New divisors in the boundary of the instanton moduli space
Marcos Jardim, Dimitri Markushevich, Alexander S. Tikhomirov
aa r X i v : . [ m a t h . AG ] J a n NEW DIVISORS IN THE BOUNDARY OF THE INSTANTONMODULI SPACE
MARCOS JARDIM, DIMITRI MARKUSHEVICH, AND ALEXANDER S. TIKHOMIROV
Abstract.
Let I ( n ) denote the moduli space of rank 2 instanton bundlesof charge n on P . We know from [2, 12, 19, 20] that I ( n ) is an irreducible,nonsingular and affine variety of dimension 8 n −
3. Since every rank 2 instantonbundle on P is stable, we may regard I ( n ) as an open subset of the projectiveGieseker–Maruyama moduli scheme M ( n ) of rank 2 semistable torsion freesheaves F on P with Chern classes c = c = 0 and c = n , and consider theclosure I ( n ) of I ( n ) in M ( n ).We construct some of the irreducible components of dimension 8 n − ∂ I ( n ) := I ( n ) \ I ( n ). These components generically lie in thesmooth locus of M ( n ) and consist of rank 2 torsion free instanton sheaves withsingularities along rational curves. Introduction A mathematical instanton of charge n is a holomorphic rank 2 vector bundle E on the projective space P with Chern classes(1) c ( E ) = 0 , c ( E ) = n, satisfying the vanishing conditions(2) h ( E ( − h ( E ( − . The epithet “mathematical”, which will be omitted in the remainder of the paper,distinguishes these objects from physical instantons. The latter are anti-self-dual SU (2)-connections on S , which give rise, by the Atiyah–Ward correspondence, tovector bundles E as above with some additional “reality” conditions.We denote by I ( n ) the moduli space of instantons of charge n . Nowadays it isknown that I ( n ) is a nice variety possessing some natural properties which wereconjectured long ago, but whose proof remained an open problem for many years: I ( n ) is affine [2], nonsingular [12], and irreducible of dimension 8 n − µ -stable and µ -stability is the same as Gieseker stabilityfor rank 2 bundles, I ( n ) can be regarded as an open subset within the projectiveMaruyama moduli space M ( n ) of Gieseker semistable rank 2 sheaves F on P with c ( F ) = c ( F ) = 0 and c ( F ) = n .Let I ( n ) denote the closure of I ( n ) within M ( n ). Our goal is to approachthe understanding of the boundary ∂ I ( n ) := I ( n ) \ I ( n ) of this compactification.Partial results in this direction are already known; in particular, it is clear that ∂ I ( n ) has, in general, several irreducible components. These components havebeen completely determined only for charges c = 1 [1] and c = 2 [14]. The case c = 3 was partially treated in [5, 15, 16]. Perrin in [16, Remarque 3.6.8] cites aconjecture of Gruson and Trautmann, suggesting that the boundary of I (3) consists of 8 irreducible divisors. He also advances towards a proof of this conjecture byconstructing two out of the eight conjectural boundary divisors, in addition to thethree ones constructed before.All the boundary components for charges c ≤
3, known or conjectural, aredivisorial and consist of non locally free sheaves . A new phenomenon occurs for c =5: in [18], Rao shows the existence of a divisorial component of ∂ I (5) whose genericpoint represents a vector bundle , for which the vanishing condition h ( E ( − n irreducible components of ∂ I ( n ), whosegeneric points represent instanton sheaves . By definition, these are rank 2 torsionfree sheaves E on P satisfying (1), (2), but also the additional vanishing conditions h ( E ( − h ( E ( − E is locally free. Thus, for example, Rao’s component of ∂ I (5) does not satisfythese vanishing conditions and is not of the type we are interested in. Also one cansee that among the eight Gruson–Trautmann (partly conjectural) components of ∂ I (3), only four parametrize instanton sheaves in our terminology. One can saythat the instanton sheaves provide a partial compactification of I ( n ), and we studythe boundary of this partial compactification.Our main result is that for any n ≥ ∂ I ( n ) contains n irreducible divisors D ( m, n ), where m = 1 , . . . , n , of I ( n ), whose generic points represent instantonsheaves and are smooth points of M ( n ). A sheaf in D ( m, n ), generically, is singularalong a normal rational curve of degree m . By [11], the singular locus of anyinstanton sheaf, if nonempty, is a scheme of pure dimension 1. An importantinvariant of a non locally free instanton sheaf E is Q E := E ∨∨ /E , which is a pure1-dimensional sheaf supported on the singular locus of E . A necessary condition fora pure 1-dimensional sheaf Q to be Q E for some E ∈ ∂ I ( n ) was determined in [17,Thm. 0.1]: Q ( −
2) should be a theta-characteristic of its supporting 1-dimensionalscheme.Our boundary components D ( m, n ) coincide with some of the components con-sidered in [1] (case n = 1), [14] (case n = 2) and [15, 16, 17] (case n = 3). For n >
3, the components here presented are new. However, our components arenot sufficient to exhaust the boundary of the partial compactification by instantonsheaves, for one of the components of ∂ I (3) presented in [16] is associated to thetacharacteristics of plane cubic curves in P . We still do not know which pure 1-dimensional sheaves do occur as singular locus of non locally free instanton sheavesin the boundary of I ( n ). Nonetheless, we expect that our examples, associated totheta characteristics of smooth rational curves, are singled out by that they aresmooth points of M ( n ). We provide an evidence to support this expectation byproving that the instanton sheaves which are singular along elliptic curves of degree ≤
4, form a separate component of M ( n ).Now we will briefly describe the content of the paper by sections. In Section 2,we introduce instanton sheaves and provide some relevant properties. In Section 3,we prove basic properties of the main tool for constructing non locally free instantonsheaves: elementary transformations along a curve in P endowed with a line bundle Q satisfying h ( Q ( − h ( Q ( − F , obtained EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 3 by an elementary transformation, with a view towards computing the infinitesimaldeformation and obstruction spaces Ext i ( F, F ) ( i = 1 , n − D ( m, n ) of M ( n ) and their open subsets D ( m, n ), parametrizing instantonsheaves F obtained by elementary transformations of instanton bundles of charge n along rational normal curves of degree m (Proposition 6.4). We also prove thatfor an instanton sheaf F from D ( m, n ), we have dim Ext ( F, F ) = 8 n − ( F, F ) = 0, so that M ( n ) is smooth of local dimension 8 n − F (Proposi-tion 6.5).In Section 7, we prove the main result of the paper, stating that D ( m, n ) aredivisors lying in the boundary ∂ I ( n ) of the instantons of charge n for each m =1 , . . . , n (Theorem 7.8). This is done by induction on m and n , using deformationsof elementary transforms of several types along 1-dimensional sheaves with supportson reducible rational curves.In Section 8, we show that the elementary transformations of null-correlationinstanton bundles along elliptic quartic curves provide a generically smooth irre-ducible component of M (5) of dimension 37, equal to the dimension of I (5) (The-orem 8.2). We also remark that somewhat easier arguments prove the followingsimilar result: the elementary transforms of O ⊕ P along plane cubic curves fill anirreducible component of M (3) of dimension 21, equal to the dimension of I (3).This property implies, in particular, that the boundary divisor of I (3) describedby Perrin in [16], associated to plane cubic curves, is a non normal singularity of M (3), given by the intersection of this new irreducible component with I (3).Throughout the paper, the base field will be the field of complex numbers C . Wework in the algebraic setting, so the term “variety” means “reduced scheme of finitetype over C ”. We will use interchangeably the terms “vector bundle” and “locallyfree sheaf”. The projectivization P ( V ) of a vector space V will be understood asthe variety of lines in V , and the projectivization P ( E ) of a vector bundle E overa variety X as the relative proj-scheme Proj X (cid:0) S ym • ( E ∨ ) (cid:1) . Acknowledgements.
MJ is partially supported by the CNPq grant number302477/2010-1 and the FAPESP grant number 2014/14743-8. MJ aknowledges thehospitality of the University of Lille 1. DM acknowledges the FAPESP grant num-ber 2011/06464-3, which allowed him to visit the University of Campinas. DM wasalso partially supported by Labex CEMPI (ANR-11-LABX-0007-01). DM and ATacknowledge the hospitality of the Max-Planck-Institut f¨ur Mathematik in Bonn,where they made a part of work on the paper.2.
Monads and instanton sheaves
In this section we set up notation and revise important results from the vastliterature on instanton sheaves on P . We also establish a couple of new propositionsthat will be relevant for the main results of the present paper. We work over thefield of complex numbers; all sheaves of modules are coherent.Recall that a monad on a projective variety X is a complex of locally free sheavesof the form(3) A α → B β → C MARCOS JARDIM, DIMITRI MARKUSHEVICH, AND ALEXANDER S. TIKHOMIROV where α is injective and β is surjective. The sheaf ker β/ im α is called the coho-mology of (3). If A = O X ( − ⊕ a , B = O ⊕ bX and C = O X (1) ⊕ c , then (3) is a called linear monad .An instanton sheaf on P is a torsion free sheaf F with trivial determinant andsatisfying the following cohomological conditions H ( F ( − H ( F ( − H ( F ( − H ( F ( − . The integer n := − χ ( F ( − F ; it is easy to check that n = h ( F ( − c ( F ). The trivial sheaf O ⊕ r P of rank r is considered as aninstanton sheaf of charge zero.Using the Beilinson spectral sequence, one can show that the instanton sheavesare precisely those that arise as cohomologies of linear monads of the form H ( F ⊗ Ω P (1)) ⊗ O P ( − → H ( F ⊗ Ω P ) ⊗ O P → H ( F ( − ⊗ O P (1) , see for instance [4, Proposition 14]. One checks that h ( F ⊗ Ω P (1)) = h ( F ( − n and h ( F ⊗ Ω P ) = 2 n + r , where r denotes the rank of F . The above monad cantherefore be written in the following simpler way:(4) M • : O P ( − ⊕ n α → O ⊕ n + r P β → O P (1) ⊕ n The following result will be relevant in what follows.
Lemma 2.1. If F is an instanton sheaf on P , then (i) E xt p ( F, F ) = 0 for p ≥ ; (ii) E xt p ( F, O P ) = 0 for p ≥ ; (iii) Ext ( F, F ) = 0 .Proof.
Since F can be realized as the cohomology of a monad of the form (4), wehave a short exact sequence0 → O P ( − ⊕ c α → K → F → , where K := ker β is a locally free sheaf. Applying the functor H om ( − , F ) we obtain E xt p − ( O P n ( − ⊕ a , F ) → E xt p ( F, F ) → E xt p ( K, F ) . Since O P ( −
1) and K are locally free sheaves, the first and the third sheaves in theprevious sequence vanish for p ≥
2, and the first item follows.The second and the third items are proved in a similar way; compare with [10,Corollary 7]. (cid:3)
Recall that the singular locus Sing( G ) of a coherent sheaf G on a nonsingularprojective variety X is given bySing( G ) := { x ∈ X | G x is not free over O X,x } , where G x denotes the stalk of G at a point x and O X,x is its local ring.It is not difficult to see that the singular locus of a non locally free instantonsheaf F on P is precisely the setSing( F ) := { x ∈ P | α ( x ) is not injective } , where α ( x ) denotes the map of fibers over the point x ∈ P .Now we focus on rank 2 instanton sheaves, which are the main object of studyin this paper. The following result is proved in [11, Main Theorem]. Theorem 2.2. If F is a non locally free instanton sheaf of rank on P , then EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 5 (i) its singular locus has pure dimension ; (ii) F ∨∨ is a (possibly trivial) locally free instanton sheaf. In particular, every relfexive instanton sheaf of rank 2 is actually locally free, asit can also be deduced from [6, Proposition 2.6]. We remark that both claims arefalse for instanton sheaves of rank at least 3, see [11] for precise examples.
Lemma 2.3.
Every nontrivial instanton sheaf F of rank on P is simple and H ( F ) = 0 . If F is locally free, then it is µ -stable. If F is torsion free, then it is µ -semistable. We will see below in Lemma 4.1 that there are non locally free rank 2 instantonsheaves that are not µ -stable, though certain non locally free rank 2 instantonsheaves are µ -stable, see Lemma 4.1 below. Proof.
The simplicity claim is [10, Lemma 23]; the vanishing of h ( F ) is [10, Propo-sition 11]. If F is locally free, the vanishing of h ( F ) is sufficient to guarantee the µ -stability. From the sequence0 → F → F ∨∨ → F ∨∨ /F → F is not µ -semistable, then neither is F ∨∨ , which is a contradiction. (cid:3) Elementary transformations of instantons
Let Σ be a reduced locally complete intersection curve of arithmetic genus g . Bythe degree of a line bundle L on Σ we understand the integer deg( L ) = χ ( L )+ g − k ∈ Z set Pic k (Σ) := { [ L ] ∈ Pic(Σ) | deg( L ) = k } .Let E be an instanton sheaf (possibly trivial); elementary transformation data (Σ , L, ϕ ) for E consist of the following:(i) an embedding ι : Σ ֒ → P of degree d ;(ii) a line bundle L ∈ Pic g − (Σ) such that h ( ι ∗ L ) = h ( ι ∗ L ) = 0;(iii) a surjective morphism ϕ : E → ( ι ∗ L )(2). Proposition 3.1.
Given an instanton sheaf E of rank r and charge n , and ele-mentary transformation data (Σ , L, ϕ ) for E as above, the sheaf F := ker ϕ is aninstanton sheaf of rank r and charge n + d . Moreover, F ∨ ≃ E ∨ and, if E is locallyfree, then F ∨∨ /F ≃ ( ι ∗ L )(2) . The sheaf F := ker ϕ is called an elementary transform of E along Σ. A similarconstruction was proposed by Maruyama and Trautmann in [13, Definition 1.7] forthe case when Σ is a union of finitely many disjoint lines. Proof.
First, applying Riemann–Roch for the immersion ι : Σ ֒ → P , we have(5) ch(( ι ∗ L )(2)) = ι ∗ (cid:0) ch( L ( 2 d pt)) · td( N Σ / P ) − (cid:1) , where L ∈ Pic g − (Σ) and N Σ / P is the normal bundle, related to the tangentbundles T Σ and T P to Σ and P respectively by the exact triple(6) 0 → T Σ → ι ∗ T P → N Σ / P → . It follows from (6) that c ( N Σ / P ) = 4 d + 2 g −
2. Plugging this back into (5), weget ch (( ι ∗ L )(2)) = c ( L ( 2 d pt)) − c ( N ) / g − d −
12 (4 d + 2 g −
2) = 0 . MARCOS JARDIM, DIMITRI MARKUSHEVICH, AND ALEXANDER S. TIKHOMIROV
Now consider the short exact sequence(7) 0 → F δ −→ E ϕ −→ ( ι ∗ L )(2) → . Since ch (( ι ∗ L )(2)) = ch (( ι ∗ L )(2)) = 0, we obtain c ( F ) = c ( F ) = 0; since alsoch (( ι ∗ L )(2)) = [Σ], we conclude that c ( F ) = c ( E ) + d = n + d .Next, twisting (7) by O P ( −
2) and passing to cohomology, we see that h ( F ( − h ( F ( − h p ( ι ∗ L ) = 0 for p = 0 , h ( E ( − h ( E ( − h ( F ( − h ( F ( − F is an instanton sheaf ofrank r and charge n + d .Finally, note that E xt (( ι ∗ L )(2) , O P ) = 0 because the sheaf ( ι ∗ L )(2) is supportedin dimension 1. Therefore, dualizing the sequence (7), we obtain the isomorphism F ∨ ≃ E ∨ .For the last claim, consider the diagram0 (cid:15) (cid:15) (cid:15) (cid:15) / / F (cid:15) (cid:15) / / E ≃ (cid:15) (cid:15) / / ( ι ∗ L )(2) / / F ∨∨ (cid:15) (cid:15) ≃ / / E ∨∨ F ∨∨ /F (cid:15) (cid:15) F ∨∨ /F ≃ ( ι ∗ L )(2). (cid:3) Note that two elementary transforms F and F ′ of the same instanton sheaf E given by elementary transformation data (Σ , L, ϕ ) and (Σ ′ , L ′ , ϕ ′ ) respectivelyare isomorphic if and only if there is an isomorphism ψ : ι ∗ L → ι ′∗ L ′ such that ψ ◦ ϕ = ϕ ′ .Consider now the case when Σ = Σ ∪ Σ , where Σ and Σ are reduced, lo-cally complete intersection curves such that Σ ∪ Σ is again a locally completeintersection curve. Take an instanton sheaf E and elementary transformation data(Σ , L , ϕ ) for E ; let F be the elementary transform of E along these data. Thentake some elementary transformation data (Σ , L , ϕ ) for F , and let F be the EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 7 corresponding elementary transform. We obtain the diagram(8) 0 (cid:15) (cid:15) / / F j (cid:15) (cid:15) i ◦ j / / E ≃ (cid:15) (cid:15) / / Q / / (cid:15) (cid:15) ✤✤✤ / / F ϕ (cid:15) (cid:15) i / / E ϕ / / ( ι ∗ L )(2) / / ι ∗ L )(2) (cid:15) (cid:15) F is an elementary transform of E along a torsion sheaf Q fitting into theshort exact sequence 0 → ( ι ∗ L )(2) → Q → ( ι ∗ L )(2) → . Note that if Σ and Σ do not intersect, then Q = ( ι ∗ L )(2) ⊕ ( ι ∗ L )(2). We willsay that F is obtained from E by a concatenation of the elementary transformationswith elementary transformation data (Σ , L , ϕ ), (Σ , L , ϕ ). Remark 3.2.
More generally, fix an instanton sheaf E of rank r and charge n , andlet Quot d ( k +2) ( E ) denote the Quot scheme of quotients ϕ : E ։ Q with Hilbertpolynomial P Q ( k ) = d ( k + 2).Take ( Q, ϕ ) ∈ Quot dk +2 d ( E ) satisfying h ( Q ( − h ( Q ( − F := ker ϕ is an instanton sheaf of rank r and charge n + d .We say that F is a transform of E along Q .In this case, one has the short exact sequence0 → F −→ E ϕ −→ Q → . Since E is an instanton sheaf and Q is supported in dimension 1, we have h ( F ( − h ( F ( − h ( Q ( − h ( Q ( − h ( F ( − h ( F ( − F is a linearsheaf. The Hilbert polynomial of F is given by P F ( k ) = P E ( k ) − ( dk + 2 d ) = r k + rk + (cid:18) r − ( n + d ) (cid:19) k + ( r − n + d )) , so that c ( F ) = 0 and c ( F ) = n + d .4. Stability of transforms
In this section we will prove several facts regarding the stability and µ -stability ofnon locally free instanton sheaves, obtained as transforms of locally free instantonsin the way described in Remark 3.2. Lemma 4.1. If E is a µ -(semi)stable instanton sheaf, then every transform of E isalso µ -(semi)stable. Moreover, every transform of the trivial sheaf is µ -semistablebut not µ -stable. MARCOS JARDIM, DIMITRI MARKUSHEVICH, AND ALEXANDER S. TIKHOMIROV
Proof. If F is a transform of a µ -(semi)stable instanton sheaf E which is not µ -(semi)stable, then it admits a subsheaf G with µ ( G ) ≥ µ ( G ) > G would also destabilize E .It follows in particular that every transform F of the trivial sheaf is µ -semistable;to see that it is not µ -stable, just note that H ( F ∨ ) = H ( O ⊕ r P ) = 0. (cid:3) As is well known, every nontrivial locally free instanton sheaf of rank 2 is µ -stable. This implies: Corollary 4.2.
Every transform of a nontrivial locally free instanton sheaf of rank is µ -stable, and hence stable. Now we will focus on the case of rank 2. We will see that certain transforms ofthe trivial rank 2 sheaf are also stable.In what follows, we denote by p F ( k ) the reduced Hilbert polynomial of a sheaf F on P . Lemma 4.3.
Every elementary transform of the trivial sheaf of rank along anirreducible curve is stable.Proof. Let F be defined by the short exact sequence0 → F → O ⊕ P ϕ −→ Q → , L, ϕ ) with irreducible Σ, where Q =( ι ∗ L )(2) and ι : Σ ֒ → P is the natural embedding. It is enough to considertorsion free sheaves G ⊂ F of rank 1 whose quotient T := F/G is also torsion free.Moreover, since F is µ -semistable, we can take deg( G ) = 0, thus G is the idealsheaf I ∆ of a subscheme ∆ ⊂ P of dimension at most 1.We thus obtain the diagram(9) 0 (cid:15) (cid:15) (cid:15) (cid:15) / / I ∆ (cid:15) (cid:15) / / O P (cid:15) (cid:15) / / ı ∆ ∗ O ∆ / / ζ (cid:15) (cid:15) ✤✤✤ / / F (cid:15) (cid:15) / / O ⊕ P (cid:15) (cid:15) / / Q / / T (cid:15) (cid:15) O P (cid:15) (cid:15) , where ı ∆ : ∆ ֒ → P is the natural embedding. It provides a map ζ : O ∆ → Q andan exact sequence 0 → ker ζ → T → O P → coker ζ → . Since T is torsion free, it follows that ker ζ = 0 and O ∆ is a subsheaf of Q . Since Σis irreducible, we must then have that either ∆ = Σ, and hence G = I Σ , or ∆ = ∅ ,and hence G = O P . EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 9
Since H ( F ) = 0 by Lemma 2.3, the second case does not occur. In the firstcase we have, p F ( k ) − p I Σ ( k ) = d k > , where d := deg(Σ) = c ( F ). Hence F is stable. (cid:3) Remark 4.4.
Note that we do not use the hypothesis h ( Q ( − h ( Q ( − Q is sheaf with Hilbert polynomial P Q ( k ) = dk + 2 d and irreducible support, then the kernel of a surjective morphism O ⊕ P ։ Q is a stable rank 2 torsion free sheaf F with c ( F ) = c ( F ) = 0 and c ( F ) = d .As a by-product of the previous proof we also obtain the following interestingfact. Corollary 4.5.
Every elementary transform of O ⊕ P along an irreducible curve Σ of genus g and degree d is an extension of an ideal sheaf I Z of a -dimensionalsubscheme Z ⊂ P of length d + g − by the ideal sheaf I Σ . We now consider another situation, given by a concatenation of two elementarytransformations along irreducible rational curves, that will be relevant later on.Let ı : ℓ ֒ → P be a line and let : Γ ֒ → P be a rational curve of degree m − m ≥ ℓ ∩ Γ = ∅ , or ℓ , Γ intersect quasi-transversely at asingle point P ; we say that the intersection is quasi-transverse when the tangentsto Γ , ℓ at the intersection point are distinct. Consider the sheaf Q given by anextension(10) 0 → ( ı ∗ O ℓ ( − pt))(2) −→ Q τ −→ ( ∗ O Γ ( − pt))(2) → . Such an extension may be nontrivial only if Γ , ℓ intersect. So, either Q ≃ ( ı ∗ O ℓ (pt)) ⊕ ( ∗ O Γ ((2 m − , or the curve Γ ∪ ℓ is connected and Q is a line bundle on Γ ∪ ℓ such that Q | ℓ ≃ O ℓ (2pt) , Q | Γ ≃ O ℓ ((2 m − . Let E be an instanton sheaf of rank r and charge c , and assume there exists asurjective map ϕ : E → Q . Let F := ker ϕ . We have the following diagram:(11) 0 (cid:15) (cid:15) ( ı ∗ O ℓ ( − pt))(2) (cid:15) (cid:15) / / F / / E ≃ (cid:15) (cid:15) ϕ / / Q / / τ (cid:15) (cid:15) / / F ′ / / E τ ◦ ϕ / / ( ∗ O Γ ( − pt))(2) / / (cid:15) (cid:15) , where F ′ := ker τ ◦ ϕ . We then obtain the short exact sequence(12) 0 → F → F ′ → ( ı ∗ O ℓ ( − pt))(2) → . Comparing with diagram (8), we see that F is obtained by concatenation of twoelementary transformations, first along Γ then along ℓ . Lemma 4.6.
Every transform of the trivial sheaf of rank along a sheaf Q givenby an extension (10) is stable.Proof. Let F be defined by the exact triple0 → F → O ⊕ P → Q → . As observed above, F also fits into the short exact sequence (12), where F ′ is anelementary transform of the trivial sheaf of rank 2 along Γ.Proceeding as in the beginning of the proof of Lemma 4.3, we conclude that anyrank 1 torsion free subsheaf G ⊂ F with torsion free quotient T := F/G and zerodegree is the ideal sheaf I ∆ of a closed subscheme ∆ ⊂ P of dimension at most 1,and that O ∆ is a subsheaf of Q . There are only four possibilities for ∆: ∅ , ℓ , Γand Γ ∪ ℓ ; then, respectively, G = O P , G = I ℓ , G = I Γ , and G = I Γ ∪ ℓ .The first possibility does not occur, since H ( F ) = 0. Let us examine the otherthree possibilities.In the case G = I ℓ , we have G ⊂ F ⊂ F ′ and p F ′ ( k ) − p I ℓ ( k ) = (3 − m )2 k + 2 − m. If m ≥
3, we get p F ′ ( k ) − p I ℓ ( k ) <
0, so G destabilizes F ′ , which is impossibleby Lemma 4.3. Assume now that G = I ℓ and m = 2. Then ℓ , Γ are two linesspanning a plane, and I ℓ does not destabilize F ′ , but p F ( k ) − p I ℓ ( k ) = − I ℓ destabilizes F . Let us see that this case is impossible.Consider the diagram (9) for our sheaf F with ∆ = ℓ . Arguing as in the proofof Lemma 4.3, we can complete it to the diagram(13) 0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / I ℓ (cid:15) (cid:15) / / O P (cid:15) (cid:15) / / ι ∗ O ℓ / / ζ (cid:15) (cid:15) / / F (cid:15) (cid:15) / / O ⊕ P (cid:15) (cid:15) / / Q (cid:15) (cid:15) / / / / T (cid:15) (cid:15) / / O P / / (cid:15) (cid:15) coker ζ / / (cid:15) (cid:15)
00 0 0If, for instance, Γ, ℓ intersect and the extension (10) is non-trivial, then Q | ℓ ≃ ( ι ∗ O ℓ )(2) and Q | Γ ≃ ( ∗ O Γ )(1). As ζ factors through Q | ℓ ( − pt) ֒ → Q ,coker ζ ≃ ( ∗ O Γ )(1) ⊕ C P ′ , where C P ′ denotes a sky-scraper sheaf of length 1 supported at a point P ′ ∈ ℓ . It isobvious that coker ζ cannot be generated by a single section, so that the surjection O P → coker ζ in the last line of (13) does not exist. By a completely similarargument, one treats the remaining cases: (a) Γ, ℓ intersect, but the extension (10) EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 11 is trivial, and (b) Γ, ℓ do not intersect. The conclusion is that G = I ℓ for m = 2cannot occur as a destabilizing subsheaf.Next, the ideal sheaf I Γ does not destabilize F when m ≥
3, for p F ( k ) − p I Γ ( k ) = m − k + 1 − m > , and it does not destabilize F for m = 2 by an analysis of a commutative diagram,similar to (13) (see also Remark 4.7 below).Finally, we check that the ideal sheaf I Γ ∪ ℓ also does not destabilize F when m ≥
2; indeed: p E ( k ) − p I Γ ∪ ℓ ( k ) = m k + (2 − m ) > . (cid:3) Remark 4.7.
Observe that we can permute the roles of ℓ, Γ in the constructionof the concatenation. In the same notation as in the paragraph preceding equation(10), let Q be given by a nontrivial extension0 → ( ∗ O Γ ( − pt))(2) −→ Q τ −→ ( ı ∗ O ℓ ( − pt))(2) → , and let F be the rank 2 instanton sheaf obtained by an elementary transformationof the trivial sheaf of rank 2 along Q :0 → F → O ⊕ P → Q → . We then have the following two exact triples:0 → F → F ′ → ( ∗ O Γ ( − pt))(2) → → F ′ → O ⊕ P → ( ı ∗ O ℓ ( − pt))(2) → . Note that F ′ is stable by Lemma 4.3, and the stability of F is proved by an argumentsimilar to that of Lemma 4.6.For example, the ideal sheaf I ℓ would destabilize F for some m ≥
2, if onemight find an embedding I ℓ ֒ → F with torsion free quotient T = F/I ℓ . Let usassume that such an embedding exists. Then we can complete it to a diagram ofthe form (13). We observe that Q | ℓ ≃ ( ι ∗ O ℓ )(1) and Q | Γ ≃ ∗ O Γ ((2 m − ζ ≃ ∗ O Γ ((2 m − Hom’s and Ext’s of elementary transforms
We start by the following general claim.
Lemma 5.1.
Let ι : Σ ֒ → P be a reduced locally complete intresection curve, M an invertible O Σ -sheaf, and E a rank 2 locally free sheaf on P equipped witha surjective map ϕ : E → ι ∗ M . For the torsion free sheaf F := ker ϕ , we have H om ( E, E ) / H om ( F, F ) ≃ ι ∗ M ⊗ det( E ) ∨ .Proof. First, apply H om ( − , E ) to the short exact sequence(14) 0 → F → E → ι ∗ M → . Since ι ∗ M is a torsion sheaf supported on a curve and E is locally free, we have H om ( ι ∗ M, E ) = E xt ( ι ∗ M, E ) = 0, and conclude that H om ( E, E ) ≃ H om ( F, E ).Next, applying H om ( ι ∗ M, − ) to the sequence (14) and again using the van-ishing of H om ( ι ∗ M, E ) and E xt ( ι ∗ M, E ), we obtain that E xt ( ι ∗ M, F ) ≃H om ( ι ∗ M, ι ∗ M ) = ι ∗ O Σ . Now, the diagram(15) 0 (cid:15) (cid:15) / / H om ( E, F ) (cid:15) (cid:15) / / H om ( E, E ) / / ≃ (cid:15) (cid:15) H om ( E, ι ∗ M ) / / / / H om ( F, F ) (cid:15) (cid:15) τ / / H om ( E, E ) / / coker τ / / ι ∗ O Σ (cid:15) (cid:15) H om ( E, − ) tothe sequence (14). The second row comes from applying H om ( F, − ) to the samesequence, and using the identification H om ( E, E ) ≃ H om ( F, E ). The left columncomes from applying H om ( − , F ) to (14), noting that E xt ( ι ∗ M, F ) ≃ ι ∗ O Σ and E xt ( ι ∗ M, E ) = 0.Now the Snake Lemma provides us with the short exact sequence0 → ι ∗ O Σ → E ∨ ⊗ ι ∗ M → coker τ → , since H om ( E, ι ∗ M ) ≃ E ∨ ⊗ ι ∗ M . Since E has rank 2, it easily follows that H om ( E, E ) / H om ( F, F ) = coker τ ≃ ι ∗ ( M ) ⊗ det( E ) ∨ , as desired. (cid:3) Next, we apply the previous Lemma to the case of elementary transformationsof locally free instanton sheaves.
Lemma 5.2.
Let E be a rank locally free instanton sheaf of charge n , and let (Σ , L, ϕ ) be elementary transformation data for E . If L is an invertible O Σ -sheafsatisfying h (( ι ∗ L )(4)) = 0 , then for the sheaf F =: ker ϕ , we have (i) Ext ( F, F ) = H ( E xt ( F, F )) ⊕ H ( H om ( F, F )) ; (ii) Ext ( F, F ) = H ( E xt ( F, F )) ≃ H ( E xt (( ι ∗ L )(2) , F )) ;Furthermore, one has (16) h ( H om ( F, F )) = 8 n + h (( ι ∗ L )(4)) − Proof.
We use the local-to-global spectral sequence for Ext’s, whose E term is ofthe form E pq = H p ( E xt q ( F, F )) . First, from Lemma (2.1(i)) we know that E xt q ( F, F ) = 0 for q = 2 ,
3, killing theterms E pq for q = 2 ,
3. Moreover, since E xt ( F, F ) is suported on Σ, it follows that E p for p = 2 , → F ( − → E ( − → ι ∗ L → , we obtain the short exact sequence(17) 0 → H om ( F, F ) → H om ( E, E ) → ( ι ∗ L )(4) → . EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 13
Since ι ∗ L is supported in dimension 1, passing to cohomology, we conclude that H ( H om ( F, F )) ≃ Ext ( E, E ) = 0 by (2.1 (iii)); thus E also vanishes.Finally, sequence (17) and the hypothesis h (( ι ∗ L )(4)) = 0 imply that H ( H om ( F, F )) ≃ H ( H om ( E, E )). The last cohomology vanishes by hypothe-sis, thus also E = 0.It then follows that the spectral sequence already degenerates at the second page,and one concludes that:Ext ( F, F ) = H ( E xt ( F, F )) ⊕ H ( H om ( F, F )) and Ext ( F, F ) = H ( E xt ( F, F )) , as desired. The isomorphism H ( E xt ( F, F )) ≃ H ( E xt (( ι ∗ L )(2) , F )) is obtainedfrom applying the functor H om ( − , F ) to the sequence (7).In order to establish formula (16), consider the cohomology exact sequence as-sociated to the exact triple of sheaves (17):0 → H ( H om ( F, F )) → H ( H om ( E, E )) → H (( ι ∗ L )(4)) → H ( H om ( F, F )) → H ( H om ( E, E )) → . Since every nontrivial rank 2 instanton sheaf is simple, we have h ( H om ( F, F )) = 1. Then, counting dimensions in the sequence above, wehave h ( H om ( F, F )) = h ( H om ( E, E )) + h (( ι ∗ L )(4)) + 1 − h ( H om ( E, E )) . There are now two cases to consider. First, if E is a nontrivial rank 2 locallyfree instanton of charge n , it follows that Ext ( E, E ) = 0 (see [12]). Moreover, h ( H om ( E, E )) = 1, since E is simple. Therefore, h ( H om ( E, E )) = 8 n −
3, hencethe desired formula follows.On the other hand, if E = O ⊕ P (i.e. if n = 0), then h ( H om ( E, E )) = 4 and h ( H om ( E, E )) = 0, hence one also obtains formula (16). (cid:3)
Lemma 5.2 prompts us to characterize the sheaf E xt ( F, F ) when F is an ele-mentary transform of a rank 2 locally free instanton sheaf. Applying the functor H om ( F, − ) to sequence (7) we obtain(18) 0 → H om ( F, F ) → H om ( F, E ) → H om ( F, ( ι ∗ L )(2)) →E xt ( F, F ) → E xt ( F, E ) → E xt ( F, ( ι ∗ L )(2)) → , since E xt ( F, F ) = 0 by Lemma 2.1. Invoking now Lemma 5.1 and the isomorphism H om ( F, E ) ≃ H om ( E, E ) (see the first paragraph of the proof of Lemma 5.1)sequence (18) can be rewritten in the following way:(19) 0 → ( ι ∗ L )(4) → H om ( F, ( ι ∗ L )(2)) → E xt ( F, F ) →E xt ( F, O P ) ⊗ E → E xt ( F, ( ι ∗ L )(2)) → . In the next section, we will carefully analyze each term of this exact sequencefor elementary transforms along rational curves.6.
Definition and properties of D ( m, n )Let R ( m ) denote the space of nonsingular rational curves of degree m on P . Itis well-known (see e. g. [3]) that R ( m ) is a nonsingular irreducible quasiprojectivevariety of dimension 4 m and that the set R ∗ ( m ) := { Γ ∈ R ( m ) | N Γ / P ≃ O Γ ((2 m − ⊕ } , where pt denotes a point of Γ, is a dense open subset of R ( m ) for m ≥ , m = 2.Besides, it is obvious that N Γ / P ≃ O Γ (2pt) ⊕ O Γ (4pt) for every Γ ∈ R (2). We set R ∗ (2) := R (2). Lemma 6.1.
Let E be an instanton sheaf. Then for any m ≥ , the restriction of E to a generic rational curve of degree m in P is trivial.Proof. For m = 1, the assertion follows from the µ -semistability of E and theGrauert–M¨ullich Theorem [9, Theorem 3.1.2]. For m >
1, we start by restriction toa generic chain of m lines and then smooth out the chain of lines to a nonsingularrational curve of degree m .By a chain of lines we mean a curve Γ = ℓ ∪ ... ∪ ℓ m in P such that ℓ , ..., ℓ m are distinct lines and ℓ i ∩ ℓ j = ∅ if and only if | i − j | ≤
1. It is well known (seee. g. [8, Corollary 1.2]) that a chain of lines Γ = ℓ ∪ ... ∪ ℓ m in P consideredas a reducible curve of degree m can be deformed in a flat family with a smoothone-dimensional base (∆ ,
0) to a nonsingular rational curve Γ ∈ R ( m ). Makingan ´etale base change, we can obtain such a smoothing with a cross-section.By the case m = 1, the restriction of E to a generic line is trivial. By inductionon m , we easily deduce that for a generic chain of lines Γ , the restriction of E to Γ is also trivial: E | Γ ≃ O ⊕ , which is equivalent to saying that E | ℓ i ≃ O ⊕ ℓ i for all i = 1 , . . . , m . Choosing a smoothing { Γ t } t ∈ ∆ of Γ with a cross-section t x t ∈ Γ t as above, we remark that E | Γ t ≃ O Γ t ( k t pt) ⊕ O Γ t ( − k t pt) for someinteger k t which may depend on t . The triviality of E | Γ t is thus equivalent to thevanishing of h ( E | Γ t ( − pt)). Using the semi-continuity of h ( E | Γ t ( − x t )), we seethat E | Γ t is trivial for generic t ∈ ∆. (cid:3) Lemma 6.2.
Let ≤ m ≤ n , and let E be an instanton sheaf of charge n . Then R ∗ ( m ) E := { Γ ∈ R ∗ ( m ) | E | Γ ≃ O ⊕ } is a nonempty open subset of R ∗ ( m ) . Moreover, B ( m, n ) := { ([ E ] , Γ) ∈ I ( n − m ) × R ∗ ( m ) | Γ ∈ R ∗ ( m ) E } is a nonempty open subset of I ( n − m ) × R ∗ ( m ) , whose projections to both factorsare surjective.Proof. To prove the first assertion, note that R ∗ ( m ) E is nonempty by Lemma 6.1.It thus suffices to show that any Γ ∈ R ∗ ( m ) E has a Zariski open neighborhood in R ∗ ( m ), entirely contained in R ∗ ( m ) E .For any Γ ∈ R ∗ ( m ) E , we can find a plane Π ≃ P in P meeting Γ transverselyin m distinct points. Then the set U = U (Π) ⊂ R ∗ ( m ) of curves Γ ′ ∈ R ∗ ( m ) thatmeet Π transversely in m points is a Zariski neighborhood of Γ as a point of R ∗ ( m ).Define ˜ U = ˜ U (Π) := { ( x, Γ) ∈ Π × U | x ∈ Γ } . Let Γ ⊂ P × R ∗ ( m ) be the universal degree- m rational curve over R ∗ ( m ), and forany morphism T → R ∗ ( m ), denote by Γ T the pullback T × R ∗ ( m ) Γ ⊂ P × T of Γ viathis morphism. For t ∈ T , we will denote by Γ t the fiber of Γ T over t . The naturalmap ˜ U → U is an ´etale covering of degree m , and the pullback Γ ˜ U → ˜ U admits m disjoint cross-sections x i : ˜ U → Γ ˜ U ( i = 1 , . . . , m ) with images in Π × ˜ U . For˜ u ∈ ˜ U , the triviality of E | Γ ˜ u is equivalent to the vanishing h ( E | Γ ˜ u ( − x (˜ u ))) = 0,so, by the semi-continuity of h , the subset ˜ V ⊂ ˜ U of points ˜ u ∈ ˜ U such that E | Γ ˜ u is trivial is open in ˜ U . By the openness of finite ´etale morphisms, the image V of EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 15 ˜ V in U is open. This is an open neighborhood of Γ contained in R ∗ ( m ) E , so thefirst assertion is proved.The second assertion is obtained by relativizing the above argument over I ( n − m ). This is straightforward when I ( n − m ) possesses a universal instan-ton bundle. By [9, Section 4.6], this happens when n − m is odd. Moreover, theassertion is trivial for n − m = 0, in which case the only instanton bundle is O ⊕ P .In the case of even n − m ≥
2, the argument extends verbatim by replacing “uni-versal” by “quasi-universal”. A quasi-universal sheaf e E of multiplicity ν ≥ P × M st ( n − m ) such that e E | P ×{ t } ≃ E ⊕ νt for every t ∈ M st ( n − m ),where M st ( n − m ) is the locus of stable sheaves in M ( n − m ), and E t denotes astable sheaf whose isomorphism class is represented by t . By loc. cit., there existsa quasi-universal sheaf of multiplicity ν = 1 if n − m is odd (and then it is indeeduniversal) and ν = 2 if n − m is even.Taking a pair ( t, Γ) ∈ B ( m, n ), we choose a plane Π in P meeting Γ transverselyand define U, ˜ U as above, and consider the relative degree- m rational curve Υ := I ( n − m ) × Γ ˜ U with a natural projection π : Υ → I ( n − m ) × ˜ U . Then the set e V = n ( t, ˜ u ) ∈ I ( n − m ) × ˜ U (cid:12)(cid:12)(cid:12) h (cid:16) e E Υ | π − ( t, ˜ u ) ( − x (˜ u )) (cid:17) = 0 o is open by Grauert’s semicontinutity. Hence its image V under the ´etale map I ( n − m ) × ˜ U → I ( n − m ) × R ∗ ( m ) is an open neighborhood of ( t, Γ), and theopenness of B ( m, n ) is proved. (cid:3) Corollary 6.3. B ( m, n ) is a nonsingular irreducible quasiprojective variety of di-mension n − m − . For ( t, Γ) ∈ B ( m, n ), let L be the line bundle O Γ ( − pt) of degree − ι the natural embedding Γ ֒ → P , we have h ( ι ∗ L ) = h ( ι ∗ L ) = 0 andHom( E t , ( ι ∗ L )(2)) ≃ Hom( O ⊕ , O Γ ((2 m − ≃ H ( O Γ ((2 m − ⊕ . We thus conclude that there exists a surjective map ϕ : E t → ( ι ∗ L )(2), so that(Γ , L, ϕ ) is a set of elementary transformation data for E t .By Proposition 3.1, Corollary 4.2 and Lemma 4.3, F := ker ϕ is a stable rank2 instanton sheaf of charge n , whose double dual is isomorphic to E t . For n ≥ m = 1 , . . . , n , consider the subset D ( m, n ) of M ( n ) consisting of theisomorphism classes [ F ] of the sheaves F obtained in this way: D ( m, n ) := { [ F ] ∈ M ( n ) | [ F ∨∨ ] ∈ I ( n − m ) , Γ = Supp( F ∨∨ /F ) ∈ R ∗ ( m ) F ∨∨ , and F ∨∨ /F ≃ ( ι ∗ L )(2) , where L = O Γ ( − pt) } . Let D ( m, n ) denote the closure of D ( m, n ) in M ( n ). Proposition 6.4.
For n ≥ and ≤ m ≤ n , D ( m, n ) is an irreducible projectivevariety of dimension n − , and D ( m, n ) is open in D ( m, n ) .Proof. We will construct a Severi–Brauer variety P = P ( m, n ), fibered over B = B ( m, n ) with fiber P m − , and an open subset Z ⊂ P together with a morphism
Z → M ( n ) mapping Z bijectively onto D = D ( m, n ).According to [9, Section 4.6], a universal rank-2 sheaf exists locally in the ´etaletopology over the stable locus M st of M = M ( n − m ). Thus there exists an open´etale covering Φ : W → M st and a rank 2 sheaf E = E W over P × W such that for any w ∈ W , E | P × w ≃ E t , where t = Φ( w ) ∈ M st and E t denotes the stablesheaf whose isomorphism class is represented by t . Refining the ´etale covering,one can define “transition functions” g : pr ∗ E −→∼ pr ∗ E over P × W × M st W ,where pr ij... denotes the projection to the product of the i -th, j -th. . . factors of aproduct of several factors. The transition functions, in general, fail to satisfy thecocycle condition, so E does not descend to M st . The failure of g to satisfy the1-cocycle condition is measured by a gerbe c g ∈ Γ( W × M st W × M st W, O ∗ ), whichis an ´etale 2-cocycle representing a ν -torsion element α = [ c g ] of the Brauer groupBr ( M st ) ⊂ H ( M st , O ∗ ), where ν = 1 or 2 depending on the parity of n − m .Over P × W × M st W × M st W , the following twisted cocycle condition holds:(20) pr ∗ g ◦ pr ∗ g = pr ∗ c g · pr ∗ g. We now choose an ´etale open covering e V → B by the ´etale open sets e V con-structed in the end of the proof of Lemma 6.2, and define e U = W × M st e V . Wealso denote by e Γ the pullback of the universal degree- m rational curve in P withtwo natural maps ˜ π : e Γ → e U , ι : e Γ ֒ → P × e U , and by e E the pullback of E to e Γ .Then ˜ π possesses m disjoint cross-sections over e U , which we denote x , . . . , x m , asbefore, and we can choose a global line bundle L on e Γ of relative degree − e U by setting L := O e Γ ( − x ). For an integer k , we denote by O e Γ ( k ) the pullback of O P ( k ), and L ( k ) := L ⊗ O e Γ ( k ).For any ˜ u ∈ e U , we have e E Γ ˜ u ≃ O ⊕ ˜ u , and Hom( e E Γ ˜ u , L (2) | Γ ˜ u ) ≃ H (Γ ˜ u , O ⊕ ˜ u ((2 m − ˜ u := ˜ π − (˜ u ), is a 4 m -dimensional vector space.These vector spaces glue into the vector bundle τ := H om e Γ / e U ( E e U , ι ∗ L (2)) over e U .Its associated projective bundle p e U : P e U := P τ → e U descends to B , that is,there exists a Severi–Brauer variety p B : P B → B with fibers P m − such that P e U ≃ e U × B P B . Indeed, as follows from (20), the failure of the transition functionsof E e U and of τ to form a 1-cocycle for the ´etale open covering e U → B is only in ascalar factor c g , so the transition functions modulo homotheties do form a 1-cocycleand define a projective bundle over B .The open subset Z ˜ u in P m − u := P Hom( e E Γ ˜ u , L (2) | Γ ˜ u ) consisting of propor-tionality classes [ φ ] of surjective homomorphisms φ : e E Γ ˜ u ։ L (2) | Γ ˜ u is thecomplement to the zero locus ∆ ˜ u of the resultant R ( f , f ) of the two com-ponents of φ , viewed as a pair of polynomials ( f , f ) on Γ ˜ u ≃ P of degree2 m − f i ∈ H (Γ ˜ u , O Γ ˜ u ((2 m − R ( f , f ) de-pends on the choice of homogeneous coordinates on Γ ˜ u ≃ P via the character GL ( C ) → C ∗ , A (det A ) (2 m − , and is multiplied by (det B ) − m +2 when theidentification e E Γ ˜ u ≃ O ⊕ ˜ u is changed by means of a matrix B ∈ GL ( C ). Thisimplies that the resultant R can be viewed as a section of a line bundle upon liftingto an appropriate GL ( C ) × GL ( C )-torsor over P e U , and the zero locus of thissection descends to a relative divisor ∆ B in P B over B .We have thus shown that the surjectivity loci Z ˜ u ⊂ P m − u glue into an opensubset Z e U ⊂ P e U which is the inverse image of the open subset Z B = P B \ ∆ B of P B . Now we are ready to construct a universal family F of the instanton sheavesof charge n over Z e U .Remark, that p e U ∗ ( O P f U / e U (1)) = τ ∨ (the equality sign meaning a canonicalisomorphism), so that the canonical Casimir section σ ∈ Γ( e U , τ ∨ ⊗ τ ) comes from EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 17 a unique section ˜ σ ∈ Γ( P e U , p ∗ e U τ ⊗ O P f U / e U (1)). The latter defines a morphism ofsheaves φ : p ∗ e U E e U → p ∗ e U ι ∗ L (2), and we set F := ker φ | Z f U . We have the followingexact triple:0 −→ F −→ p ∗ e U E e U | Z f U φ | Z f U −−−→ p ∗ e U ( ι ∗ L (2)) ⊗ O P f U / e U (1) | Z f U −→ . By construction, the classifying map Z e U → M ( n ) of F factors through a bijection B → D ( m, n ), and we are done. (cid:3) Now we will prove the smoothness of M ( n ) along D ( m, n ). Proposition 6.5.
Let [ F ] ∈ D ( m, n ) with n ≥ and ≤ m ≤ n , where [ F ] denotes, as before, the point of the moduli space representing the isomorphism classof a stable sheaf. Then dim Ext ( F, F ) = 8 n − and Ext ( F, F ) = 0 , i.e. [ F ] is anonsingular point of M ( n ) .Proof. Recall that, if ι : Γ ֒ → P is a smooth rational curve of degree m belongingto R ∗ ( m ), then by definition(21) N Γ / P ≃ (cid:26) O Γ ((2 m − ⊕ if m = 2 O Γ (2pt) ⊕ O Γ (4pt) if m = 2 . Thus(22) det N Γ / P ≃ O Γ ((4 m − m ≥ . Moreover,(23) E xt ( ι ∗ O Γ , ι ∗ O Γ ) ≃ ι ∗ N Γ / P , E xt ( ι ∗ O Γ , ι ∗ O Γ ) ≃ ι ∗ det N Γ / P , (24) E xt ( ι ∗ O Γ , O P ) = 0 and E xt ( ι ∗ O Γ , O P ) ≃ ι ∗ det N Γ / P . We have the short exact sequence(25) 0 → F → F ∨∨ → ( ι ∗ L )(2) → , where L := O Γ ( − pt) and F ∨∨ is a rank 2 locally free instanton sheaf of charge n − m . It follows that Ext ( F ∨∨ , F ∨∨ ) = 0 (see [12]) and H ( P , ( ι ∗ L )(4)) ≃ H (Γ , O Γ ((4 m − , thus we can apply Lemma 5.2.We start by computing h ( H om ( F, F )). Since h ( P , ( ι ∗ L )(4)) = h (Γ , O Γ ((4 m − m − , we obtain from formula (16) that either(26) h ( H om ( F, F )) = 8 n − m − , if n > m, i.e. if F ∨∨ is nontrivial, or(27) h ( H om ( F, F )) = 4 m − , if n = m, i.e. if F ∨∨ is trivial.Next, we characterize the sheaf E xt ( F, F ). To do this, we first apply the con-travariant functor H om ( − , F ∨∨ ) to the sequence (25). Since H om ( ι ∗ L, F ∨∨ ) = 0(the first sheaf is torsion, while the second is locally free), one concludes that H om ( F ∨∨ , F ∨∨ ) ∼ → H om ( F, F ∨∨ ) and E xt (( ι ∗ L )(2) , F ∨∨ ) ≃ E xt ( ι ∗ O Γ , O P ) ⊗ ( F ∨∨ ⊗ ( ι ∗ L − ))( −
2) = 0 , by (24).Notice that H om ( F, ( ι ∗ L )(2)) ≃ ι ∗ H om ( ι ∗ F, O Γ ((2 m − ι ∗ ( − ⊗ ι ∗ O Γ ) to the sequence (25) one obtains the following exact sequenceof sheaves on Γ:(28) 0 → ι ∗ T or (( ι ∗ L )(2) , ι ∗ O Γ ) → F | Γ → F ∨∨ | Γ → O Γ ((2 m − → . Breaking it into two short exact sequences, one obtains, since F ∨∨ | Γ ≃ O ⊕ :(29) 0 → ι ∗ T or (( ι ∗ L )(2) , ι ∗ O Γ ) → F | Γ → G → → G → O ⊕ → O Γ ((2 m − → . One concludes from sequence (30) that G ≃ O Γ ( − (2 m − ι ∗ T or (( ι ∗ L )(2) , ι ∗ O Γ ) ≃ N ∨ Γ / P ⊗ O Γ ((2 m − ≃ O ⊕ for m = 2 , and ι ∗ T or (( ι ∗ L )(2) , ι ∗ O Γ ) ≃ N ∨ Γ / P ⊗ O Γ (3pt) ≃ O Γ (pt) ⊕ O Γ ( − pt) for m = 2 , it follows from sequence (29) that F | Γ ≃ O ⊕ ⊕ O Γ ( − (2 m − m = 2 ,F | Γ ≃ O Γ (pt) ⊕ O Γ ( − pt) ⊕ O Γ ( − m = 2 . We then conclude that(31) H om ( F, ( ι ∗ L )(2)) ≃ ι ∗ O Γ ((2 m − ⊕ ⊕ ι ∗ O Γ ((4 m − m = 2 , (32) H om ( F, ( ι ∗ L )(2)) ≃ ι ∗ O Γ (2pt) ⊕ ι ∗ O Γ (4pt) ⊕ ι ∗ O Γ (6pt) for m = 2 . Now, by applying the functor H om ( − , F ∨∨ ) to (25), one concludes that E xt ( F, F ∨∨ ) ≃ E xt (( ι ∗ L )(2) , F ∨∨ ). However, E xt (( ι ∗ L )(2) , F ∨∨ ) ≃ E xt ( ι ∗ O Γ ( − pt) , O P ) ⊗ F ∨∨ ( − ≃E xt ( ι ∗ O Γ , O P ) ⊗ ι ∗ O Γ (pt) ⊗ F ∨∨ ( − . Thus, by (22), (24) and since F ∨∨ | C ≃ O ⊕ , we obtain(33) E xt ( F, F ∨∨ ) ≃ ι ∗ O Γ ((2 m − ⊕ . Finally, we apply the functor H om ( − , ( ι ∗ L )(2)) to (25), using (23) and the iso-morphism E xt (( ι ∗ L )(2) , ( ι ∗ L )(2)) ≃ E xt ( ι ∗ O Γ , ι ∗ O Γ ), and we see that(34) E xt ( F, ( ι ∗ L )(2)) ≃ E xt (( ι ∗ L )(2) , ( ι ∗ L )(2)) ≃ ι ∗ det N Γ / P . Thus, in view of (22), we have(35) E xt ( F, ( ι ∗ L )(2)) ≃ ι ∗ O Γ ((4 m − . Substituting (31)-(33) and (35) into (19), we obtain the exact sequence0 → ι ∗ O Γ ((4 m − → ι ∗ O Γ ((2 m − ⊕ ⊕ ι ∗ O Γ ((4 m − →E xt ( F, F ) → ι ∗ O Γ ((2 m − ⊕ → ι ∗ O Γ ((4 m − → m = 2 , EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 19 and0 → ι ∗ O Γ (6pt) → ι ∗ O Γ (2pt) ⊕ ι ∗ O Γ (4pt) ⊕ ι ∗ O Γ (6pt) → E xt ( F, F ) → ι ∗ O Γ (3pt) ⊕ → ι ∗ O Γ (6pt) → m = 2 . Breaking these into three short exact sequences, one obtains:(36) 0 → G → ι ∗ O Γ ((2 m − ⊕ → ι ∗ O Γ ((4 m − → m ≥ , (37) 0 → G → E xt ( F, F ) → G → , (38) 0 → ι ∗ O Γ ((4 m − → ι ∗ O Γ ((2 m − ⊕ ⊕ ι ∗ O Γ ((4 m − → G → m = 2 , and(39) 0 → ι ∗ O Γ (6pt) → ι ∗ O Γ (2pt) ⊕ ι ∗ O Γ (4pt) ⊕ ι ∗ O Γ (6pt) → G → m = 2 . From (36) one concludes that G ≃ ι ∗ O Γ , while (38) and (39) impliy that G ≃ ι ∗ O Γ ((2 m − ⊕ for m = 2 and G ≃ ι ∗ O Γ (2pt) ⊕ ι ∗ O Γ (4pt) for m = 2. Then(37) yields E xt ( F, F ) ≃ ι ∗ O Γ ((2 m − ⊕ ⊕ ι ∗ O Γ , m = 2 , E xt ( F, F ) ≃ ι ∗ O Γ (2pt) ⊕ ι ∗ O Γ (4pt) ⊕ ι ∗ O Γ , m = 2 . It then follows that h ( E xt ( F, F )) = 4 m + 1 and h ( E xt ( F, F )) = 0 . We conclude from Lemma 5.2 that Ext ( F, F ) = 0 anddim Ext ( F, F ) = 4 m + 1 + 8 n − m − n − n > m, dim Ext ( F, F ) = 4 m + 1 + 4 m − n − n = m, as desired. (cid:3) Sheaves in D ( m, n ) as limits of instantons The goal of this section is to prove that the varieties D ( m, n ) are contained inthe instanton boundary ∂ I ( n ). We begin with the following technical lemma. Lemma 7.1.
Let C be a smooth irreducible curve with a marked point , and set B := C × P . Let F and G be O B -sheaves, flat over C and such that F is locallyfree along Supp( G ) . Denote P t = { t } × P , and G t = G | { t }× P , F t = F | { t }× P for t ∈ C. Assume that, for each t ∈ C , (40) H i ( H om ( F t , G t )) = 0 , i ≥ . Assume that s : F → G is an epimorphism. Then, after possibly shrinking C , s extends to an epimorhism s : F ։ G . Proof.
The condition that F is locally free along Supp( G ) implies that E xt ( F , G ) =0. Thus applying the functor H om ( F , − ) to the short exact sequence0 → G · ζ t → G → G t → , t ∈ C, where ζ t is the local parameter of O C,t , we obtain the short exact sequence(41) 0 → H · ζ t → H → H om ( F , G t ) → , t ∈ C, in which H denotes the sheaf H om ( F , G ). Since F is locally free along Supp( G t ) ⊂ Supp( G ), we obtain that(42) H om ( F , G t ) ≃ H om ( F t , G t ) ≃ H | P t , t ∈ C. This means that (41) can be rewritten as the short exact sequence(43) 0 → H · ζ t → H → H | P t → − ⊗ H to the short exact sequence0 → O B · ζ t → O B → O P t → . This yields T or ( H , O P t ) = 0 for any t ∈ C .The last equality means that the H is flat over C . Furthermore, the condition(40) in view of (42) can be rewritten as H i ( H om ( F t , G t )) = 0 , i ≥ . Thus,by Base Change, we obtain that R j p ∗ H = 0 for j ≥
1, where p : B → C is theprojection. As a corollary, applying R j p ∗ to (43) for t = 0 and using (42), we obtain0 → p ∗ H → p ∗ H e → p ∗ H om ( F , G ) → . Since p ∗ H om ( F , G ) is supported at 0, it follows that, after possibly shrinking C ,the homomorphism of sections h ( e ) : H ( p ∗ H ) → H ( p ∗ H om ( F , G ))is surjective. However, this epimorphism canonically coincides with the epimor-phism Hom( F , G ) ։ Hom( F , G ). Hence, the homomorphism s ∈ Hom ( F , G ),from the hypothesis of the lemma extends to a morphism s ∈ Hom( F , G ). In addi-tion, as s is an epimorphism, again, shrinking C if necessary, we may assume that s is an epimorphism as well. (cid:3) Proposition 7.2. D (1 , n ) ⊂ ∂ I ( n ) for n ≥ .Proof. Fix a disjoint union of n + 1 lines Λ = F ≤ i ≤ n ℓ i in P . The extensions of O P -sheaves of the form(44) 0 → O P ( − → E → I Λ , P (1) → V = Ext ( I Λ , P (1) , O P ( − E ; when E is locally free,it is called a ’t Hooft instanton . A standard computation gives an isomorphism(45) V ≃ ⊕ ≤ i ≤ n H ( O ℓ i ) . Under this isomorphism, any point x in V can be represented by its coordinates x = ( t , ..., t n ) , t i ∈ H ( O ℓ i ) = C . By [7, Proposition 3.1], there exists a universal(flat) family of extensions (44) parametrized by the affine space V = A n +1 . Restrictit to the line defined by A = { ( t, , ..., | t ∈ C } , and denote by E = { E t } t ∈ A the thus obtained family of instanton sheaves with base A . It follows from the EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 21 construction of the isomorphism (45) that E t is locally free and [ E t ] ∈ I ( n ) for0 = t ∈ A . For t = 0, the sheaf E is not locally free. By Theorem 2.2, E ∨∨ is aninstanton bundle, [ E ∨∨ ] ∈ I ( n − → O P ( − → E ∨∨ → I Λ , P (1) → , Λ = G ≤ i ≤ n ℓ i , so it is a ’t Hooft instanton.We thus obtain a morphism φ : A → I ( n ) , t [ E t ], and this proves that[ E ] ∈ ∂ I ( n ) . The exact triples (44) for E = E and (46) yield the exact triple0 → E → E ∨∨ → O ℓ (pt) → , which shows that [ E ] ∈ D (1 , n ). By Proposition 6.5, Ext ( E , E ) = 0. So[ E ] is a smooth point of M ( n ), in particular, [ E ] does not lie in the inter-section of two components of M ( n ). Hence D (1 , n ), D (1 , n ) are entirely con-tained in the component I ( n ). As none of the sheaves in D (1 , n ) is locally free, D (1 , n ) ⊂ D (1 , n ) ⊂ ∂ I ( n ). (cid:3) Let now n ≥ m ≥
2. Fix a disjoint union Γ = Γ ′ ⊔ ℓ in P , where Γ ′ is a smoothirreducible rational curve of degree m − ℓ is a line, and let ι : Γ ֒ → P be the embedding. Consider the O Γ -sheaf L = O Γ ′ ( − pt) ⊕ O ℓ ( − pt). Fix aninstanton bundle E ∈ I ( n − m ) such that(47) E | Γ ′ ≃ O ⊕ ′ , E | ℓ ≃ O ⊕ ℓ ;the existence of such a bundle E follows from the above. Let us fix an epimorphism e : E ։ ( ι ∗ L )(2), and let the sheaf F be defined by the short exact sequence(48) 0 → F → E e −→ ( ι ∗ L )(2) → . Proposition 7.3.
Let n ≥ m ≥ , and assume that D ( m − , n − ⊂ ∂ I ( n − .Let F be the O P -sheaf defined by (48). Then [ F ] ∈ ∂ I ( n ) .Proof. Let F ′ be the kernel of the composition ε : E e / / / / ( ι ∗ L )(2) e ′ / / / / ι ∗ O Γ ′ ((2 m − , where e ′ is the projection onto the direct summand. We thus obtain a short exactsequence(49) 0 → F ′ → E ε −→ ι ∗ O Γ ′ ((2 m − → , which shows, since [ E ] ∈ I ( n − m ), that[ F ′ ] ∈ D ( m − , n − . Furthermore, since Γ is a disjoint union of Γ ′ and ℓ , the last isomorphism in (47)yields(50) F ′ | ℓ ≃ O ⊕ ℓ . Besides, (48) and (49) imply the exact triple(51) 0 → F → F ′ s ′ → ι ∗ O ℓ (pt) → . By hypothesis, [ F ′ ] is in ∂ I ( n − C with marked point 0 and a O C × P -sheaf F ′ , flat over C , such that(52) [ F ′ | { y }× P ] ∈ I ( n − , y ∈ C r { } , F ′ | { }× P ≃ F ′ . By (47), F ′ is locally free along ℓ , and by (52), E := F ′ is locally free alongSupp( G ), where G = ι ∗ ( O C ⊠ O ℓ (pt)) and ι = id × ( ι | ℓ ) : C × ℓ ֒ → C × P is the embedding. Moreover, (47) implies that, after possibly shrinking C , we mayassume that E | { y }× ℓ ≃ O ⊕ ℓ for all y ∈ C . This together with the definition of G yields (40). Hence E , G satisfy the hypothesis of Lemma 7.1, and there exists anepimorphism s ′ | : E ։ G extending the epimorphism s ′ in (51). We thus obtainthe short exact sequence(53) 0 → F → F ′ s ′ → ι ∗ ( O C ⊠ O ℓ (pt)) → F | { }× P ≃ F . Restricting (53) to { y }× P for y ∈ C , we obtain, by (52), that [ F | { y }× P ] ∈ D (1 , n )for y ∈ C r { } . Moreover, F is stable by Corollary 4.2 and Lemma 4.6, so [ F ] ∈M ( n ). This together with (54) implies that [ F ] ∈ D (1 , n ), and the propositionfollows from Proposition 7.2. (cid:3) Let now n ≥ m ≥
2, [ E ] ∈ I ( n − m ), let Γ = Γ ′ ∪ ℓ , where Γ ′ is a smoothrational curve of degree m − ℓ is a line intersecting Γ ′ quasi-transversely at onepoint, say, x , and let the following properties hold:(55) N Γ ′ / P ≃ O Γ ′ ((2 m − ⊕ , (56) E | Γ ′ ≃ O ⊕ ′ , E | ℓ ≃ O ⊕ ℓ . Note that (55) and the first two isomorphism in (56) mean, in the notation ofSection 6, that Γ ′ ∈ R ∗ ( m − E .We will now treat the transform E of the instanton bundle E along the non-invertible sheaf O ℓ ( − pt) ⊕ O Γ ′ ( − pt) on Γ . Proposition 7.4.
Let n ≥ m ≥ , and let E , Γ be as above. Set L = O Γ ′ ( − pt) ⊕O ℓ ( − pt) and consider a sheaf E fitting into an exact sequence (57) 0 → E → E → ( ι ∗ L )(2) → , where ι : Γ ֒ → P is the natural embedding. Assume that D ( m − , n − ⊂ ∂ I ( n − . Then [ E ] ∈ ∂ I ( n ) .Proof. We can include Γ in a one-parameter family of curves Γ U = { Γ u = Γ ′ ∪ ℓ u } u ∈ U over a smooth irreducuble curve U such that Γ ′ ∩ ℓ u = ∅ for any u ∈ U \ { } . Shrinking U , if necessary, we may assume by semicontinuity and the lastisomorphism in (55) that(58) E | ℓ u ≃ O ⊕ ℓ u , u ∈ U. The union ℓ U = S u ∈ U ℓ u is a surface meeting Γ ′ only at x . We may assume thatthis intersection is transverse and that the lines ℓ u do not intersect each other (forexample, we can start the construction from a family of lines ℓ u which is one of thetwo families of rectilinear generators of a smooth quadric, and then ℓ U is an opensubset of this quadric). We have Γ U = ( U × Γ ′ ) ∪ ℓ U . EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 23
Consider the following O Γ U -sheaf:(59) L U = O U ⊠ O Γ ′ ( − pt) ⊕ ( O P ( − | ℓ U ) . Then for each u ∈ U , we have L u := L U | u × P ≃ O Γ ′ ( − pt) ⊕ O ℓ u ( − pt) . Let : Γ U ֒ → U × P be the natural embedding. Then | { }× P = ι , and by (58), asurjection E e ։ ι ∗ L (2) from (57) extends to a surjection e : O U ⊠ E ։ ∗ L U (2),where twisting a sheaf on Γ U or on U × P by an integer k means tensoring it by O U ⊠ O P ( k ). The O U × P -sheaf E U := ker e fits in the exact triple0 → E U → O U ⊠ E e −→ ∗ L U (2) → . Let E u := E U | { u }× P for u ∈ U . If u ∈ U \ { } , then Γ u is a disjoint unionΓ ′ ⊔ ℓ u , so the result of Proposition 7.3 holds for F = E u . Hence [ E u ] ∈ ∂ I ( n )for all u ∈ U \ { } . For u = 0, the stability of E is assured by Corollary 4.2and Lemma 4.6, so that [ E ] ∈ M ( n ), and by closedness of ∂ I ( n ), we also have[ E ] ∈ ∂ I ( n ) . (cid:3) Remark 7.5.
The family of curves Γ u used in the proof is not flat, but it carriesa U -flat family of pure rank-1 sheaves L u that is a part of degeneration data of a U -flat family of instanton sheaves E u .Note that, by construction, E fits in the exact triple(60) 0 → E → E → ι ∗ ( O Γ ′ ((2 m − ⊕ O ℓ (pt)) → . Proposition 7.6.
Under the hypothesis of Proposition 7.4,
Ext ( E , E ) = 0 , sothat [ E ] is a smooth point of M ( n ) .Proof. Denote L ′ := ι ∗ ( O Γ ′ ((2 m − L := ι ∗ ( O ℓ (pt)). Then (60) impliesthe exact triples(61) 0 → F → E → L ′ → , (62) 0 → E → F → L → , where F is the kernel of the composition E ։ L ′ ⊕ L pr / / / / L ′ . Applying to (62)the functor Ext • ( − , E ) we obtain the exact sequence(63) Ext ( F, E ) → Ext ( E , E ) → Ext ( L , E ) , By Serre–Grothendieck duality, Ext ( L , E ) = Hom( E , L ( − ∨ . Applying H om ( − , L ( − → O ℓ ( − → O ℓ ( − → H om ( E , O ℓ ( − → O ℓ ( − ⊕ ⊕ k ( x ) → k ( x ) ≃ E xt ( L ′ , L ( − ≃ E xt ( L ′ , L ( − x = Γ ′ ∩ ℓ . Since O ℓ ( − O ℓ -module, H om ( E , O ℓ ( − O ℓ -module, hence H om ( E , O ℓ ( − ≃ O ℓ ( − pt) ⊕ O ℓ ( − ⊕ and(64) Ext ( L , E ) = Hom( E , L ( − ∨ = H ( H om ( E , O ℓ ( − ∨ = 0 . Next, since L ′ has homological dimension 2 and E is locally free, F has homologicaldimension 1. Therefore,(65) E xt ( F, E ) = 0 . Also, since E is locally free, (61) implies that(66) E xt ( F, E ) = E xt ( L ′ , E ) . Next, (55) implies that E xt ( L ′ , L ′ ) ≃ N Γ ′ / P ≃ O Γ ′ ((2 m − ⊕ , E xt ( L ′ , L ′ ) ≃ det N Γ ′ / P ≃ O Γ ′ ((4 m − . (67)Similarly, by (56), we have(68) E xt ( L ′ , E ) ≃ det N Γ ′ / P ⊗ (( L ′ ) ⊕ ) ∨ ≃ O Γ ′ ((2 m − ⊕ . Now applying E xt • ( L ′ , − ) to (60), we obtain the exact sequence E xt ( L ′ , L ′ ⊕ L ) γ → E xt ( L ′ , E ) → E xt ( L ′ , E ) δ → E xt ( L ′ , L ′ ⊕ L ) . Using (67), (68) and the relations E xt ( L ′ , L ) ≃ E xt ( L ′ , L ) ≃ k ( x ), we rewritethe last sequence as follows: O Γ ′ ((2 m − ⊕ ⊕ k ( x ) γ → E xt ( L ′ , E ) → O Γ ′ ((2 m − ⊕ δ →O Γ ′ ((4 m − ⊕ k ( x ) → . It implies that ker( δ ) is an O Γ ′ -sheaf of degree ≥ −
1, so h (ker( δ )) = 0. Similarly, h (im( γ )) = 0. Hence H ( E xt ( L ′ , E )) = 0, and by (66),(69) H ( E xt ( F, E )) = 0 . Next, as E is locally free, (60) yields the exact triple H ( H om ( E, L ′ ⊕ L )) → H ( H om ( E, E )) → H ( H om ( E, E )) . From (56) and the definition of L ′ and L , it follows that H ( H om ( E, L ′ ⊕ L )) = 0.Besides, since [ E ] ∈ I ( n − m ), it follows that H ( H om ( E, E )) = 0. Thus we have(70) H ( H om ( E, E )) = 0 . Applying H om ( − , E ) to (61), we obtain the exact triple 0 → H om ( E, E ) →H om ( F, E ) ε → E xt ( L ′ , E ). As Supp( L ′ ) = Γ ′ , it follows that H (im( ε )) = 0, andthe last exact triple together with (70) yields H ( H om ( F, E )) = 0. This equal-ity together with (65), (69) and the spectral sequence E pq = H p ( E xt q ( F, E )) ⇒ Ext • ( F, E ) implies that Ext ( F, E ) = 0. By (63) and (64), we also haveExt ( E , E ) = 0. (cid:3) Proposition 7.7.
Let n ≥ m ≥ , and assume that D ( m − , n − ⊂ ∂ I ( n − .Then D ( m, n ) ⊂ I ( n ) .Proof. Take E , Γ = Γ ′ ∪ ℓ as in Propositions 7.4 and 7.6 and consider the exten-sions of O P -sheaves of the form(71) 0 → ι ∗ O ℓ (pt) → M → ι ∗ O Γ ′ ((2 m − → . We have E xt O P ( ι ∗ O Γ ′ ((2 m − , ι ∗ O ℓ (pt)) ≃ k ( x ), where x = Γ ′ ∩ ℓ , andExt O P ( ι ∗ O Γ ′ ((2 m − , ι ∗ O ℓ (pt)) ≃ k . There is a universal family of extensions(71) over the affine line A = V (Ext P ( ι ∗ O Γ ′ ((2 m − , ι ∗ O ℓ (pt)) ∨ ) (see [7,Proposition 3.1]):(72) 0 → ι ∗ ( O A ⊠ O ℓ (pt)) → M → ι ∗ ( O A ⊠ O Γ ′ ((2 m − → , where ι = id T × ι and ι : Γ ֒ → P is the embedding. EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 25
Let M t = M | { t }× P , t ∈ A . For t = 0 the extension (71) splits, i.e.(73) M ≃ ι ∗ O Γ ′ ((2 m − ⊕ ι ∗ O ℓ (pt) , and we may rewrite the triple (60) in the form(74) 0 → E → E s → M → . On the other hand, for any t ∈ A \ { } , the sheaf M := M t is a locally free O Γ -sheaf fitting in the exact triple (71). Take an arbitrary open subset T of A containing 0. Then the curve T and the sheaves F := O T ⊠ E and G := M satisfythe hypothesis of Lemma 7.1, hence by this Lemma, after possibly shrinking T ,the epimorphism s in (74) extends to an epimorphism s : O T ⊠ E ։ M . We thusobtain the exact triple 0 → E T → O T ⊠ E s −→ M → . Restricting this triple to any point t ∈ T and denoting E t = E T | { t }× P , we obtainan exact triple(75) 0 → E t → E → M t → , t ∈ T. By Corollary 4.2 and Lemma 4.6, E t is stable for any t ∈ T . We thus have awell-defined morphism φ : T → M ( n ) given by t [ E t ].For t = 0, the sheaf M t is a locally free O Γ -sheaf fitting in (71). Hence(76) M t | Γ ′ ≃ ι ∗ O Γ ′ ((2 m − , M t | ℓ ≃ ι ∗ O ℓ (2pt) . Next, by Proposition 7.6 [ E ] is a smooth point of M ( n ). In addition, sincethe triples (60) and (74) coincide, it follows that [ E ] ∈ ∂ I ( n ) by Proposition 7.4.Thus,(77) [ E ] ∈ ∂ I ( n ) \ Sing M ( n ) . Fix a point t ∈ T \ { } , denote ˜ M := M t , ˜ E := E t , and rewrite the triple(75) for t = t :(78) 0 → ˜ E → E ε → ˜ M → . Now, as in Lemma 6.1, one can see that there exists a family of curves { Γ b } b ∈ B ,parametrized by a curve B with a marked point 0 ∈ B , such that Γ = Γ ′ ∪ ℓ andΓ b is smooth for b ∈ B \ { } , and such that ˜ M deforms into an invertible sheaf˜ M b of degre 2 m − b for b ∈ B \ { } . Denote by ι b the embedding Γ b ֒ → P for b ∈ B . Let Γ → B be the family { Γ b } b ∈ B and ι = { ι b } b ∈ B : Γ ֒ → B × P theembedding; for b = 0, we have the embedding ι : Γ ֒ → P defined earlier. Thefamily of sheaves { ˜ M b } b ∈ B constitutes an invertible O Γ -sheaf ˜ M , so that ι ∗ ˜ M isa O B × P -sheaf, flat over B , and the sheaves ˜ M b := ι ∗ ˜ M | { b }× P , b ∈ B , satisfy therelations(79) ˜ M = M t , ˜ M b ≃ ι b ∗ ( O Γ b (2 m − , b ∈ B \ { } . Since E | Γ ≃ O ⊕ by (56), we may assume, after shrinking B if necessary, that(80) E | Γ b ≃ O ⊕ b , b ∈ B. Formulas (79) and (80) show that the curve T = B and the sheaves F := O B ⊠ E and G := ι ∗ ˜ M satisfy the hypothesis of Lemma 7.1. Hence by this Lemma, after possibly shrinking B , the epimorphism ε in (74) extends to an epimorphism ε : O B ⊠ E ։ ι ∗ ˜ M , providing a short exact sequence0 → ˜ E → O B ⊠ E ε → ι ∗ ˜ M → . Restricting it to any b ∈ B and denoting ˜ E b = ˜ E | { b }× P , we obtain a short exactsequence(81) 0 → ˜ E b → E → ˜ M b → , b ∈ B. Since [ E ] ∈ I ( n − m ), we deduce from (75) that ˜ E b is stable for any b ∈ B byCorollary 4.2 and Lemmas 4.3, 4.6. We thus have a well-defined morphism φ : B →M ( n ), given by b [ ˜ E b ]. Now (79)-(81) imply that [ ˜ E b ] ∈ D ( m, n ) for 0 = b ∈ B .Hence B ⊂ D ( m, n ). In particular, [ E t ] = [ ˜ E ] ∈ D ( m, n ) , t ∈ T \ { } . Therefore, T ⊂ D ( m, n ) and, in particular, [ E ] ∈ D ( m, n ). This together with (77) yields D ( m, n ) ∩ ( ∂ I ( n ) \ Sing M ( n )) = ∅ . From the irreducibility of D ( m, n ), we deduce that D ( m, n ) ⊂ ∂ I ( n ). (cid:3) We are finally ready to complete the proof of the main result of this section.
Theorem 7.8.
For each n ≥ and each m = 1 , . . . , n − , D ( m, n ) ⊂ ∂ I ( n ) .Proof. The result follows by induction on m . For m = 1, the assertion is true byProposition 7.2. The induction step m − m is provided by Proposition 7.7.The theorem is proved. (cid:3) Elementary transformations along elliptic quartic curves
In this section, we consider the case in which the curve Σ (notation from Section3) is an elliptic quartic curve, that is a complete intersection of two hypersurfaces f = 0 and f = 0 of degree 2. The minimal locally free resolution of its structuresheaf has the form(82) 0 → O P ( − → O P ( − ⊕ → O P → ι ∗ O Σ → , where ι : Σ → P is the inclusion map.Following the procedure outlined by Proposition 3.1, let L ∈ Pic (Σ). If L isnontrivial, we have h p ( ι ∗ L ) = h p (Σ , L ) = 0, for p = 0 ,
1, as desired.Let E be a locally free instanton sheaf of rank 2 and charge n . In order to performan elementary transformation of E along Σ, we must find out whether there exists asurjective map E → ( ι ∗ L )(2). In Lemma 8.1 below, we give the affirmative answerin the case when E is a null-correlation bundle, that is an instanton of charge 1.Breaking (82) into short exact sequences and tensoring by any instanton bundle E , we obtain:0 → E ( − → E ( − ⊕ → E ⊗ I Σ → → E ⊗ I Σ → E → E | Σ → , where I Σ denotes the ideal sheaf of Σ.Since h ( E ( − h ( E ( − H ( E ⊗ I Σ ) ≃ H ( E ( − E is notthe trivial instanton, then also h ( E ⊗ I Σ ) = 0. Now, moving to the second sequenceabove, we obtain for a nontrivial instanton E :0 → H ( E | Σ ) → H ( E ( − → H ( E ) → H ( E | Σ ) → . It is not difficult to check that h ( E ( − h ( E ) = 2 n − EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 27
Lemma 8.1. If E is a null-correlation bundle on P , then its restriction to anynonsingular elliptic quartic curve Σ ֒ → P is either of the form L ⊕ L ∨ for somenontrivial L ∈ Pic(Σ) , or the nontrivial extention of a nontrivial L ∈ Pic (Σ) oforder by itself. In either case, we have Hom( E, ( ι ∗ L )(2)) ≃ H (Σ , L ⊗ L (8pt)) ⊕ H (Σ , L ∨ ⊗ L (8pt)) . Proof.
The restriction E | Σ is a rank 2 bundle on Σ with trivial determinant; if E is a null-correlation bundle, then also h (Σ , E | Σ ) = h (Σ , E | Σ ) = 0. The first claimnow follows from Atiyah’s classification of rank 2 bundles on nonsingular ellipticcurves.Next, note that Hom( E, ( ι ∗ L )(2)) ≃ H (Σ , E ∨ | Σ ⊗ L (8pt)). If E | Σ ≃ L ⊕ L ∨ ,the second claim follows immediately. On the orther hand, if E | Σ is an extensionof the form 0 → L → E | Σ → L → L ≃ O Σ , then, twisting by L (8pt), we obtain the cohomology exact sequence0 → H (Σ , L ⊗ L (8pt)) → H (( E | Σ ) ∨ ⊗ L (8pt)) → H (Σ , L ⊗ L (8pt)) → , and the second claim also follows. (cid:3) In particular, we conclude that for any null-correlation bundle E and any ellipticquartic curve ι : Σ ֒ → P , there exists a surjective map ϕ : E → ( ι ∗ L )(2). Thekernel sheaf E ′ := ker ϕ is a rank 2 instanton of charge 5, since Σ has degree 4;note, in addition, that E ′ is µ -stable by Lemma 4.1, thus in particular [ E ′ ] ∈ M (5).Our next goals are evaluating the dimension and proving the generic smoothnessof the locus of all the instanton sheaves obtained by elementary transformationsfrom null-correlation bundles along elliptic quartic curves.Let E denote the set of nonsingular elliptic quartic curves in P . It can beregarded as an open subset of the Grassmannian G (2 , S V ), where V is the 4-dimensional complex vector space such that P = P ( V ). This is a family of non-singular elliptic curves, so let j : J → E denote the relative Jacobian variety, i.e. j − (Σ) = Pic (Σ), and denote J o4 := J \ { zero section } . A point of J o4 can bethought of as a pair (Σ , [ L ]), in which Σ is a smooth elliptic quartic curve and L isa nontrivial line bundle of degree 0 on Σ, so that [ L ] ∈ Pic (Σ). Equivalently, it canbe thought of as the isomorphism class of the sheaf ι ∗ L on P , where ι : Σ ֒ → P isthe natural embedding.Note that J o4 is an irreducible, quasiprojective variety of dimension 17.Consider now the following subset of M (5): Q := { [ E ] ∈ M (5) | [ E ∨∨ ] ∈ I (1) , [( E ∨∨ /E )( − ∈ J o4 } . Let Q denote the closure of Q in M (5). Theorem 8.2. Q is an irreducible component of M (5) of dimension , distinctfrom the instanton component I (5) .Proof. Consider the map ̟ : Q → I (1) × J o4 given by E (cid:0) [ E ∨∨ ] , [( E ∨∨ /E )( − (cid:1) , and observe that it is surjective. Indeed, given a null-correlation bundle F ∈ I (1), and a nonsingular elliptic quartic curve ι : Σ ֒ → P equipped with a nontrivial L ∈ Pic (Σ), so that [ ι ∗ L ] ∈ J o4 , we know from Lemma8.1 that there exists a surjective map ϕ : F → ( ι ∗ L )(2). Then, as we have seen above, E := ker ϕ is an instanton sheaf from M (5) such that E ∨∨ ≃ F and E ∨∨ /E ≃ ( ι ∗ L )(2), i.e. [ E ] ∈ Q .The fiber ̟ − ( F, ι ∗ L ) consists precisely of all surjective maps F → ( ι ∗ L )(2) upto homothety, so it forms an open subset of P (Hom( F, ( ι ∗ L )(2))), which, again byLemma 8.1, has dimension 15 for every [ F ] ∈ I (1) and every [ ι ∗ L ] ∈ J o4 .It follows that Q is an irreducible quasiprojective variety of dimensiondim I (1) + dim J o4 + dim P Hom( F, ( ι ∗ L )(2)) = 37 . We have concluded that Q is an irreducible subvariety of M (5) of the same di-mension as I (5) and whose generic point represents a non locally free instantonsheaf. This implies that Q and I (5) are distinct components of M (5). (cid:3) In particular, we have the following interesting consequence.
Corollary 8.3.
The moduli space of instanton sheaves of rank and charge isreducible. Remark 8.4.
It is not clear, however, whether the moduli space of instantonsheaves of rank and charge is connected; in other words, we do not know whetherthe intersection Q ∩ I (5) contains instanton sheaves.
Finally, it is interesting to note that the generic point of Q is a smooth pointof M (5). Proposition 8.5.
Let F be an instanton sheaf obtained, as above, by an elementarytransformation of a null-correlation bundle along a pair (Σ , L ) where Σ is an ellipticquartic curve in P and [ L ] ∈ J o4 , L
6≃ O Σ . Then dim Ext ( F, F ) = 37 and dim Ext ( F, F ) = 0 , so [ F ] is a smooth point of M (5) .Proof. First, note that, as Σ is a complete intersection of two quadrics, the isomor-phism (34) yields(83) E xt ( F, ( ι ∗ L )(2)) ≃ ( ι ∗ O Σ )(4) . Next, applying the functor H om ( − , E ) to the sequence0 → F → E → ( ι ∗ L )(2) → , where E := F ∨∨ is a null-correlation bundle, and using the isomorphism E xt ( O Σ , O P ) ≃ ι ∗ det N Σ / P ≃ ( ι ∗ O Σ )(4), we obtain(84) E xt ( F, E ) ≃ E xt (( ι ∗ L )(2) , E ) ≃ ( ι ∗ L − )( − ⊗ E | Σ ⊗ E xt ( O Σ , O P ) ≃ ( ι ∗ L − )(2) ⊗ E | Σ . Next, using the isomorphism T or (( ι ∗ L )(2) , ι ∗ O Σ ) ≃ ( ι ∗ L )(2) ⊗ N ∨ Σ / P ≃ ι ∗ L ⊕ ,similarly to (28), we obtain an exact sequence0 → ι ∗ L ⊕ → F | Σ → E | Σ → ( ι ∗ L )(2) → . Since det E | Σ ≃ O Σ , it follows that(85) ker { E | Σ ։ ( ι ∗ L )(2) } ≃ ( ι ∗ L − )( − . Hence the above exact sequence yields the following exact triple:0 → ι ∗ L ⊕ → F | Σ → ( ι ∗ L − )( − → . As [ ι ∗ L ] ∈ J o4 , it follows that h (( ι ∗ L )(2)) = h (Σ , L (8pt)) = 0. There-fore the last sequence splits, i.e. F | Σ ≃ ι ∗ L ⊕ ⊕ ( ι ∗ L − )( − EW DIVISORS IN THE BOUNDARY OF THE INSTANTON MODULI SPACE 29 H om ( F, ( ι ∗ L )(2)) ≃ ( ι ∗ O ⊕ )(2) ⊕ ( ι ∗ L )(4). Substituting this relation togetherwith (83) and (84) into (19), we obtain the exact sequence0 → ( ι ∗ L )(4) → ι ∗ O Σ (2) ⊕ ⊕ ( ι ∗ L )(4) → E xt ( F, F ) → ( ι ∗ L − )(2) ⊗ E | Σ → ( ι ∗ O Σ )(4) → . Similarly to (85), one has ker (cid:8) ( ι ∗ L − )(2) ⊗ E | Σ ։ ( ι ∗ O Σ )(4) (cid:9) ≃ ι ∗ L − . Thus thelast exact sequence provides the exact triple0 → ( ι ∗ O Σ )(2) ⊕ → E xt ( F, F ) → ι ∗ L − → . Moreover, since h (( ι ∗ L )(2) ⊕ ) = 0, this triple splits, and E xt ( F, F ) ≃ ( ι ∗ O Σ )(2) ⊕ ⊕ ι ∗ L − . Since by assumption L −
6≃ O Σ , it follows that h (Σ , L − ) = 0, and the aboveisomorphism implies h ( E xt ( F, F )) = 0. This together with Lemma 5.2 yields theProposition. (cid:3)
We conclude by remarking that one can also perform elementary transformationsof the trivial bundle along plane cubic curves, a fact also realized by Perrin in[16, 17], as mentioned in the Introduction.Indeed, let ι : Σ → P be a plane cubic curve. For any L ∈ Pic (Σ) onecan find surjective morphisms ϕ : O ⊕ P → ( ι ∗ L )(2), since Hom( O ⊕ P , ( ι ∗ L )(2)) ≃ H (Σ , L (6pt)) ⊕ . Assuming that L is nontrivial, the kernel sheaf E ′ := ker ϕ is aninstanton sheaf of charge 3; it is also stable, by Lemma 4.3, therefore [ E ′ ] ∈ M (3). Remark 8.6.
One can also consider surjective morphisms ϕ : O ⊕ P → ( ι ∗ O Σ )(2) .In this case, the kernel E ′ := ker ϕ is a stable rank torsion free sheaf with c ( E ′ ) = c ( E ′ ) = 0 and c ( E ′ ) = 3 , but is not an instanton sheaf. Next, let now E be the set of nonsingular plane cubic curves, regarded as anopen subset of P ( S ( T ( P ) ∨ ( − J be the relative Jacobian over E , and J o3 the complement of the zero section. Consider the following set: Q := { [ E ] ∈ M (3) | E ∨∨ ≃ O ⊕ P , [( E ∨∨ /E )( − ∈ J o3 } . Similarly to Theorem 8.2, one can show that Q is an irreducible component of M (3) of dimensiondim J o3 + h (Σ , L (6pt) ⊕ ) − dim Aut( O ⊕ P ) = 21 . It follows that Q does not coincide with the instanton component I (3), sincedim I (3) = 21 as well.The proof of Proposition 8.5 also works in this case, and one can show that theinstanton sheaves obtained as above, by elementary transformations of the trivialrank 2 bundle along a pair (Σ , L ) where Σ is a plane cubic curve in P and [ L ] ∈ J o3 , L
6≃ O Σ , are smooth points of M (3).Furthermore, as mentioned at the Introduction, Perrin provides additional infor-mation on the intersection Q ∩ I (3) of these two irreducible components of M (3).In fact, he has shown that if L is a theta characteristic on Σ (i.e. if L = O Σ ), thenthe sheaves E given by short exact sequences of the form0 → E → O ⊕ P → ( ι ∗ L )(2) → I (3), and form an irreducible compo-nent of the instanton boundary ∂ I (3), see [16, Thm. 0.1]. References [1] Barth, W.: Some properties of stable rank-2 vector bundles on P n . Math. Ann. (1977),125–150.[2] Costa, L., Ottaviani, G.: Nondegenerate multidimensional matrices and instanton bundles.Trans. Amer. Math. Soc. (2003), 49–55.[3] Eisenbud, D., Van de Ven, A.: On the Normal Bundles of Smooth Rational Space Curves.Math. Ann. (1981), 453–463.[4] Frenkel, I. B., Jardim, M.: Complex ADHM equations, and sheaves on P . J. Algebra (2008), 2913–2937.[5] Gruson, L., Skiti, M.: 3-instantons et r´eseaux de quadriques. Math. Ann. (1994),253–273.[6] Hartshorne, R.: Stable reflexive sheaves. Math. Ann. (1980), 121–176.[7] Hartshorne, R., Hirschowitz, A.: Cohomology of a general instanton bundle, Ann. Sci. EcoleNorm. Sup. (4) , 365–390 (1982).[8] Hartshorne, R., Hirschowitz, A.: Smoothing algebraic space curves. Springer Lecture Notesin Math. Vol. 1124 (1985), 98–131.[9] Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd ed., CambridgeMath. Lib., Cambridge University Press, Cambridge, 2010.[10] Jardim, M.: Instanton sheaves on complex projective spaces. Collec. Math. (2006), 69–91.[11] Jardim, M., Gargate, M.: On the singular locus of rank 2 instanton sheaves. Preprint (2014),arXiv:1407.0897.[12] Jardim, M., Verbitsky, M.: Trihyperk¨ahler reduction and instanton bundles on P . Compo-sitio Math. (2014), 1836–1868.[13] Maruyama, M., Trautmann, G.: Limits of instantons. Internat. J. Math (1992), 213–276.[14] Narasimhan, M. S., Trautmann, G.: Compactification of M P (0 ,
2) and Poncelet pairs ofconics. Pacific J. Math. (1990), 255–365.[15] Perrin, N.: Une composante du bord des instantons de degr´e 3. C. R. Acad. Sci. Paris S´er. IMath. (2000), no. 3, 217-220.[16] Perrin, N.: Deux composantes du bord de I . Bull. Soc. Math. France (2002), 537–572.[17] Perrin, N.: D´eformations de fibr´es vectoriels sur les vari´et´es de dimension 3. ManuscriptaMath. (2005), 449–474.[18] Rao, A. P.: A family of vector bundles on P . Space curves (Rocca di Papa, 1985), 208–231,Lecture Notes in Math., 1266, Springer, Berlin, 1987.[19] Tikhomirov, A.S. Moduli of mathematical instanton vector bundles with odd c on projectivespace. Izvestiya: Mathematics :5 (2012), 991-1073.[20] Tikhomirov, A.S. Moduli of mathematical instanton vector bundles with even c on projectivespace. Izvestiya: Mathematics :6 (2013), 1331-1355. IMECC - UNICAMP, Departamento de Matem´atica, Caixa Postal 6065, 13083-970Campinas-SP, Brazil
E-mail address : [email protected] Math´ematiques – bˆat. M2, Universit´e Lille 1, F-59655 Villeneuve d’Ascq Cedex,France
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