Dré91, Proposition 5.1]) allows one to resolve any µ H m -semistable sheaf V of character v as(3.8.1) → O ( − , − → O ( − , ⊕ O (0 , − → V → . Note that this is precisely the L -Gaeta type resolution (2.18.1) with L = O . Thus we consider the family V t /T of O (1 , -prioritary sheaves admitting an O -Gaeta type resolution → O ( − , − ψ t −→ O ( − , ⊕ O (0 , − → V t → , where T ⊂ H = Hom (cid:0) O ( − , − , O ( − , ⊕ O (0 , − (cid:1) is the open subset parameterizing injective sheaf maps with torsion-free cokernel. By Proposition 2.20,the subset T is not empty, codim H ( H \ T ) ≥ and the family V t /T is complete. We conclude that any H m -semistable V ∈ M H m ( v ) is equal to some V t for t ∈ T .We demonstrate that S H m ( v , v ) is nonempty in this complete family as follows. Note that we alsohave DLP < H (cid:18) − E (cid:19) = DLP < H (cid:18) − F (cid:19) = 12 . We then take F ∈ M H (cid:18) , − F, (cid:19) and F ∈ M H (cid:18) , − E, (cid:19) , and consider their direct sum F ⊕ F , which is a µ H -semistable sheaf. Since the Beilinson-type spectral sequence is insensitive to smallvariations in the polarization, this sheaf is still resolved by (3.8.1) and, therefore, appears as V τ for some τ ∈ T . For the H m -polarization it is, however, no longer semistable: µ H m ( F ) > µ H m ( F ⊕ F ) . Note that F and F are in fact µ H -stable ( v i is primitive), and since slope stability is open in thepolarization, they remain µ H m -stable by our choice of H m . It follows that the H m -Harder-Narasimhanfiltration of V τ is ⊂ F ⊂ V τ , so that V τ belongs to S H m ( v , v ) and this divisorial ∆ i = -stratum is nonempty in T . Example 3.9. Generalizing the previous example, we can generate an infinite sequence of bad H m -semistable Chern characters w k such that analogous complete families of O (1 , -prioritary sheaves ofcharacter w k arising from O -Gaeta type resolutions all contain a non-empty ∆ i = -stratum.Set w := v = v + v , where v , v , v are the characters from the previous example. Inductivelydefine(3.9.1) w k := v + w k − for k ≥ . One inductively checks that for m = 1 + ε, < ε (cid:28) , the character w k is H m -semistable and allthe conditions of Definition 3.6 are satisfied for the decomposition of w k as in (3.9.1). Below we listcharacters w k = ( r k , ν k , ∆ k ) for small values of k and plot their total slopes in the ( ε, ϕ ) -plane alongwith the top-down projection of various branches of the DLP-surface (compare to Figure 1 (A)): k ( r k , ν k , ∆ k )1 (4 , − / E − / F, / , − / E − / F, / , − / E − / F, / , − / E − / F, / , − / E − / F, / As in the previous example, the same Beilinson-type spectral sequence ([Dré91, Proposition 5.1])allows one to resolve any µ H m -semistable sheaf W of character w k as → O ( − , − α → O ( − , β ⊕ O (0 , − γ → W → for some positive integers α, β, γ . Arguing as above, one considers the complete family W t /T of O (1 , -prioritary sheaves of character w k admitting an O -Gaeta type resolution → O ( − , − α ψ t −→ O ( − , β ⊕ O (0 , − γ → W t → , where T ⊂ H = Hom (cid:16) O ( − , − α , O ( − , β ⊕ O (0 , − γ (cid:17) is the open subset parameterizing injective sheaf maps, and one shows that the ∆ i = -stratum S H m ( v , w k − ) is nonempty.Finally, note that χ ( O , v ) = χ ( O , v ) = 0 = ⇒ χ ( O , w k ) = 0 and ∆( w k ) = DLP Example 3.10. Furthermore, we can repeat the constructions of the previous two examples starting withany pair of of primitive characters of the form v = ( r, ϕE + εF, ) , v = ( r, εE + ϕF, ) that lie onthe branch of the DLP-surface given by the line bundle O µ H m ( O ) > µ H m ( v i ) , DLP 2) (2 , − / E, − / 2) (4 , − / E − / F, / , − / E − / F, / 2) (12 , − / E − / F, / 2) (24 , − / E − / F, / , − / E − / F, / 2) (70 , − / E − / E, / 2) (140 , − / E − / F, / , − / E − / F, / 2) (408 , − / E − / F, / 2) (816 , − / E − / F, / Further still, one can easily replace the line bundle O by an arbitrary line bundle L and generateanalogous infinite sequences of bad characters lying on the branch of the DLP-surface given by L . Question 3.11. Note that the characters from the previous three examples lie on a branch of theDLP Let v , v , ..., v l be H -semistable Chern characters with p H, v > p H, v > ... > p H, v l . We conclude this section by discussing how to build an irreducible family of sheaves containing all torsion-free sheaves whose quotients in the H -Harder-Narasimhan filtration have invariants v , v , ..., v l . Welater use these results to show the irreducibility of Shatz strata in certain complete families of O (1 , -prioritary sheaves. The statements of this subsection are briefly mentioned in [Yos96b] without proof,and the outline of the proof of Proposition 3.13 was communicated to us by Yoshioka directly (also seethe Appendix to [Yos95] for some similar constructions).Given H -semistable Chern characters v , v , ..., v l with p H, v > p H, v > ... > p H, v l , consider the family F ( v , v , ..., v l ) of isomorphism classes of torsion-free sheaves V whose H -Harder-Narasimhan filtration ⊂ F ⊂ F ⊂ ... ⊂ F l = V is of length l and whose quotients satisfy v ( gr i ) = v i . Note that when l = 1 , F ( v ) is just the family ofisomorphism classes of H -semistable sheaves with Chern character v .We first recall how to construct irreducible families for H -semistable sheaves of Chern character v . Lemma 3.12. Let v = ( r, ν, ∆) be an H -semistable Chern character. Then there exists a family V s /S of sheaves over an irreducible base S with the following property: (P) V s ∈ F ( v ) for every s ∈ S, and for any V ∈ F ( v ) there exists s ∈ S with V = V s . Proof. When r = 1 , the moduli space M H ( v ) for v = (1 , ν, n ) is a fine moduli space with a universalfamily U . For v = (1 , ν, n ) the moduli space M H ( v ) is isomorphic to the Hilbert scheme X [ n ] of n points on X . Therefore, it is irreducible and we take S := M H ( v ) .When r ≥ , take S to be an open subset of the Quot scheme parameterizing H -semistable quotients V ⊗ O ( − N H m ) (cid:16) V for N (cid:29) as in the GIT construction of M H ( v ) ([HL10, paragraph 4.3]). Waltershows in the proof of [Wal93, Theorem 1] that S is irreducible as a consequence of his more general resultwhich says that the stack of O (0 , -prioritary sheaves is irreducible (see our discussion in §2.6). (cid:3) Now we prove the analogous result for the family F ( v , v , ..., v l ) . Proposition 3.13. Let v , v , ..., v l be H -semistable Chern characters with (3.13.1) p H, v > p H, v > ... > p H, v l . Then there exists a family V s /S of sheaves over an irreducible base S with the following property: (P) V s ∈ F ( v , v , ..., v l ) for every s ∈ S, and for any V ∈ F ( v , v , ..., v l ) there exists s ∈ S with V = V s . Proof. We use induction on l . Case l = 1 is Lemma 3.12.For l ≥ , take V ∈ F ( v , v , ..., v l ) . It fits into a short exact sequence(3.13.2) → F → V → E → with F ∈ F ( v ) and E ∈ F ( v , ..., v l ) . By the induction assumption we have a family F t /T over anirreducible base T satisfying Property (P) with respect to F ( v ) , and a family E r /R over an irreduciblebase R satisfying Property (P) with respect to F ( v , ..., v l ) . Intuitively, we want to build S by taking allpossible extensions of E r by F t for all possible t ∈ T and r ∈ R . However, since ext ( E r , F t ) may notbe constant for different t ∈ T, r ∈ R , we will have to enlarge S in a certain sense.To this end, since by the induction assumption F ( v ) and F ( v , ..., v l ) are bounded families, we canchoose N (cid:29) so that H i ( X, E r ( N H m )) = 0 for i > and all r ∈ R and(3.13.3) H i ( X, F t ( N H m )) = 0 for i > and all t ∈ T. Taking V to be a vector space of dimension h ( X, E r ( N H m )) , we have a surjection(3.13.4) V ⊗ O ( − N H m ) (cid:16) E r for each r ∈ R. Since hom( V ⊗ O ( − N H m ) , E r ) = h ( X, V ∨ ⊗ E r ( N H m )) is constant as a function of r , we get that p ∗ ( H om ( V ⊗ q ∗ O ( − N H m ) , E ) is a vector bundle on R .Let V π → R be the corresponding geometric vector bundle. Note that V remains irreducible. On V × X we have auniversal morphism V ⊗ π ∗ X q ∗ O ( − N H m ) Φ → π ∗ X E , HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 27 and we let U ⊂ V be the open subset parameterizing surjective morphisms. Observe that due to (3.13.4) U π → R remains surjective and that U is irreducible. Let A = ker(Φ | U × X ) , B = V ⊗ π ∗ X q ∗ O ( − N H m ) ,and consider the exact sequence of sheaves over U × X :(3.13.5) → A Ξ → B Φ → π ∗ X E → . By (3.13.3) we have Ext i ( B u , F t ) = 0 for i > and all u ∈ U , t ∈ T . By (3.13.1) and semistabilitywe also have that Ext ( E r , F t ) = Hom ( F t , E r ⊗ K X )) ∨ = 0 for all r ∈ R, t ∈ T. Applying Hom ( _ , F t ) to (3.13.5) at point u ∈ U we get that Ext i ( A u , F t ) = 0 for i > and all u ∈ U , t ∈ T . Thus hom( A u , F t ) is constant for all u ∈ U , t ∈ T and we have (3.13.6) → Hom ( E π ( u ) , F t ) → Hom ( B u , F t ) → Hom ( A u , F t ) → Ext ( E π ( u ) , F t ) → for all u ∈ U , t ∈ T. Recall that in our intuitive explanation we mentioned that parameterising extensions of E r by F t mightbe problematic due to jumping values of ext ( E r , F t ) for different r ∈ R, t ∈ T . Now (3.13.6) showsthat Hom ( A u , F t ) is a vector space of constant dimension for different u ∈ U , t ∈ T , so we can build anirreducible space parameterizing all homomorphisms A u → F t for all u ∈ U , t ∈ T . Since Hom ( A u , F t ) surjects onto Ext ( E π ( u ) , F t ) , this irreducible space will be a "cover" for the naive "space of extensions."To this end, consider the natural projections pr U × X : U × T × X → U × X,pr T × X : U × T × X → T × X,pr U × T : U × T × X → U × T. By the above discussion, pr U × T, ∗ ( H om ( pr ∗ U × X A, pr ∗ T × X F ) is a vector bundle over the irreducible base U × T , therefore the associated geometric vector bundle S ρ → U × T is irreducible too. Consider the universal morphism over S × Xρ ∗ X pr ∗ U × X A Ψ → ρ ∗ X pr ∗ T × X F , as well as the induced morphism ρ ∗ X pr ∗ U × X A ρ ∗ X pr ∗ U × X (Ξ) (cid:44) −−−−−−−−→ ρ ∗ X pr ∗ U × X B, where ρ X := ρ × Id X . Taking the direct sum of these maps and calling the resulting cokernel sheaf by V , we obtain the following short exact sequence of sheaves on S × X :(3.13.7) → ρ ∗ X pr ∗ U × X A ρ ∗ X pr ∗ U × X (Ξ) ⊕ Ψ −−−−−−−−−−→ ρ ∗ X pr ∗ U × X B ⊕ ρ ∗ X pr ∗ T × X F Ω → V → . For a point s ∈ S , this short exact sequence can be expanded into the following commutative diagram: F pr T ( ρ ( s )) F pr T ( ρ ( s )) A pr U ( ρ ( s )) B pr U ( ρ ( s )) ⊕ F pr T ( ρ ( s )) V s A pr U ( ρ ( s )) B pr U ( ρ ( s )) E π ( pr U ( ρ ( s ))) 00 0 0 Ξ pr U ( ρ ( s )) ⊕ Ψ s Ω s Ξ pr U ( ρ ( s )) Φ pr U ( ρ ( s )) The row in the middle corresponds to (3.13.7), while the row at the bottom corresponds to (3.13.5).The column in the middle is a trivial extension. By construction, the fiber of S over point ( u, t ) ∈ U × T is the vector space Hom ( A u , F t ) which by (3.13.6) surjects onto Ext ( E π ( u ) , F t ) . For a given s ∈ S with corresponding Ψ s ∈ Hom ( A u , F t ) , the resulting extension in Ext ( E π ( u ) , F t ) is displayed in the rightcolumn in the above diagram. This way, as s varies over S , we parameterize all possible extensions(3.13.2) and the Property (P) is satisfied. (cid:3) Group actions and Gaeta-type resolutions In this section, we recall some basic facts about the Picard group of G -linearized line bundles on avariety Y , and discuss how to compute with the Donaldson homomorphism when working with the familyof O (1 , -prioritary sheaves admitting an L -Gaeta type resolution constructed above in Propositions 2.20and 2.21.4.1. Characters and linearized line bundles. Let G be an algebraic group acting on a variety Y . A crossed morphism is a morphism of varieties θ : G × Y → C ∗ , satisfying θ ( gg (cid:48) , y ) = θ ( g, g (cid:48) y ) θ ( g (cid:48) , y ) for any g, g (cid:48) ∈ G, y ∈ Y. Crossed morphisms are in bijection with the linearizations of the trivial bundle O Y . Indeed, given acrossed morphism θ define the action of G on the total space Y × C of O Y over the action of G on Y by g · ( y, a ) = ( g · y, θ ( g, y ) a ) . A crossed morphism θ is said to be principal if there exists f ∈ O ∗ ( Y ) such that θ ( g, y ) = f ( g · y ) f ( y ) for any g ∈ G, y ∈ Y. Observe that for a principal crossed morphism θ coming from f ∈ O ∗ ( Y ) the trivial line bundle O Y witha trivial linearization is isomorphic as G -bundles to the bundle ( O Y , θ ) via ( y, a ) (cid:55)→ ( y, f ( y ) a ) , which is easily seen to be a G -equivariant map.In summary, we get an exact sequence O ∗ ( Y ) → CrMor ( Y, G ) → Pic G ( Y ) → Pic ( Y ) G , HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 29 where the second term is the group of crossed morphisms and the last term denotes G -invariant linebundles. Now note that any character η ∈ Char ( G ) can be viewed as a crossed morphism via θ η ( g, y ) = η ( g ) . Drezét shows in [Dré87, Proposition 14] that for those algebraic groups G for which any invertible functionon G can be written as a product of a constant and a character of G we in fact have an isomorphismChar ( G ) → CrMor ( Y, G ) . Therefore, for such groups we have the following result. Proposition 4.1. Let Y be an integral variety equipped with an action of an algebraic group G . Furthersuppose that any invertible function on G can be written as a product of a constant and a character of G . Then we have the following exact sequence (4.1.1) O ∗ ( Y ) → Char ( G ) → Pic G ( Y ) → Pic ( Y ) G . We remark that in the first map ( f (cid:55)→ η f ) the resulting character η f is described by the equality η f ( g ) = f ( g · y ) f ( y ) for any g ∈ G, y ∈ Y. Characters of the general linear group. In the context of the Gaeta-type resolutions we willbe interested in the action of the general linear group and groups closely related to it. These groupswill satisfy the assumption of Proposition 4.1. In view of exact sequence (4.1.1), we now recall how todescribe characters for such groups.For a fixed positive integer n , consider the homomorphism Z → Char ( GL ( n )) a (cid:55)→ [ η a : A (cid:55)→ det( A ) a ] . Since the coordinate ring of GL ( α ) is the localization C [ { x ij } ] det , the only invertible functions mapping Id ∈ GL ( n ) to ∈ C are of the form [ A → det( A ) a ] for some a ∈ Z . It follows that the abovehomomorphism is in fact an isomorphism.More generally, for k positive integers n , n , ..., n k let G := GL ( n ) × GL ( n ) × ... × GL ( n k ) . Wehave an isomorphism(4.1.2) Z k → Char ( G ) given by ( a , a , ..., a k ) (cid:55)→ [ η ( a ,a ,...,a k ) : ( A , A , ..., A k ) (cid:55)→ det( A ) a det( A ) a ... det( A k ) a k ] . Finally, let G := ( GL ( n ) × GL ( n ) × ... × GL ( n k )) / C ∗ ( Id, Id, ..., Id ) . Under the above isomor-phism Char ( G ) can be described as(4.1.3) Char ( G ) = { ( a , a , ..., a k ) ∈ Z k | a n + a n + ... + a k n k = 0 } ⊂ Z k . Natural action of G on Gaeta-type resolutions. We return back to the case X = P × P . Con-sider the family V t /T of O (1 , -prioritary sheaves of Chern character v with r ( v ) ≥ over T = U ⊂ H admitting an L -Gaeta type resolution (2.18.1) → L ( − , − α ψ t → L ( − , β ⊕ L (0 , − γ ⊕ L δ → V t → , as in Proposition 2.20. We first treat the case where all integers α, β, γ, δ are not zero and say how tomodify the argument when some of the exponents vanish later. In this case, there is a natural group action of G = GL ( α ) × GL ( β ) × GL ( γ ) × GL ( δ ) on T : for point ψ t ∈ T the point ( g α , g β , g γ , g δ ) · ψ t corresponds to the morphism ( g β ⊕ g γ ⊕ g δ ) ◦ ψ t ◦ ( g α ) − . Note that since c ( Id, Id, Id, Id ) · ψ t = ψ t , c ∈ C , there is also an induced action of G = ( GL ( α ) × GL ( β ) × GL ( γ ) × GL ( δ )) / C ∗ ( Id, Id, Id, Id ) on T .We extend both actions onto T × X . On T × X , there is a universal short exact sequence of sheaves(4.1.4) → q ∗ ( L ( − , − α ψ → q ∗ ( L ( − , β ⊕ q ∗ ( L (0 , − γ ⊕ q ∗ L δ → V → . We endow the trivial families with a natural G -linearization as follows. Let g = ( g α , g β , g γ , g δ ) ∈ G. Theaction of g is described as ( q ∗ ( L ( − , − α ) t = L ( − , − α g α −→ L ( − , − α = ( q ∗ ( L ( − , − α ) g · t ( q ∗ ( L ( − , β ) t = L ( − , β g β −→ L ( − , β = ( q ∗ ( L ( − , β ) g · t ( q ∗ ( L (0 , − γ ) t = L (0 , − γ g γ −→ L (0 , − γ = ( q ∗ ( L (0 , − γ ) g · t ( q ∗ L δ ) t = L δ g δ −→ L δ = ( q ∗ L δ ) g · t There is then a unique G -linearization of V making (4.1.4) a short exact sequence of G - linearized sheaves.For g ∈ G, ψ t ∈ T as above, it is described as the unique isomorphism Φ ( g,t ) completing the diagram L ( − , − α L ( − , β ⊕ L (0 , − γ ⊕ L δ V t L ( − , − α L ( − , β ⊕ L (0 , − γ ⊕ L δ V g · t . ψ t g α g β ⊕ g γ ⊕ g δ Φ ( g,t ) ψ g · t This allows us to use Lemma 2.5 (3) and compute the Donaldson homomorphism λ V t : K ( X ) → Pic G ( T ) explicitly, taking into account that both K ( X ) and Pic G ( T ) are free Z -modules. Specifically, we identify K ( X ) ∼ = Z by choosing the following Z -basis e := [ L ∨ ( − , − , e := [ L ∨ ( − , , e := [ L ∨ (0 , − , e := [ L ∨ ] . On the other hand, by Proposition 2.20 codim H ( H \ T ) ≥ and, since H is an affine space, it follows that O ∗ ( T ) = C ∗ and Pic ( T ) = 0 . Note that G satisfies the assumptions of Proposition 4.1, so we get(4.1.5) Char ( G ) ∼ → Pic G ( T ) and the former group was shown to be Z in (4.1.2). HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 31 Proposition 4.2. Consider the family V t /T of O (1 , -prioritary sheaves of character v with r ( v ) ≥ admitting an L -Gaeta type resolution → L ( − , − α ψ t → L ( − , β ⊕ L (0 , − γ ⊕ L δ → V t → where T ⊂ H = Hom (cid:16) L ( − , − α , L ( − , β ⊕ L (0 , − γ ⊕ L δ (cid:17) is the open subset parameterizing injective sheaf maps with torsion-free cokernel and all the exponents α, β, γ, δ are nonzero.Then the Donaldson homomorphism λ V t : K ( X ) → Pic G ( T ) is an isomorphism, and the image of v ⊥ is equal to Char ( G ) ⊂ Char ( G ) ∼ = Pic G ( T ) . Proof. Let A = q ∗ ( L ( − , α = q ∗ ( L ( − , − ⊗ V α and B = q ∗ ( L ( − , β ⊕ q ∗ ( L (0 , − γ ⊕ q ∗ L δ = ( q ∗ ( L ( − , ⊗ V β ) ⊕ ( q ∗ ( L (0 , − ⊗ V γ ) ⊕ ( q ∗ L ⊗ V δ ) , where V α , V β , V γ , V δ are vector spaces of dimension α, β, γ and δ respectively. Since the universal shortexact sequence (4.1.4) is a sequence of G -linearized sheaves, we have that for u ∈ K ( X ) ,λ V ( u ) = λ B ( u ) ⊗ λ A ( u ) ∨ as elements of Pic G ( T ) , or λ V ( u ) = λ B ( u ) − λ A ( u ) as elements of Char ( G ) under the isomorphism (4.1.5).Using this, one readily checks that p ! ( B ⊗ q ∗ L ∨ ( − , − p ! (( q ∗ O ( − , − ⊗ V β ) ⊕ ( q ∗ O ( − , − ⊗ V γ ) ⊕ ( q ∗ O ( − , − ⊗ V δ )) = 0 and p ! ( A ⊗ q ∗ L ∨ ( − , − p ! ( q ∗ O ( − , − ⊗ V α ) = [ O T ⊗ V α ] , viewed as elements in K G ( X ) . Thus λ V t ( e ) = det( O T ⊗ V α ) ∨ , which corresponds to character η ( − , , , ∈ Char ( G ) under the isomorphism (4.1.5). Similar calculations show that λ V ( e ) corresponds to η (0 , − , , ,λ V ( e ) corresponds to η (0 , , − , ,λ V ( e ) corresponds to η (0 , , , . In summary, the Donaldson homomorphism λ V t viewed as a map K ( X ) → Char ( G ) is given by(4.2.1) u = a e + a e + a e + a e (cid:55)→ η = η ( − a , − a , − a ,a ) Alternatively, it has matrix − − − when viewed as a map Z → Z .Now we turn to the second statement in the proposition. One checks that v = − α [ L ( − , − β [ L ( − , γ [ L (0 , − δ [ L ] in K ( X ) by applying χ ( _ · e i ) to both sides and using (2.19.1) along with the fact that e = [ L ( − , − , e = [ L ( − , , e = [ L (0 , − , e = [ L ] is a χ ( _ · _ ) -orthogonal basis to e , e , e , e : χ ( e i · e j ) = 0 for i (cid:54) = j and χ ( e · e ) = χ ( e · e ) = 1 , χ ( e · e ) = χ ( e · e ) = − . One further checks that the condition u ∈ v ⊥ ⇐⇒ χ ( v · u ) = 0 is equivalent to − a α − a β − a γ + a δ = 0 . By (4.1.3), this last condition is precisely equivalent to η = η ( − a , − a , − a ,a ) ∈ Char ( G ) . (cid:3) The above proof easily carries over to the case when one of the exponents in an L -Gaeta-type resolutionis zero. In particular, we will later work with the case when δ = 0 . In this case, set(4.2.2) G ˆ δ = GL ( α ) × GL ( β ) × GL ( γ ) G ˆ δ = ( GL ( α ) × GL ( β ) × GL ( γ )) / C ∗ ( Id, Id, Id ) . Proposition 4.3. Consider the family V t /T of O (1 , -prioritary sheaves of character v with r ( v ) ≥ admitting an L -Gaeta-type resolution → L ( − , − α φ t → L ( − , β ⊕ L (0 , − γ → V t → , where T ⊂ H = Hom (cid:16) L ( − , − α , L ( − , β ⊕ L (0 , − γ (cid:17) is the open subset parameterizing injective sheaf maps with torsion-free cokernel and the exponents α, β, γ are nonzero.Then the Donaldson homomorphism λ V t : K ( X ) → Pic G ˆ γ ( T ) is an epimorphism, and the image of v ⊥ is equal to Char ( G ˆ δ ) ⊂ Char ( G ˆ δ ) ∼ = Pic G ˆ δ ( T ) . Finally, we can repeat the discussion of this subsection for the "dual version" of a Gaeta-type resolu-tion. Consider the family V t /T of O (1 , -prioritary sheaves of Chern character v with r ( v ) ≥ over T = U ⊂ H admitting an L -Gaeta type resolution (2.20.1) → V t → L (1 , α ⊕ L (0 , β ⊕ L γ ψ t −→ L (1 , δ → as in Proposition 2.21. There is a natural action of G = GL ( α ) × GL ( β ) × GL ( γ ) × GL ( δ ) and G = ( GL ( α ) × GL ( β ) × GL ( γ ) × GL ( δ )) / C ∗ ( Id, Id, Id, Id ) on T and T × X if all the exponents α, β, γ, δ are nonzero, and of G ˆ γ = GL ( α ) × GL ( β ) × GL ( δ ) and G ˆ γ = ( GL ( α ) × GL ( β ) × GL ( δ )) / C ∗ ( Id, Id, Id ) if α, β, δ > , but γ = 0 . HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 33 As before, the action of G (resp. G ˆ γ ) on T × X lifts to a linearization of the universal families ofsheaves and we have the obvious analogues of Propositions 4.2 and 4.3.5. The Picard group of the moduli space Associated exceptional bundles and the Donaldson homomorphism. Exceptional bundles as-sociated to an H m -semistable character v (see Definition 2.16) give rise to easy-to-describe classes inthe kernel of the the Donaldson homomorphism λ : v ⊥ → Pic ( M H m ( v )) .Specifically, suppose E is associated to a nonsemiexceptional H m -semistable character v and µ H m ( E ) ≥ µ H m ( v ) . By semistability and Serre duality, for any semistable V of character v we haveHom ( E, V ) = Ext ( E, V ) = 0 , and since ∆( V ) = DLP H m ,E ( ν ( V )) , we also have χ ( E, V ) = 0 and Ext ( E, V ) = 0 . This way, we see that if U r /R is the family of H m -semistable sheaves parameterized by a subset R ofthe Quot scheme used in the GIT construction of M H m ( v ) , then in the notation of §2.4 we have p ! ( q ∗ [ E ∨ ] · [ U ]) = 0 . Proposition 2.7 then shows that λ ([ E ∨ ]) = 0 . Similarly, if µ H m ( E ) < µ H m ( v ) , then λ ([ E ∨ ⊗ K X ]) = 0 . For that matter, we introduce the following uniform notation: for an exceptional bundle E associatedto character v , we define the following class in K ( X )[ E ] = (cid:40) [ E ∨ ] if µ H m ( E ) ≥ µ H m ( v ) , [ E ∨ ⊗ K X ] if µ H m ( E ) < µ H m ( v ) . The main theorem. Finally, we are ready to state our first main result about the Picard group of themoduli space M H m ( v ) . We recall that λ : v ⊥ → Pic ( M H m ( v )) denotes the Donaldson homomorphismconstructed in Proposition 2.7. Theorem 5.1. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ ≥ . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) . (1) If ∆ > , ∆ > DLP Chern character from Definition 3.6. This assumption is substantial: as we show in Theorem6.3 below for certain bad characters lying on a single branch of the DLP-surface the Picard number dropsto . We also emphasize that determining which statement of the theorem applies to a given character v = ( r, ν, ∆) is a finite computational procedure and, therefore, can be implemented on a computer: thecomputation of DLP DLP H m , O ( a,b ) ( aE + bF ) we have Pic ( X [ n ] ) ∼ = Z by the Theorem of Fogarty [Fog73].The proof of Theorem 5.1 occupies the rest of this section. For the convenience of the reader, we willprove the theorem in a series of propositions according to how the theorem is stated. Cases (1.a), (1.b)and (3) of the theorem have relatively simple proofs. We then prove a part of case (2) so that at thatpoint the theorem will be proved for characters v in a large region in the ( r, ν, ∆) -space. The remainingcharacters v have their discriminant in a narrow range < ∆ < and have no line bundles associatedto them. These conditions allow us to deal with case (1.c) and the remainder of case (2) in a uniformfashion though the proofs become considerably more involved.5.3. Proof of the main theorem. We start by proving case (3.b) of the theorem to be able to assume ∆ > in the rest of the proof and use the surjectivity of the Donaldson morphism from Theorem 2.8. Proposition 5.2. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ = . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If v is H m -semistable, then M H m ( v ) is a projective space and Pic ( M H m ( v )) ∼ = Z . Proof. First, if v = ( r, ν, ) = ( r, c , χ ) is a primitive character, then for an appropriate (generic) choiceof m = pq one checks that gcd( r, c · ( qH m ) , χ ) = 1 . In this case, M H m ( v ) = M sH m ( v ) is a smooth projective variety of dimension dim M H m ( v ) = exp dim M H m ( v ) = 1 . HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 35 Moreover, Walter shows in [Wal93] that M sH m ( v ) is irreducible and unirational. It follows that in thiscase M H m ( v ) ∼ = P .Now, assume v is not primitive. We can write v = N v (cid:48) with N ∈ N and v (cid:48) primitive. In this case M H m ( v ) consists of strictly semistable sheaves. But since M H m ( v (cid:48) ) = M sH m ( v (cid:48) ) carries a universalfamily of H m -stable sheaves U , we can take its N -fold sum to get a morphism M H m ( v (cid:48) ) × M H m ( v (cid:48) ) × ... × M H m ( v (cid:48) ) → M H m ( v ) . This morphism is surjective on closed points and invariant under permutation of factors, i.e. factorsthrough the symmetric product(5.2.1) S N ( M H m ( v (cid:48) )) → M H m ( v ) , which is now bijective at closed points. Note that since M H m ( v (cid:48) ) ∼ = P , the symmetric product is justa projective space S N ( M H m ( v (cid:48) )) ∼ = P N . Now recall that M H m ( v ) is a good quotient R // G of a smooth subvariety R of the Quot scheme.In particular, R is normal. Since normality is preserved under taking categorical quotients (see [MFK94,Page 5]), M H m ( v ) is normal too. It follows that (5.2.1) is an isomorphism. (cid:3) From now on, we will be working with characters v with ∆( v ) > . Note that by Proposition 2.17(1) the stable locus M sH m ( v ) will be nonempty for H m -semistable Chern characters with ∆( v ) > .Applying Theorem 2.8, we know that the Donaldson homomorphism is surjective λ : v ⊥ (cid:16) Pic ( M H m ( v )) , and we need to study its kernel.The next proposition corresponds to cases (1.a) and (1.b) of Theorem 5.1. Proposition 5.3. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ > . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If ∆ − r ≥ DLP Proposition 5.4. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ > . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If ∆ = DLP In this case, the discussion in §5.1 shows that the subgroup Z [ E ] + Z [ E ] lies in the kernel ofthe Donaldson homomorphism, which now factors as v ⊥ / Z [ E ] ⊕ Z [ E ] (cid:16) Pic ( M H m ( v )) . Since the ample bundle generates a free Z -submodule inside Pic ( M H m ( v )) , it follows that the Picardnumber is equal to one ρ ( M H m ( v )) = 1 . If v is primitive, then for a generic choice of m = pq we have gcd( r, c · ( qH m ) , χ ) = 1 . Applying Theorem 2.11 we get that Pic ( M H m ( v )) is torsion-free and, therefore,Pic ( M H m ( v )) ∼ = Z . Now assume ( E , E ) or ( E , E ) forms an exceptional pair. One checks that ( E , E ) or ( E , E ) isstill an exceptional pair. Zyuzina [Zyu94] shows that any exceptional pair on P × P can be completedto a full exceptional collection. Since a full exceptional collection forms a Z -basis for K ( X ) , we see that Z [ E ] ⊕ Z [ E ] is a primitive lattice inside v ⊥ ⊂ K ( X ) . This way, the Donaldson homomorphism induces Z ∼ = v ⊥ / Z [ E ] ⊕ Z [ E ] (cid:16) Pic ( M H m ( v )) . The result follows. (cid:3) The arguments above worked equally well for both good and bad H m -semistable Chern characters.However, for the remaining cases (1.c) and (2) of Theorem 5.1 the assumption that character v is goodis essential.It will be convenient to separate the proof of case (2) of Theorem 5.1 into the following two subcases(keeping the notation and the assumptions of the theorem):(2.a) If v is a good character with ∆ = DLP Proposition 5.5. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ > . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If v is a good character with ∆ = DLP We treat the case µ H m ( v ) ≤ µ H m ( L ) and say how to modify the argument in the other case at the end of the proof.In this situation, the general H m -semistable sheaf V admits an L -Gaeta type resolution with exponents α = − χ ( v ⊗ L ∨ ( − , − > β = − χ ( v ⊗ L ∨ ( − , > γ = − χ ( v ⊗ L ∨ (0 , − > δ = χ ( v ⊗ L ∨ ) = 0 Note that none of α, β, γ can be equal to . For otherwise, one checks using the semistability of V that one of the bundles L (1 , , L (1 , , L (0 , or their Serre twists would also be associated to v ,contradicting our assumption.Consider the family V t /T of O (1 , -prioritary sheaves admitting the L -Gaeta type resolution(5.5.1) → L ( − , − α ψ t → L ( − , β ⊕ L (0 , − γ → V t → , where T ⊂ H = Hom (cid:16) L ( − , − α , L ( − , β ⊕ L (0 , − γ (cid:17) is the open subset parameterizing injective sheaf maps with torsion-free cokernel from Proposition 2.20.Since we are assuming that v is a good character, there is no ∆ i = strata in this family by Proposition3.5. The other potential divisorial Shatz stratum should consist of sheaves V t admitting the H m -Harder-Narasimhan filtration ⊂ L ⊂ V t . Applying Hom ( L, _ ) to the short exact sequence (5.5.1), we see thatHom ( L, V t ) = 0 for all ψ t ∈ T , so this potential stratum is empty. Combined with Proposition 2.20, we get thatcodim T ( T \ T ss ) ≥ . Functorial properties of the Donaldson homomorphism from Lemma 2.5 give the following commutativediagram (recall our notation from (4.2.2) and Proposition 4.3): v ⊥ Pic ( M H m ( v )) K ( X ) Pic G ˆ δ ( T ss ) Pic G ˆ δ ( T ) . λ φ ∗V t | Tss λ V t λ V t | Tss ∼ = res By Proposition 4.3 we know that the image of ( res − ◦ φ ∗V t | Tss ) isChar ( G ˆ δ ) ⊂ Char ( G ˆ δ ) ⊂ Pic G ˆ δ ( T ) , which is a free Z -module of rank . On the other hand, by the discussion in §5.1 we know that [ L ] liesin the kernel of the Donaldson homomorphism λ . Putting these together, we get that φ ∗V t | Tss ◦ λ factorsas Z ∼ = v ⊥ / Z [ L ] (cid:16) Pic ( M H m ( v )) (cid:16) Z , so both maps are isomorphisms and Pic ( M H m ( v )) ∼ = Z .In the other case when µ H m ( v ) > µ H m ( L ) , one modifies the above proof by using the dual version of a Gaeta-type resolution → V → L (1 , β ⊕ L (0 , γ → L (1 , δ → and replacing Propositions 2.20 and 4.3 by Proposition 2.21 and the dual version of Proposition 4.3. (cid:3) At this point let us make a couple of useful observations. First, note that the previous cases fullyestablish Theorem 5.1 for characters v = ( r, ν, ∆) with r = 2 . Indeed, ( r = 2 , ∆ = ) was covered byProposition 5.2 and for ( r = 2 , ∆ > ) we have four different cases:(1) v = (2 , ν, ∆) = (2 , εE + ϕF, ∆) with ε, ϕ ∈ Z and ∆ > DLP < H m ( εE + ϕF ) = DLP H m , O ( ε,ϕ ) ( εE + ϕF ) = 1 . The inequality implies that ∆ ≥ . Therefore, ∆ − ≥ DLP H m , O ( ε,ϕ ) ( εE + ϕF ) = 1 and this is covered by Proposition 5.3.(2) v = (2 , ν, ∆) = (2 , εE + ϕF, ∆) with ε, ϕ ∈ Z and ∆ = DLP < H m ( εE + ϕF ) = DLP H m , O ( ε,ϕ ) ( εE + ϕF ) = 1 . The line bundle O ( ε, ϕ ) is the only exceptional bundle associated to v in this case. Since characters v with r = 2 are always good characters (see Definition 3.6), this case is covered by Proposition5.5 above.(3) v = (2 , ν, ∆) = (2 , εE + ϕF, ∆) with ε ∈ ( Z [ ] \ Z ) , or ϕ ∈ ( Z [ ] \ Z ) (or both) and ∆ > DLP < H m ( εE + ϕF ) . As before, one shows that then ∆ − ≥ DLP < H m ( εE + ϕF ) and DLP < H m ( εE + ϕF ) ≥ . In case one of the inequalities is strict, we have ∆ − > . If ∆ − = , then v (cid:48) = (2 , ν, ) =(2 , c , χ ) is primitive, because by our assumption c = (2 ε ) E + (2 ϕ ) F has an odd component andtherefore is not divisible by . We conclude that this case is covered by Proposition 5.3 above. HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 39 (4) v = (2 , ν, ∆) = (2 , εE + ϕF, ∆) with ε ∈ ( Z [ ] \ Z ) , or ϕ ∈ ( Z [ ] \ Z ) (or both) and ∆ = DLP < H m ( εE + ϕF ) . From Figure 1, one sees that v lies on two branches of the DLP < H m -surface ∆ = DLP H m ,L ( εE + ϕF ) = DLP H m ,L ( εE + ϕF ) = 34 with L = O (( (cid:98) ε (cid:99) + 1) E + (cid:98) ϕ (cid:99) F )) , L = O ( (cid:98) ε (cid:99) E + ( (cid:98) ϕ (cid:99) + 1) F ) , so there are two line bundlesassociated to v and they form an exceptional pair. This is covered by Proposition 5.4 above.It remains to prove statements (1.c) and (2.b). By the above discussion we can assume that r ( v ) ≥ .We show next that for the remaining characters v = ( r, ν, ∆) there is a strong upper bound on thediscriminant ∆ . Lemma 5.6. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ > . Let (cid:15) ∈ Q be sufficientlysmall (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If v is a character satisfying either • ∆ > DLP Suppose first ∆ − r ≤ . We immediately get ∆ ≤ 12 + 1 r ≤ 12 + 13 = 56 < recall r ≥ . Now suppose that ∆ > DLP 59 + 13 = 89 < . Finally, suppose character v = ( r, ν, ∆) satisfies ∆ = DLP Let v = ( r, ν, ∆) = ( r, εE + ϕF, ∆) ∈ K ( X ) be a character satisfying conditions (5.6.2) .Let (cid:15) ∈ Q be sufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) ≤ ε + ϕ < (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) + 1 , then a general H m -semistable sheaf V of character v admits an L -Gaeta-type resolution → L ( − , − α → L ( − , β ⊕ L (0 , − γ ⊕ L δ → V → where L := L (cid:98) (cid:15) (cid:99) , (cid:98) φ (cid:99) and all the exponents are nonzero.If (cid:98) φ (cid:99) + 1 < (cid:15) + φ < (cid:98) (cid:15) (cid:99) + (cid:98) φ (cid:99) + 2 , then a general H m -semistable sheaf V of character v admits a dual L -Gaeta-type resolution → V → L (1 , α ⊕ L (0 , β ⊕ L γ → L (1 , δ → where L := L (cid:98) (cid:15) (cid:99) , (cid:98) φ (cid:99) and all the exponents are nonzero.Proof. We prove the case(5.7.1) (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) ≤ ε + ϕ < (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) + 1 , and say how to modify the argument for the other case at the end of the proof. HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 41 With the notation of §2.8, we use the bound (5.6.2) (2) and formally compute using Riemann-Roch: χ ( v ( − L ε,ϕ )) = r (1 − ∆) > . Thus, in the ( a, b ) -plane R the point ( ε, ϕ ) lies below the lower-left branch Q of the hyperbola χ ( v ( − L a,b )) = 0 . Therefore, the integral point ( (cid:98) ε (cid:99) , (cid:98) ϕ (cid:99) ) also lies below Q and we have that χ ( v ( − L (cid:98) ε (cid:99) , (cid:98) ϕ (cid:99) )) > . For a sufficiently small (cid:15) condition (5.7.1) translates into a condition on µ H m -slopes: µ H m ( L (cid:98) ε (cid:99) , (cid:98) ϕ (cid:99) ) ≤ µ H m ( v ) < µ H m ( L (cid:98) ε (cid:99) + k, (cid:98) ϕ (cid:99) + l ) + 1 for ( k, l ) ∈ { (1 , , (0 , , (1 , } . Therefore, for V ∈ M H m ( v ) we haveHom ( L (cid:98) ε (cid:99) + k, (cid:98) ϕ (cid:99) + l , V ) = Ext ( L (cid:98) ε (cid:99) + k, (cid:98) ϕ (cid:99) + l , V ) = 0 , resulting in − χ ( v ( − L (cid:98) ε (cid:99) + k, (cid:98) ϕ (cid:99) + l )) = − χ ( L (cid:98) ε (cid:99) + k, (cid:98) ϕ (cid:99) + l , v ) = ext ( L (cid:98) ε (cid:99) + k, (cid:98) ϕ (cid:99) + l , V ) ≥ . In fact, the inequalities are strict, for otherwise L (cid:98) ε (cid:99) + k, (cid:98) ϕ (cid:99) + l (or their Serre twists) would be associated to v contradicting assumption (5.6.2) (4). This shows that the line bundle L := L (cid:98) ε (cid:99) , (cid:98) ϕ (cid:99) satisfies (2.19.1)of Theorem 2.19 with all integers α, β, γ, δ being nonzero.In the case (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) + 1 < ε + ϕ < (cid:98) (cid:15) (cid:99) + (cid:98) φ (cid:99) + 2 , one can first pass to the dual character v (cid:48) , resolve a generic µ H m -stable vector bundle by a Gaeta-typeresolution with all exponents nonzero as above, and then take the dual of the whole resolution. Here weuse Proposition 2.17 to guarantee the existence µ H m -stable bundles. (cid:3) Thus, to study M H m ( v ) for v satisfying conditions (5.6.2) with (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) ≤ ε + ϕ < (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) + 1 we consider the complete family V t /T of O (1 , -prioritary sheaves admitting an L -Gaeta type resolution(5.7.2) → L ( − , − α ψ t → L ( − , β ⊕ L (0 , − γ ⊕ L δ → V t → , where L = L (cid:98) ε (cid:99) , (cid:98) ϕ (cid:99) , all the exponents are nonzero, and(5.7.3) T ⊂ H = Hom (cid:16) L ( − , − α , L ( − , β ⊕ L (0 , − γ ⊕ L δ (cid:17) is the open subset parameterizing injective sheaf maps with torsion-free cokernel. By Proposition 2.20(5.7.4) codim H ( H \ T ) ≥ . Likewise, to study M H m ( v ) for v satisfying conditions (5.6.2) with (cid:98) ϕ (cid:99) + 1 < ε + φ < (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) + 2 we consider the complete family V t /T of O (1 , -prioritary vector bundles admitting the dual version ofan L -Gaeta type resolution → V t → L (1 , α ⊕ L (0 , β ⊕ L γ ψ t −→ L (1 , δ → , where L = L (cid:98) ε (cid:99) , (cid:98) ϕ (cid:99) , all the exponents are nonzero, and T ⊂ H = Hom (cid:16) L (1 , α ⊕ L (0 , β ⊕ L γ , L (1 , δ (cid:17) is the open subset parameterizing surjective sheaf maps. By the dual version of Proposition 2.21codim H ( H \ T ) ≥ . Next, we analyze the Shatz stratification of T and apply the irreducibility results of §3.4 to computethe kernel of the Donaldson homomorphism in the remaining cases. In what follows, we treat the case (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) ≤ ε + ϕ < (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) + 1 , leaving the necessary modifications of the proof in the other case to the reader.The next proposition proves case (1.c) of Theorem 5.1. Proposition 5.8. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ > . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If v is a good character with ∆ > DLP Proposition 5.9. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ > . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If v is a good character with ∆ = DLP Proof. By the discussion above, we can again assume that character v satisfies (5.6.2) with (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) ≤ ε + ϕ < (cid:98) ε (cid:99) + (cid:98) ϕ (cid:99) + 1 . We further assume µ H m ( v ) ≤ µ H m ( E ) , leaving the necessary modifications of the proof in the other case to the reader. By the discussion in§5.1, we know that [ E ] lies in the kernel of the Donaldson homomorphism. We show that in fact ker( λ ) = Z [ E ] . Step 1 . Once again we analyze the Shatz stratification of the family V t /T from (5.7.3) which param-eterizes (1 , -prioritary sheaves admitting an L -Gaeta type resolution (5.7.2):(5.7.2) → L ( − , − α ψ t → L ( − , β ⊕ L (0 , − γ ⊕ L δ → V t → . This time, Proposition 3.5 says that there is at most one possible divisorial Shatz stratum S T since ∆ i = -strata are excluded by the assumption that v is good. In fact, it must be nonempty for ourfamily V t /T . For otherwise arguing as in the Proposition 5.8, we would show that λ is injective, incontradiction to Z [ E ] ⊂ ker( λ ) . According to Proposition 3.5, this stratum S T = S T,H m ( v , v ) consists of points ψ t ∈ T such thatthe corresponding sheaf V t admits the H m -Harder-Narasimhan filtration of length l = 2 (5.9.1) ⊂ E ⊂ V t , where E is the exceptional bundle associated to v and v := v ( E ) , v := v ( V t /E ) . Step 2 . We claim that this Shatz stratum is irreducible. Set B = L ( − , β ⊕ L (0 , − γ ⊕ L δ .Consider the Quot scheme Quot ( B, v ) parameterizing quotients q = [ B (cid:16) E q ] , q ∈ Quot ( B, v ) , where v ( E q ) = v . First, restrict the family {E q } to the open subset of Quot ( B, v ) parameterizingtorsion-free E q . Over this subset we have a universal family of sheaves → K → q ∗ B → E → . Note that by (5.7.2) v ( K q ) = v ( L ( − , − α ) . We further restrict to the open subset Q ⊂ Quot ( B, v ) parameterizing those quotients q for which K q is a semistable vector bundle. Since v ( L ( − , − α ) is a semiexceptional Chern character, K q ∼ = L ( − , − α for each q ∈ Q. By the universal property of Quot schemes the family of quotients { B → V t } ψ t ∈ T of (5.7.2) gives asurjective morphism T Ω (cid:16) Q ⊂ Quot ( B, v ) , whose fibers are isomorphic to GL ( α ) . Denote the Shatz stratum of points q ∈ Q such that thecorresponding E q has the H m -Harder-Narasimhan filtration ⊂ E ⊂ E q by S Q = S Q,H m ( v , v ) . From the Cartesian diagram S T TS Q Q, Ω it follows that the irreducibility of S T is equivalent to the irreducibility of S Q . Indeed, S T is equidi-mensional and the fibers of S T (cid:16) S Q are all irreducible and isomorphic to GL ( α ) . We can apply thefollowing version of the irreducibility criterion: if Y → X is a finite type surjective morphism from anequidimensional Noetherian C -scheme to an irreducible Noetherian C -scheme, and all fibers over theclosed points are irreducible of the same dimension, then Y is irreducible.To show the irreducibility of S Q , consider the family W s /S over irreducible S having Property (P)with respect to F ( v , v ) that was constructed in Proposition 3.13 (we denote the sheaves in this familyby W s instead of V s to avoid confusion with the sheaves V t from (5.7.2)). Recall that heuristically W s /S parameterizes all torsion-free sheaves whose H m -Harder Narasimhan filtration is of lenght andhas quotients of characters v , v , possibly with repetition. Intuitively, we are going to build a family ofquotients over an irreducible base out of W s /S that will surject onto S Q under the universal morphismto the Quot scheme Q .Note that for ψ t ∈ T the Gaeta-type resolution (5.7.2) impliesExt i ( B, V t ) = 0 for i > ⇒ hom( B, V t ) = χ ( B, v ) . Thus, consider the open subset U ⊂ S parameterizing those W s for whichExt i ( B, W s ) = 0 for i > . It is non-empty because we concluded above that S T is non-empty, and irreducible. It follows that ( p U ) ∗ H om ( q ∗ B, W ) is a vector bundle on U . Denote the corresponding geometric vector bundle by V π → U, so that over V × X we have a universal morphism π ∗ q ∗ B Ψ → π ∗ W . We further restrict to an open subset U ⊂ V parameterizing surjective maps with an H m -semistablekernel, so that for u ∈ U we have an exact sequence → L ( − , − α → B Ψ u → W π ( u ) → . By the universal property of Quot schemes, we obtain a map U → Q, whose image is equal to S Q because of the Property (P). Since U is irreducible, it follows that S Q isirreducible too. We summarize the discussion in the following diagram GL ( α ) GL ( α ) S T T U S Q Q. Step 3 . We return to the problem of describing the kernel of the Donaldson homomorphism. Let T (cid:48) = T \ S T and note that this is a G -invariant open subset of T . Because of the irreducibility provedat the previous step, S T = V ( f ) for some irreducible polynomial(5.9.2) f ∈ C [ { x ij } ] , HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 45 where C [ { x ij } ] is the coordinate algebra of H . The sequence (4.1.1) for Y = T (cid:48) (5.9.3) { af k | a ∈ C ∗ , k ∈ Z } = O ∗ ( T (cid:48) ) af k (cid:55)→ kη f −−−−−−→ Char ( G ) → Pic G ( T (cid:48) ) → now implies that Pic G ( T (cid:48) ) ∼ = Char ( G ) / Z · η f . Since S T was the only divisorial Shatz stratum, we have thatcodim T (cid:48) ( T (cid:48) \ T ss ) ≥ and we obtain the following commutative diagram v ⊥ Pic ( M H ( v )) K ( X ) Pic G ( T ss ) K ( X ) Pic G ( T (cid:48) ) .K ( X ) Pic G ( T ) . λ φ ∗V t | Tss λ V t | Tss λ V t | T (cid:48) ∼ = resλ V t ∼ = res Chasing this diagram shows that integer multiples of [ E ] ∈ v ⊥ are the only elements in the kernel of λ . For if there was u ∈ ( v ⊥ \ Z [ E ]) with λ ( u ) = 0 , then going around the outer lower part of the diagram would imply that two Z -linearly independentelements, λ V t ( u ) and λ V t ([ E ]) , lie in the kernel of the restrictionPic G ( T ) res −→ Pic G ( T (cid:48) ) . But this then contradicts the fact that the kernel of this restriction is a cyclic subgroup, that could beseen from looking at sequence (4.1.1) for the inclusion T (cid:48) ⊂ T : C ∗ = O ∗ ( T ) Char ( G ) Pic G ( T ) 0 { af k | a ∈ C ∗ , k ∈ Z } = O ∗ ( T (cid:48) ) Char ( G ) Pic G ( T (cid:48) ) 0 . resaf k (cid:55)→ kη f This finishes the proof of statement (3.b) of the theorem. (cid:3) The main Theorem 5.1 is now fully proved. Remark 5.10. Note that we can describe the polynomial f appearing in (5.9.2) in such a way so that onecan explicitly compute the character η f appearing in (5.9.3). Recall that for ψ t ∈ S T the correspondingsheaf V t comes equipped with a filtration ⊂ E ⊂ V t , while for an H m -semistable V τ we haveHom ( E, V τ ) = Ext ( E, V τ ) = 0 . Therefore, S T ⊂ { ψ t ∈ T | Hom ( E, V t ) (cid:54) = 0 } . The long exact sequence in cohomology coming from(5.7.2) → Hom ( E, V t ) → Ext ( E, L ( − , − α ) ( ψ t ) ∗ −−−→ Ext ( E, B ) → Ext ( E, V t ) → shows that { ψ t ∈ T | Hom ( E, V t ) (cid:54) = 0 } is a determinantal divisor given as the vanishing locus of ψ t (cid:55)→ det(( ψ t ) ∗ ) . As S T = { ψ t ∈ T | Hom ( E, V t ) (cid:54) = 0 } , this describes f as f ( ψ t ) = det(( ψ t ) ∗ ) . Since η f is defined by the equation f ( g · ψ t ) = η f ( g ) f ( ψ t ) , we can explicitly recover η f from the following computation f ( g · ψ t ) = det(( g · ψ t ) ∗ )= det (cid:2)(cid:0) ( g β ⊕ g γ ⊕ g δ ) ◦ ψ t ◦ ( g α ) − (cid:1) ∗ (cid:3) = (cid:2) det(( g β ) ∗ ) det(( g γ ) ∗ ) det(( g γ ) ∗ ) det(( g α ) ∗ ) − (cid:3) det(( ψ t ) ∗ )= (cid:104) det( g β ) − χ ( E,L ( − , β ) det( g γ ) − χ ( E,L (0 , − γ ) det( g δ ) − χ ( E,L δ ) det( g α ) χ ( E,L ( − , α ) (cid:105) det(( ψ t ) ∗ ) . Thus η f = η a,b,c,d with a = χ ( E, L ( − , α ) ,b = − χ ( E, L ( − , β ) ,c = − χ ( E, L (0 , − γ ) ,d = − χ ( E, L δ ) . Corollaries of Theorem 5.1. We conclude this section by exploring some immediate corollaries ofTheorem 5.1.First, we can get rid of some of the assumptions in Proposition 5.4 at the expense of loosing theinformation about torsion in Pic ( M H m ( v )) . Corollary 5.11. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ and ∆ > . Let (cid:15) ∈ Q besufficiently small (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If ∆ = DLP Let v = ( r, ν, ∆) be a good H m -semistable Chern character with ∆ > , where m = 1 + (cid:15) and (cid:15) ∈ Q is a sufficiently small number depending on r , < | (cid:15) | (cid:28) .If v lies above the DLP Suppose on the contrary that ∆ = DLP H m ,F ( ν ) for an exceptional bundle F with r ( F ) > r and | ( ν − ν ( F )) · H m | ≤ − K X · H m . By §5.1, the class [ F ] lies in the kernel of the Donaldson homomorphism. Case 1 : ∆ > DLP Let v = ( r, ν, ∆) = ( r, εE + ϕF, ∆) ∈ K ( X ) be a good H m -semistable Chern characterwith r ≥ , ∆ > satisfying either • ∆ = DLP Pic G ( T ss ) . But in the proofs of Propositions 5.5, 5.8, 5.9 we showed that the above map is injective, a contradiction. (cid:3) Remark 5.14. The restrictions on the numerical invariants in Corollary 5.13 are substantial conditions.When these conditions are not satisfied some of the exponents in a Gaeta-type resolution may becomezero. As a result, we can no longer ensure that we can find L such that for the resulting complete family V t /T of O (1 , -prioritary sheaves admitting an L -Gaeta-type resolution we haverk ( Pic G ( T ss )) ≥ ρ ( M H m ( v )) . This way, the homomorphism Pic ( M H m ( v )) φ ∗V t | Tss −−−−−→ Pic G ( T ss ) may no longer be injective. 6. Bad Chern characters In this section we show that when an H m -semistable character v is bad , the Picard number of M H m ( v ) is no longer controlled only by the position of v relative to the DLP Let m = 1 + (cid:15), where (cid:15) ∈ Q and < (cid:15) (cid:28) . Consider character v = (4 , − E − F, ) from Example 3.8. In that example we considered the family V t /T of O (1 , -prioritary sheaves ofcharacter v admitting an O -Gaeta-type resolution(6.1.1) → O ( − , − ψ t −→ O ( − , ⊕ O (0 , − → V t → , where T ⊂ H = Hom (cid:0) O ( − , − , O ( − , ⊕ O (0 , − (cid:1) HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 49 is the open subset parameterizing injective sheaf maps with torsion-free cokernel. We showed that T isnot empty, codim H ( H \ T ) ≥ , the family V t /T is complete and any H m -semistable V ∈ M H m ( v ) isequal to V t for some t ∈ T . This last property implies that the classifying morphism T ss φ V t | Tss −−−−−→ M H m ( v ) realizes M H m ( v ) = M sH m ( v ) as a geometric quotient of T ss under the action of G = ( GL (2) × GL (3) × GL (3)) / C ∗ ( Id, Id, Id ) = G/ C ∗ ( Id, Id, Id ) , see [DLP85, Proposition 2.6]. Thus, by [MFK94, p. 32]Pic ( M H m ( v )) ∼ = Pic G ( T ss ) . As before, we can compute the latter group using the exact sequence from Proposition 4.1:(6.1.2) O ∗ ( T ss ) → Char ( G ) → Pic G ( T ss ) → . We claim that the first map is not zero. Take the ∆ i = -stratum S T = S H m ( v , v ) described inExample 3.8. Its closure S T is a Weil divisor in H , so it is given by a polynomial f : S T = V ( f ) , f ∈ C [ { x ij } ] . Since S T is G -invariant, the complement H \ S T is G -invariant too, and the polynomial f defines aninvertible function on it, which by a remark after Proposition 4.1 satisfies f ( gh ) = η f ( g ) f ( h ) for some η f ∈ Char ( G ) and any g ∈ G, h ∈ H \ S T . Note that since f ( h ) = f ( gh ) = 0 for h ∈ S the above equation in fact holds for all h ∈ H .We show that η f is a nontrivial character, which would establish our claim. Assume, on the contrary,that η f is a trivial character so that f is G -invariant and, consequently, G -invariant: f ∈ C [ { x ij } ] G . As the closure of any G -orbit contains the zero morphism ∈ H , all G -invariant functions are constant C [ { x ij } ] G = C ∗ . But f defines a non-empty divisor, so this is a contradiction.Now, sequence (6.1.2) gives Z / Z η f (cid:16) Pic G ( T ss ) ∼ = Pic ( M H m ( v )) . Since the ample bundle generates a free Z -submodule inside Pic ( M H m ( v )) , it follows that the Picardnumber is equal to one ρ ( M H m ( v )) = 1 . Note, that an explicit computation of η f along the lines of remark 5.10 does not work in this case.The closure of the divisorial Shatz stratum is now described as S T = { ψ t ∈ T | Hom ( F , V t ) (cid:54) = 0 for some F ∈ M H m ( v ) } , and compared to Remark 5.10 the computation is obstructed by the fact that F is not a fixed bundle,but varies along its one-dimensional moduli space.However, note that by Remark 3.7, character v = ( r, c , χ ) is primitive, so for a generic choice of m = pq we have gcd( r, c · ( qH m ) , χ ) = 1 . Applying Proposition 2.11 we get that Pic ( M H m ( v )) is torsion-free and thereforePic ( M H m ( v )) ∼ = Z . Let us also remark that using Proposition 3.13 we can argue as in the proof of Proposition 5.9 andshow that S T is an irreducible subvariety of T . Example 6.2. Let m = 1 + (cid:15), where (cid:15) ∈ Q and < (cid:15) (cid:28) . Let { w k } k ∈ N be one of the infinitesequences of bad Chern characters constructed in Examples 3.9 and 3.10. The same argument can beapplied verbatim to the complete family W t /T from Example 3.9 to conclude thatPic ( M H m ( w k )) ∼ = Z for any k ∈ N .It turns out that the techniques of the previous two examples allow us to tackle bad H m -semistableChern characters whenever they lie on a branch of the DLP-surface given by a line bundle . Note that H m -semistable characters v with r = 2 are always good (see Definition 3.8), so we can assume r ≥ . Theorem 6.3. Let v = ( r, ν, ∆) ∈ K ( X ) be a character with r ≥ , ∆ > . Let (cid:15) ∈ Q be sufficientlysmall (depending on r ), < | (cid:15) | (cid:28) , and set m = 1 + (cid:15) .If v is a bad character with ∆ = DLP Assume first that µ H m ( L ) ≥ µ H m ( v ) . For a semistable V of character v we haveExt i ( L, V ) = 0 , i = 0 , , , by semistability and the fact that L is associated to v .Using this, one checks that the Beilinson-type resolution from [Dré91, Proposition 5.1] coincides withthe L -Gaeta-type resolution and every H m -semistable sheaf V of character v is resolved as → L ( − , − α → L ( − , β ⊕ L (0 , − γ → V → . We can then repeat the argument of Example 6.1 to concludePic ( M H m ( v )) ∼ = Z . When µ H m ( L ) < µ H m ( v ) , the Beilinson-type resolution coincides with the dual version of the L -Gaetatype resolution and every H m -semistable sheaf V of character v is resolved as → V → L (1 , α ⊕ L (0 , β → L (1 , δ → . We can also repeat the argument of Example 6.1 with straightforward modifications. (cid:3) It is interesting to further explore the geometry of M H m ( v ) for bad characters v as in the previoustheorem, taking into account that these are unirational varieties (see §2.6) with Picard number ρ = 1 .As a step in this direction, we consider the character v from example 6.1. Example 6.4. We claim that for v = (4 , − E − F, ) and m = 1 + (cid:15) with (cid:15) ∈ Q , < (cid:15) (cid:28) , we infact have M H m ( v ) ∼ = P . First, note that for a generic V ∈ M H m ( v ) with the corresponding Gaeta-type resolution → O ( − , − ψ −→ O ( − , ⊕ O (0 , − → V → the map pr O (0 , − ◦ ψ : O ( − , − → O ( − , HE PICARD GROUP OF THE MODULI SPACE OF SHEAVES ON A QUADRIC SURFACE 51 is an injective map of vector bundles. Therefore, we can expand the Gaeta-type resolution of V into thefollowing commutative diagram O (0 , − O (0 , − O ( − , − O ( − , ⊕ O (0 , − V O ( − , − O ( − , O ( − , 2) 0 . ψ pr O ( − , So, next we consider extensions ξ := [0 → O (0 , − → V ξ → O ( − , → with ξ = ( ξ , ξ , ξ ) ∈ Ext ( O ( − , , O (0 , − ⊕ . We assert that V ξ is H m -semistable if and only ifthe corresponding vectors ξ , ξ , ξ are linearly independent.Indeed, suppose without loss of generality that ξ = aξ + bξ . Consider the morphism O (0 , − A −→ O (0 , − given by the matrix A = (cid:20) a b (cid:21) . Then the induced mapExt ( O ( − , , O (0 , − ⊕ A ∗ −−→ Ext ( O ( − , , O (0 , − ⊕ sends ( ξ , ξ ) to ( ξ , ξ , ξ ) . This fact translates into the following commutative diagram O (0 , − V ( ξ ,ξ ) O ( − , 2) 00 O (0 , − V ( ξ ,ξ ,ξ ) O ( − , 2) 0 . A Since µ H m ( V ( ξ ,ξ ) ) = − − ε > − − ε µ H m ( V ( ξ ,ξ ,ξ ) ) , we conclude that V ( ξ ,ξ ,ξ ) is unstable.Conversely, suppose V ξ is unstable. A rank bundle can be destabilized by subbundles or quotientbundles of rank or . We will only sketch the argument in the case of a destabilizing subbundle of rank and leave the similar routine checks for the other cases to the reader. Suppose there is a destabilizingsubbundle W ⊂ V ξ with r ( W ) = 2 . Since(6.4.1) µ H m ( W ) ≥ µ H m ( V ξ ) > µ H m ( O (0 , − ) , there is no maps W → O (0 , − and, therefore, the composition W → V ξ → O ( − , is not zero. Wewill further assume that this composition is surjective, leaving the check in the other case to the reader.In this case we have the following commutative diagram O ( a, b ) W O ( − , 2) 00 O (0 , − V ξ O ( − , 2) 0 B with a ≤ and b ≤ − . Furthermore, one checks that (6.4.1) is satisfied only if ( a, b ) = (0 , − . Inthis case, denote the extension defining W by ζ and write B = b b b , b i ∈ C . The induced map Ext ( O ( − , , O (0 , − B ∗ −−→ Ext ( O ( − , , O (0 , − ⊕ sends ζ to ξ = ( ξ , ξ , ξ ) = ( b η, b η, b η ) . Thus, we see that ξ , ξ , ξ are linearly dependent.Denote the locus of ξ with linearly independent component vectors ξ , ξ , ξ by U ⊂ Ext ( O ( − , , O (0 , − ⊕ . By the above discussion, the universal extension over U × X defines a dominant morphism U → M H m ( v ) . Note that the isomorphism class of V ξ only depends on the hyperplane spanned by ξ , ξ , ξ in thefour-dimensional space Ext ( O ( − , , O (0 , − , so the above map factors through P ( Ext ( O ( − , , O (0 , − ∨ ) → M H m ( v ) , as claimed.It is also interesting to note that the extensions → F → V → F → with F i ∈ M H m ( v i ) , v = (2 , − F, ) , v = (2 , − E, ) , give an embedding of a quadric into themoduli space M H m ( v ) ∼ = P : P × P (cid:44) → P . Indeed, M H m ( v i ) ∼ = P by Theorem 5.1 (3.b). The isomorphism class of V is uniquely determined by F and F because ext ( F , F ) = 1 . Finally, the stability of V follows from [CH19, Lemma 10.8] and µ H m ( F ) < µ H m ( V ) < µ H m ( F ) . Question 6.5. 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