aa r X i v : . [ m a t h . AG ] D ec A NOTE ON STABLE SHEAVES ON ENRIQUES SURFACES
K ¯OTA YOSHIOKA
Abstract.
We shall give a necessary and sufficient condition for the existence of stable sheaves on Enriquessurfaces based on results of Kim, Yoshioka, Hauzer and Nuer. For unnodal Enriques surfaces, we also studythe relation of virtual Hodge “polynomial” of the moduli stacks. Introduction
Studies of moduli spaces of stable sheaves on Enriques surfaces were started by a series of works of Kim[5], [6], [7], [8], [9]. In particular, he studied exceptional bundles and the singular locus of the moduli spaces.Recently the type of singularities are investigated by Yamada [17]. For the topological properties of themoduli spaces, the author [19] computed the Hodge polynomials of the moduli spaces if the rank is odd. Inparticular, the condition for the non-emptiness of the moduli spaces are known. For the even rank case, byextending our arguments, Hauzer [4] related the virtual Hodge “polynomial” of the moduli spaces to thosefor rank 2 or 4. Then Nuer [12] gave the condition for the non-emptiness by studying the non-emptiness forrank 2 and 4 cases. The main purpose of this note is to give another proof of his result on the non-emptiness.
Theorem 1.1.
Let X be an unnodal Enriques surface over C . For r, s ∈ Z and L ∈ NS( X ) such that r − s is even, let M H ( r, L, − s ) be the stack of semi-stable sheaves E of rank r > , det E = L and χ ( E ) = r − s ,where the polarization is H . Assume that gcd( r, c ( L ) , r − s ) = 1 , i.e., the Mukai vector is primitive. Then M H ( r, L, − s ) = ∅ for a general H if and only if (i) gcd( r, c ( L ) , s ) = 1 and ( c ( L ) ) + rs ≥ − or (ii) gcd( r, c ( L ) , s ) = 2 and ( c ( L ) ) + rs ≥ or (iii) gcd( r, c ( L ) , s ) = 2 , ( c ( L ) ) + rs = 0 and L ≡ r K X mod 2 .If r = 0 , then by assuming L to be effective, the same claim holds. Since v is primitive and H is general, semi-stability implies stability.In order to explain the difference of the proofs, we first mention the results in [19] and [4]. In [19], weintroduced the virtual Hodge “polynomial” e ( M H ( r, L, − s )) of the moduli stacks, which is an extension ofthe virtual Hodge polynomial of an algebraic set and showed that it is preserved under a special kind ofFourier-Mukai transform. As an application, we showed that e ( M H ( r, L, − s )) is the same as e ( M H (1 , , − n )) if r is odd, where 2 n = ( c ( L ) )+ rs +1 [19, Thm. 4.6]. In particular we get the condition ( c ( L ) )+ rs ≥− e ( M H ( r, L, − s )) is the sameas e ( M H ( r ′ , L ′ , − s ′ )) where r ′ = 2 , c ( L ′ ) ) + r ′ s ′ = ( c ( L ) ) + rs . For the rank 2 case, the conditionof non-emptiness follows by Kim’s results [9]. Thus the remaining problem is to treat the rank 4 case.For this problem, Nuer [12, Thm. 5.1] constructed µ -stable vector bundles of rank 4 by Serre construction,and got the condition for the non-emptiness. On the other hand, we shall reduce the rank 4 case to the rank 2case by improving Hauzer’s argument (Theorem 2.6). Combining Kim’s results [9], Theorem 1.1 follows. Forconvenience sake, we also give another argument for the rank 2 case using a relative Fourier-Mukai transformassociated to an elliptic fibration. Replacing virtual Hodge “polynomial” by numbers of F q -rational points,our result also holds for unnodal Enriques surfaces over an algebraically closed field of characteristic p = 2.As a corollary of Theorem 1.1, by adding a deformation argument, we shall treat the nodal case in Section3. Finally I would like to remark another approach in Appendix. For our argument, main tool is a specialkind of Fourier-Mukai transforms. For the case of K3 surfaces, Toda [16] proved a certain counting invariantof the moduli stack of Bridgeland semi-stable objects are invariant under Fourier-Mukai transforms. SinceGieseker stability corresponds to the large volume limit of Bridgeland stability, it is possible to get Theorem1.1 by a more sophisticated method, i.e., Bridgeland theory of stability conditions [2]. For a more generaltreatment, we recommend a reference [13]. Key words and phrases.
Enriques surfaces, stable sheaves.The author is supported by the Grant-in-aid for Scientific Research (No. 26287007, 24224001), JSPS.2010
Mathematics Subject Classification . Primary 14D20. . Proof of Theorem 1.1
Notation and some tools.
We prepare several notation and results which will be used.The Mukai vector v ( x ) of x ∈ K ( X ) is defined as an element of H ∗ ( X, Q ): v ( x ) := ch( x ) p td X = rk( x ) + c ( x ) + (cid:18) rk( x )2 ̺ X + ch ( x ) (cid:19) ∈ H ∗ ( X, Q ) , (2.1)where ̺ X is the fundamental class of X . We also introduce Mukai’s pairing on H ∗ ( X, Q ) by h x, y i := − R X x ∨ ∧ y . Then we have an isomorphism of lattices:(2.2) ( v ( K ( X )) , h , i ) ∼ = (cid:18) − (cid:19) ⊕ (cid:18) (cid:19) ⊕ E ( − . Definition 2.1.
We call an element of v ( K ( X )) by the Mukai vector. A Mukai vector v is primitive, if v isprimitive as an element of v ( K ( X )).We denote the torsion free quotient of NS( X ) by NS f ( X ), that is, NS f ( X ) = NS( X ) / Z K X . Lemma 2.2.
Let v = ( r, c , − s ) ( r, s ∈ Z , | r − s , c ∈ NS f ( X ) ) be a Mukai vector. (1) v is primitive if and only if gcd( r, c , r − s ) = 1 . (2) Assume that v is primitive. We set ℓ := gcd( r, c , s ) . Then ℓ = 1 , . (a) If ℓ = 1 , then gcd( r, c ,
2) = 1 . (b) If ℓ = 2 , then | r , | c , | s and r + s ≡ .Proof. (1) For E = r O X + F ∈ K ( X ) with rk F = 0, v ( E ) = ( r, , r ) + (0 , D, t ), where D ∈ NS f ( X ) and t ∈ Z . Then v ( E ) is primitive if and only if gcd( r, D, t ) = 1. If v = v ( E ), then c = D and t + r = − s .Hence gcd( r, c , r − s ) = gcd( r, D, t ), which shows the claim.(2) It is [4, Lem. 2.5]. For convenience sake, we give a proof. Since s = r + 2 s − r , ℓ = 1 ,
2. If ℓ = 1, thengcd( r, c ,
2) = 1. If ℓ = 2, then 2 | r , 2 | c . Since gcd( r, c , s − r ) = 1, r + s ≡ (cid:3) For a variety Y over C , the cohomology with compact support H ∗ c ( Y, Q ) has a natural mixed Hodgestructure. Let e p,q ( Y ) := P k ( − k h p,q ( H kc ( Y )) be the virtual Hodge number and e ( Y ) := P p,q e p,q ( Y ) x p y q the virtual Hodge polynomial of Y .For α ∈ NS( X ) Q , a torsion free sheaf E is α -twisted semi-stable with respect to H , if(2.3) χ ( F ( − α + nH ))rk F ≤ χ ( E ( − α + nH ))rk E ( n ≫ F of E [10]. M αH ( v ) denotes the moduli stack of α -twisted semi-stable sheaves E with v ( E ) = v , where H is the polarization. ( H, α ) is general with respect to v , if equality in (2.3) implies v ( F )rk F = v ( E )rk E .
In particular, if v is primitive, then M αH ( v ) consists of α -twisted stable objects for a general pair ( H, α ). If α = 0, then we write M H ( v ). Then M αH ( v ) is described as a quotient stack [ Q ss /GL ( N )], where Q ss is asuitable open subscheme of Quot O ⊕ NX /X . We define the virtual Hodge “polynomial” of M αH ( v ) by(2.4) e ( M αH ( v )) = e ( Q ss ) /e ( GL ( N )) ∈ Q ( x, y ) . It is easy to see that e ( Q ss ) /e ( GL ( N )) does not depend on the choice of Q ss . The following was essentiallyproved in [18, Sect. 3.2] (see also [20, Sect. 2.2]). Proposition 2.3.
Let X be a surface such that K X is numerically trivial. Let ( H, α ) be a pair of ampledivisor H and a Q -divisor α . Then e ( M αH ( v )) does not depend on the choice of H and α , if ( H, α ) is generalwith respect to v . By using a special kind of Fourier-Mukai transform called ( − Proposition 2.4 ([19, Prop. 4.5]) . Let X be an unnodal Enriques surface. Assume that r, s > . Then (1) e (cid:0) M αH (cid:0) r, c , − s (cid:1)(cid:1) = e (cid:0) M αH (cid:0) s, − c , − r (cid:1)(cid:1) for a general ( H, α ) , if ( c ) < , i.e, h v i < rs , where v = ( r, c , − s ) . In particular, if r > h v i ,then we get our claim. If we specify the first Chern class as an element of
Pic( X ) ∼ = NS( X ) , then we also have e (cid:0) M αH (cid:0) r, L + r K X , − s (cid:1)(cid:1) = e (cid:0) M αH (cid:0) s, − ( L + s K X ) , − r (cid:1)(cid:1) for a general ( H, α ) , if ( c ( L ) ) < , i.e, h v i < rs , where v = ( r, c ( L ) , − s ) .Remark . (1) For the proof of Proposition 2.4 (2), we use the description of the ( − L + r K X is replaced by − [( L + r K X ) + h v, v ( K X ) i K X ] = − ( L + s K X ).(2) The same claim also holds for nodal case (see Appendix).2.2. Reduction to the rank 2 case.
From Subsection 2.2 to Subsection 2.5, we assume that X is anunnodal Enriques surface and r is even (and hence s is also even). We also assume that α = 0, that is, weconsider the moduli stack of ordinary Gieseker semi-stable sheaves M H ( v ). We shall prove the followingresult in this subsection. Theorem 2.6.
Let v = ( r, c , − s ) be a primitive Mukai vector such that r > is even. (1) If gcd( r, c , s ) = 1 , then e ( M H ( r, c , − s )) = e ( M H (2 , ξ, − s ′ )) for a general H , where ξ is aprimitive element of NS f ( X ) and ( ξ ) + 2 s ′ = ( c ) + rs . (2) If gcd( r, c , s ) = 2 , then e ( M H ( r, c , − s )) = e ( M H (2 , , − s ′ )) for a general H , where s ′ = ( c ) + rs . For the proof of this result, we shall slightly improve Hauzer’s argument. Let Z σ + Z f be a hyperboliclattice in NS( X ): ( σ ) = ( f ) = 1 , ( σ, f ) = 1 . The main difference of [19] and [4] is the case M H ( r, c , − s ) such that r is even and c = r bf + r b ′ σ + ξ , b, b ′ = 0 , ξ ∈ E ( − r, r bf + ξ, − s ) ( b = 0 , − , , ξ ∈ E ( − r, ξ, s ) = 1 ,
2. Indeed1 = gcd( r, r bf + ξ, r − s ) = gcd( r , s , ξ ) implies gcd( r, ξ, s ) = 1 , Lemma 2.7.
For a primitive Mukai vector v = ( r, r bf + ξ, − s ) ( b = 0 , − , , ξ ∈ E ( − ), we set l :=gcd( r, ξ, s ) . (1) e ( M H ( r, r bf + ξ, − s )) = e ( M H ( r ′ , r bf + ξ ′ , − s ′ )) for a general H , where r ′ ≡ r mod 2 l , s ′ ≡ s mod 2 l , l = gcd( r ′ , ξ ′ , s ′ ) , ξ ′ /l ∈ E ( − is primitive and r ′ s ′ ≥ r ′ > h v i . (2) e ( M H ( r, r bf + ξ, − s )) = e ( M H ( s ′′ , − ( r bf + ξ ′′ ) , − r ′ )) for a general H , where r ′ ≡ r mod 2 l , s ′′ ≡ s mod 2 l , l = gcd( s ′′ , ξ ′′ , r ′ ) , ξ ′′ /l ∈ E ( − is primitive and r ′ s ′′ ≥ s ′′ > h v i .Proof. We first note that the choice of H is not important by Proposition 2.3. So we do not explainabout the choice of H . (1) We set p := ( r, ξ ). For v = ( r, r bf + ξ, − s ), we take D ∈ E ( −
1) such that ve D = ( r, r bf + ξ , − s ′ ) satisfies ξ /p is primitive and s ′ > h v i . Since s ′ = s − ξ, D ) − r ( D ), s ′ ≡ s mod 2 l .By Proposition 2.4, e ( M H ( v )) = e ( M H ( s ′ , − ( r bf + ξ ) , − r )). Since l = ( s ′ , p ), we take D ∈ E ( −
1) suchthat ( s ′ , − ( r bf + ξ ) , − r ) e D = ( s ′ , − ( r bf + ξ ′ ) , − r ′ ) satisfies ξ ′ /l is primitive and r ′ > h v i . We also have r ′ = r + 2( ξ , D ) − s ′ ( D ) ≡ r mod 2 l . Applying Proposition 2.4, we have e (cid:0) M H (cid:0) s ′ , − ( r bf + ξ ) , − r (cid:1)(cid:1) = e (cid:16) M H (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17)(cid:17) . (2) For ( r ′ , r bf + ξ ′ , − s ′ ) in (1), we take D ∈ E ( −
1) such that ( r ′ , r bf + ξ ′′ , − s ′′ ) = ( r ′ , r bf + ξ ′ , − s ′ ) e D satisfies ξ ′′ /l ∈ E ( −
1) is primitive, s ′′ > h v i . Then we have e (cid:16) M H (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17)(cid:17) = e (cid:16) M H (cid:16) s ′′ , − ( r bf + ξ ′′ ) , − r ′ (cid:17)(cid:17) by Proposition 2.4. (cid:3) Lemma 2.8.
For a primitive Mukai vector v = ( r, r bf + ξ, − s ) ( b = 0 , − , , ξ ∈ E ( − ), there exist somezeta and t such that e (cid:0) M H (cid:0) r, r bf + ξ, − s (cid:1)(cid:1) = e (cid:0) M H (cid:0) , ζ, − t (cid:1)(cid:1) for a general H .Proof. (1) We first assume that r ≡ s ≡ e (cid:0) M H (cid:0) r, r bf + ξ, − s (cid:1)(cid:1) = e (cid:16) M H (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17)(cid:17) for a general H , where r ′ ≡ l , s ′ ≡ l , ξ ′ /l ∈ E ( −
1) is primitive and r ′ > h v i . For η ∈ E ( − D := σ − ( η )2 f + η . Then ( D ) = 0. Since r ≡ l , we can choose η such that(2.5) s ′ − rb − ξ ′ , η ) . hen ( r bf + ξ ′ , D ) = r b + ( ξ ′ , η ) = s ′ − (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17) e D = (cid:16) r ′ , r bf + ξ ′ + r ′ D, − s ′ − r bf + ξ ′ ,D )2 (cid:17) = (cid:0) r ′ , r bf + ξ ′ + r ′ D, − (cid:1) . (2.6)Hence e (cid:16) M H (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17)(cid:17) = e (cid:16) M H (cid:16) , ζ, − r ′ (cid:17)(cid:17) for a general H , where ζ = − ( r bf + ξ ′ + r ′ D ).(2) We next assume that r ≡ b = 0 and l = 2, then by using Lemma 2.7 (2), we have e (cid:0) M H (cid:0) r, ξ, − s (cid:1)(cid:1) = e (cid:16) M H (cid:16) s ′′ , − ξ ′′ , − r ′ (cid:17)(cid:17) for a general H . Since r ′ ≡ l , it is reduced to the case (1).Assume that b = ± l = 1. By Lemma 2.7 (1), we have e (cid:0) M H (cid:0) r, r bf + ξ, − s (cid:1)(cid:1) = e (cid:16) M H (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17)(cid:17) for a general H , where r ′ ≡ l , ξ ′ /l is primitive and r ′ > h v i . Since ξ ′ /l is primitive and r is odd, wetake η ∈ E ( −
1) such that r b + ( ξ ′ , η ) = 1. We set D := σ − ( η )2 f + η . Then ( D, r bf + ξ ′ ) = r b + ( ξ ′ , η ) = 1.Hence (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17) e ( s ′ − D = (cid:16) r ′ , r bf + ξ ′ + r ′ ( s ′ − D, − (cid:17) . Applying Proposition 2.4, we get e (cid:16) M H (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17)(cid:17) = e (cid:16) M H (cid:16) , ζ, − r ′ (cid:17)(cid:17) for a general H , where ζ = − ( r bf + ξ ′ + r ′ ( s ′ − D ).(3) Finally we assume that r ≡ s ≡ l = 2, then 2 | ( r bf + ξ ). By Lemma 2.2(2), v is not primitive. Hence l = 1. By using Lemma 2.7 (1) again, we have e (cid:0) M H (cid:0) r, r bf + ξ, − s (cid:1)(cid:1) = e (cid:16) M H (cid:16) r ′ , r bf + ξ ′ , − s ′ (cid:17)(cid:17) for a general H , where ξ ′ is primitive and r ′ > h v i . Since we can take η ∈ E ( −
1) with r b + ( ξ ′ , η ) = 1, asin the case (2), we get the claim. (cid:3) We shall next treat the general case. We use induction on r . We set c := d σ + d f + ξ , ξ ∈ E ( − v by v exp( kσ ), we may assume that − r < d ≤ r . We first assume that d = 0 , r . We note that( c , f ) = d . Replacing v by v exp( η ), η ∈ E ( − s > h v i . Then by Proposition 2.4, e ( M H ( v )) = e ( M H ( s, − c , − r )) for a general H . We take an integer k such that 0 < r + 2 d k ≤ | d | < r .Then v exp( kf ) = ( s, ( − c + skf ) , − r ′ ), where r ′ = r + 2 d k . Since s > h v i , Proposition 2.4, implies that e ( M H ( s, ( − c + skf ) , − r ′ )) = e ( M H ( r ′ , ( c − skf ) , − s )) for a general H . By induction hypothesis, we getour claim.If d = 0 , r , then we may assume that − r < d ≤ r . If d = 0 , r , then we can apply the same argumentand get our claim. If ( d , d ) = (0 , , ( r , , (0 , r ), then the claim follows from Lemma 2.8.Assume that ( d , d ) = ( r , r ). We may assume that ξ = kξ ′ , ξ ′ is primitive and 0 ≤ k ≤ r .For η ∈ E ( − σ ′ := σ − ( η )2 f + η . Then σ ′ and f spans a hyperbolic lattice and (cid:0) r ( σ + f ) + ξ, f (cid:1) = r (cid:0) r ( σ + f ) + ξ, σ ′ (cid:1) = r (cid:18) − ( η )2 (cid:19) + ( ξ, η ) . (2.7)Replacing η by − η if necessary, we can take η such that(2.8) ( ξ ′ , η ) = ( − , | ( η ) / , ∤ ( η ) / . (2.9) r (cid:18) − ( η )2 (cid:19) + ( ξ, η ) ≡ ( r − k mod r | ( η ) / k mod r ∤ ( η ) / . If k = r ,
0, then we can reduce to the case where | d | < r . If k = 0, then choosing η with ( η ) = −
2, we canreduced to the case d = 0. If k = r , then we choose η satisfying (( ξ ′ − η ) ) ≡ ( ξ ′ ) + 2 mod 4. Then(2.10) r (cid:18) − ( η )2 (cid:19) + r ξ ′ , η ) ≡ r. ence we can also reduce to the case where d = 0. Therefore Theorem 2.6 holds. (cid:3) Remark . In [4], Hauzer takes a hyperbolic lattice spanned by σ and σ + f + e , where e ∈ E ( −
1) is a( − c = ( r + ( ξ, e )) σ ′ + ( r/ f ′ + ξ ′ .By Theorem 2.6, Theorem 1.1 for r > Proposition 2.10 (Kim [9]) . Assume that v := (2 , c ( L ) , − s ) is primitive. Then M H (2 , L, − s ) = ∅ if andonly if (i) gcd(2 , c ( L )) = 1 and h v i ≥ − or (ii) gcd(2 , c ( L )) = 2 and h v i ≥ or (iii) gcd(2 , c ( L )) = 2 , h v i = 0 and L ≡ K X mod 2 . For the case of Proposition 2.10 (iii), by using Proposition 2.4 (2), we have Theorem 1.1 (iii). In the nextsubsection, we shall give another proof of Kim’s result.2.3.
Relative Fourier-Mukai transform.
For G ∈ K ( X ) with rk G >
0, we define G -twisted semi-stability replacing the Hilbert polynomial χ ( E ( nH )) by the G -twisted Hilbert polynomial χ ( G ∨ ⊗ E ( nH )). M GH ( r, L, − s ) denotes the moduli scheme of G -twisted semi-stable sheaves E with v ( E ) = ( r, c ( L ) , − s ) anddet E = L . If G = O X , then we also denote M GH ( r, L, − s ) by M H ( r, L, − s ). The G -twisted semi-stability isthe same as the α -twisted semi-stability, where α = c ( G ) / rk G .We have an elliptic fibration X → P such that 2 f is the divisor class of a fiber. Let G be a locally freesheaf on X such that v ( G ) = v ( O X ) + v ( O X ( σ )) + (0 , , k ). We set Y := M G H + nf (0 , f, H is anample divisor on X and n ≥
0. Then χ ( G , E ) = −h v ( G ) , v ( E ) i = 0 for E ∈ M G H + nf (0 , f, Lemma 2.11. Y consists of G -twisted stable sheaves.Proof. If E ∈ M G H + nf (0 , f,
1) is properly G -twisted semi-stable, then there is a proper subsheaf E of E such that χ ( G , E ) = 0 and E/E is also purely 1-dimensional. We set v ( E ) = (0 , ξ , a ), a ∈ Z .Then ( ξ , c ( G )) = 2 a ∈ Z . Since ( c ( E ) , c ( G )) , ( c ( E/E ) , c ( G )) ≥ c ( E ) , c ( G )) = 2,( c ( E ) , c ( G )) = 0 or ( c ( E/E ) , c ( G )) = 0. If every singular fiber is irreducible, then ( c ( E ) , c ( G )) > c ( E/E ) , c ( G )) >
0. Therefore Y consists of G -twisted stable sheaves. (cid:3) By [1], Y is a smooth projective surface which is a compactification of Pic X/C . Hence Y ∼ = X . Let E bea universal family. Let Ψ : D ( X ) → D ( Y ) be a contravariant Fourier-Mukai transform defined by(2.11) Ψ( E ) := R Hom p Y ( p ∗ X ( E ) , E ) , where p X and p Y are the projections from X × Y to X and Y respectively.Let L be a line bundle on C ∈ | H | and set G := Ψ( L )[1] (see the above of [22, Lem. 3.2.3]). We alsoset b H := − c (Ψ( G )) ([22, Lem. 3.2.1]). Proposition 2.12 ([22, Prop. 3.4.5]) . Assume that ( c ( L ) , f ) = r ∈ Z and χ ( E, L ) < . Ψ induces anisomorphism M G H + nf (cid:16) r, L, − s (cid:17) ∼ = M G b H + nf (cid:18) , D, − s ′ (cid:19) for n ≫ , where D is an effective divisor such that ( D ) = ( c ( L ) ) + rs and ( D, f ) = r .Remark . Replacing E by E ( mf ) ( m ≫ χ ( E, L ) < Remark . Although G is fixed, H is not fixed. So we can change H to be general. Corollary 2.15.
Assume that ∤ c ( L ) . Then M G H + nf (2 , L, − s ) = ∅ if and only if ( c ( L ) ) + 2 s ≥ .Proof. Let D be the divisor in Proposition 2.12. Since ( D ) = ( c ( L ) ) + 2 s , we shall prove that thecondition is ( D ) ≥
0. Obviously the condition is necessary. Conversely assume that ( D ) ≥
0. Since Y ∼ = X is unnodal, | D | contains a reduced and irreducible curve C by [3, Thm. 3.2.1], where we also use( D, f ) = 1 if ( D ) = 0. Then a line bundle F on C with χ ( F ) = − s ′ is a member of M G b H + nf (0 , D, − s ′ ). (cid:3) Rank 2 case.Proposition 2.16.
Assume that ∤ c ( L ) is primitive. Then M H (2 , L, − s ) = ∅ for a general H if and onlyif ( c ( L ) ) + 2 s ≥ .Proof. If 2 ∤ ( c ( L ) , f ) or 2 ∤ ( c ( L ) , σ ), then the claim follows from Corollary 2.15. Otherwise we mayassume that c ( L ) ∈ E ( −
1) and c ( L ) is primitive. Then there is η ∈ E ( −
1) with ( c ( L ) , η ) = 1. We set σ ′ := σ − ( η )2 f + η . Then Z σ ′ + Z f spans a hyperbolic lattice and ( σ ′ , c ( L )) = 1. Since X is unnodaland f is effective, σ ′ is effective and 2 σ ′ defines an elliptic fibration. Therefore the claim also holds for thiscase. (cid:3) roposition 2.17. Assume that | c ( L ) . Then M H (2 , L, − s ) = ∅ if and only if (i) ( c ( L ) ) + 2 s > or (ii) ( c ( L ) ) + 2 s = 0 and L ≡ K X mod 2 .Proof. We may assume that L = 0 , K X . If there is a stable sheaf E , then E ∼ = E ( K X ) and ( c ( L ) )+2 s ≥ − E = E ( K X ) and ( c ( L ) ) + 2 s ≥ −
1. Since 4 | s , ( c ( L ) ) + 2 s = 2 s ≥ c ( L ) ) + 2 s >
0, we first prove M H (2 , L, − s ) = ∅ for a general H . We set k := s >
0. Then E ⊕ E with v ( E ) = (1 , , − k − ) , v ( E ) = (1 , , − k + ) belongs to the moduli stack M H (2 , L, − s ) µ - ss of µ -semi-stable sheaves. Let F ( v , v ) be the substack of M H (2 , L, − s ) µ - ss consisting of E whose Harder-Narasimhan filtration 0 ⊂ F ⊂ F = E satisfies v ( F ) = v and v ( F/F ) = v . Thendim F ( v , v ) = h v , v i + dim M H ( v ) + dim M H ( v )= h v i − h v , v i . (2.12)We set v = (1 , ξ , − s ), v = (1 , ξ , − s ). Then ξ and ξ are numerically trivial, s < s and s + s = s .Then h v , v i = s + s = s >
0. By the deformation theory, each irreducible component M of M H ( v ) µ - ss satisfies dim M ≥ h v i . Hence there is a stable sheaf.We next treat the case where ( c ( L ) ) + 2 s = 0. By [19], M H (2 , K X , ∼ = X and E ( K X ) ∼ = E for all E ∈ M H (2 , K X , E with v ( E ) ≡ v mod K X , we see that E ∈ M H (2 , K X , M H (2 , ,
0) = ∅ . (cid:3) Therefore Proposition 2.10 holds by Proposition 2.16, 2.17, and we complete the proof of Theorem 1.1 for r > Remark . Nuer constructed µ -stable vector bundles of rank 4 in [12, Thm. 5.1]. This reuslt ([12, Thm.5.1]) does not follow from our method.2.5. Rank 0 case.
We shall prove Theorem 1.1 for r = 0. We first note that if M H (0 , L, − s ) = ∅ , then L is effective. For the proof of Theorem 1.1, we use Proposition 2.12. By choosing a suitable elliptic fibration,we may assume that ( c ( L ) , f ) >
0. Then we have e (cid:16) M H (cid:16) , L, − s (cid:17)(cid:17) = e (cid:18) M H (cid:18) r, L ′ , − s ′ (cid:19)(cid:19) , where ( c ( L ′ ) , f ) = r . Then the case of r = 0 is reduced to the case of r > c ( L ) , s ) = 1or ( c ( L ) ) >
0. Assume that gcd( c ( L ) , s ) = 2 and ( c ( L ) ) = 0. Then M H (0 , L, − s ) = ∅ or M H (0 , L + K X , − s ) = ∅ . If L ≡ r C ∈ | L | such that C is a smooth fiber of the ellipticfibration, and a stable vector bundle F of rank r and χ ( F ) = − s on C is a member of M H (0 , L, − s ). Hence M H (0 , L, − s ) = ∅ if and only if L ≡ Remark . It is easy to see that [21, Thm. 1.7] holds for Enriques surfaces. Indeed a similar claim to[21, Prop. 2.7] (see Appendix) holds and [21, Prop. 2.8, Prop. 2.11] hold if we modify the number N in theclaims suitably.Then Theorem 1.1 for r = 0 can also be reduced to the claim for r > Remark . Since X is unnodal, effectivity implies ( c ( L ) ) ≥ c ( L ) , H ) >
0. Conversely if( c ( L ) ) ≥ c ( L ) , H ) >
0, then L is effective by the Riemann-Roch theorem.3. A nodal case
We shall treat the nodal case by adding a deformation argument and results of Kim [5] and [8].
Theorem 3.1.
Let X be a nodal Enriques surface over C . We take r, s ∈ Z ( r > and L ∈ NS( X ) such that r − s is even. Assume that gcd( r, c ( L ) , r − s ) = 1 , i.e., the Mukai vector is primitive. Then M H ( r, L, − s ) = ∅ for a general H if and only if (i) gcd( r, c ( L ) , s ) = 1 and ( c ( L ) ) + rs ≥ − or (ii) gcd( r, c ( L ) , s ) = 2 and ( c ( L ) ) + rs ≥ or (iii) gcd( r, c ( L ) , s ) = 2 , ( c ( L ) ) + rs = 0 and L ≡ r K X mod 2 or (iv) ( c ( L ) ) + rs = − , L ≡ D + r K X mod 2 , where D is a nodal cycle, i.e., D is effective, ( D ) = − and | D + K X | = ∅ .Remark . If ( c ( L ) , H ′ ) > H ′ , then the same claim holds for r = 0.Obviously ( c ( L ) )+ rs ≥ − c ( L ) ) + rs ≥ −
1. In his case, the existence is a consequence of Theorem 1.1. Let (
X, H ) be an Enriquessurface X and an ample divisor H on X . By [3, Prop. 1.4.1], H ( X, T X ) ∼ = C ⊕ and H ( X, T X ) = 0. We alsohave H ( X, O X ) = 0. Hence a polarized deformation of the pair ( X, H ) is unobstructed. Let ( X , H ) → S e a general deformation of ( X, H ) such that a general member is not nodal and ( X , H ) = ( X, H ) (0 ∈ S ).Then we have a family of moduli spaces of semi-stable sheaves f : M ( X , H ) ( v ) → S . Under the assumption(i), (ii), (iii) in Theorem 1.1, M ( X , H ) ( v ) s = ∅ for unnodal X s . Hence f is dominant. By the projectivity of f , im f = S . Hence M ( X , H ) ( v ) s = ∅ for all s . Proposition 3.3.
Let X be an Enriques surface. Under the conditions (i), (ii), (iii) of Theorem 1.1, M H ( r, L, − s ) = ∅ for a general H . If gcd( r, c ( L ) , a ) = 2, ( c ( L ) ) + rs = 0 and L r K X mod 2, then M H ( r, L, − s ) = ∅ . Indeed since M H ( r, L + K X , − s )( = ∅ ) is an Enriques surface for a general H and the universal family induces a Fourier-Mukai transform, we see that every stable sheaf E with v ( E ) = ( r, c ( L ) , − s ) belongs to M H ( r, L + K X , − s ).Therefore Theorem 3.1 holds if ( c ( L ) ) + rs ≥ − Remark . If r is odd and H is general, then Ext ( E, E ) = 0 for E ∈ M H ( r, c , − s ). In this case, f is asmooth morphism in a neighborhood of 0.We treat the remaining case, i.e., ( c ( L ) ) + rs = −
2. This case is completely studied by Kim in [5] and[8]. For completeness of the proof, we add an outline of the proof in [8]. Let π : e X → X be the universalcover of X . e X is a K3 surface. We need the following elementary fact. Lemma 3.5.
For a locally free sheaf F of rank r on e X , det π ∗ ( F ) ∼ = det( π ∗ (det F ))(( r − K X ) . Proof.
Let H be an ample divisor on X . Since π ∗ ( H ) is ample, we have an exact sequence(3.1) 0 → O e X ( − nπ ∗ ( H )) ⊕ ( r − → F → I Z ( D ) → , where D is a divisor, Z is a 0-dimensional subscheme of e X and n is sufficiently large. Since π ∗ ( O e X ) = O X ⊕ O X ( K X ) and O e X ( D − ( r − nπ ∗ ( H )) = det F , we get the claim. (cid:3) We also need the following result of Kim [8, Thm. 1].
Lemma 3.6.
Assume that r ∈ Z > , a ∈ Z and L ∈ NS( X ) satisfy ( c ( L ) ) − ra = − . Then M H ( r, L, a ) = ∅ for a general H if and only if M H (2 , L − ( r − K X , ra ) = ∅ .Proof. Since the formulation of the claim is slightly different from [8, Thm. 1], we write the proof. We set v := ( r, c ( L ) , a ). Since gcd( r, c ( L )) = 1, there is an ample divisor H with gcd( r, ( c ( L ) , H )) = 1. Indeed wefirst take a divisor η with gcd( r, ( c ( L ) , η )) = 1. Then we have an ample divisor H = η + rλ , λ ∈ Amp( X ),which satisfies the claim. We may prove the claim for this polarization.For E ∈ M H ( r, L, a ), we have E ∼ = E ( K X ). By the proof of [15, Lem. 1.12], there is a simple vector bundle F such that E = π ∗ ( F ). Since E is rigid, F is also rigid (see the proof of [8, Thm. 1]). By the stability of E , F is stable with respect to π ∗ ( H ). We have π ∗ ( E ) ∼ = F ⊕ ι ∗ ( F ). We set C := det( F ). Then v ( F ) = ( r , C, a )and C + ι ∗ ( C ) = π ∗ ( L ). We see that ( C ) = ( C, ι ∗ ( C )) − C ) − ra = −
2. We set E ′ := π ∗ ( O e X ( C )).Since π ∗ ( E ) ∼ = O e X ( C ) ⊕ O e X ( ι ∗ ( C )), we see that v ( E ′ ) = (2 , c ( L ) , ( C )2 + 1) = (2 , c ( L ) , ra ). By Lemma3.5, det E ′ = (det E ) (cid:16) − (cid:16) r − (cid:17) K X (cid:17) = O X (cid:16) L − (cid:16) r − (cid:17) K X (cid:17) . Obviously E ′ is semi-stable with respect to H . Since gcd( r, ( c ( E ′ ) , H )) = 1, it is µ -stable. Therefore M H (2 , L − ( r − K X , rs ) = ∅ .Conversely for E ′ ∈ M H (2 , L − ( r − K X , ra ), there is a divisor C with π ∗ ( O e X ( C )) = E ′ . Since π ∗ ( E ′ ) ∼ = O e X ( C ) ⊕ O e X ( ι ∗ ( C )), we see that ( C ) = ( C, ι ∗ ( C )) − C ) + 2 = ra . For u := ( r , C, a ),we have h u i = −
2. By gcd( r, ( c ( L ) , H )) = 1 and π ∗ ( L ) = C + ι ∗ ( C ), gcd( r, ( C, π ∗ ( H ))) = 1. Let F be a µ -stable locally free sheaf such that v ( F ) = u with respect to π ∗ ( H ). Then E := π ∗ ( F ) is a µ -stable locallyfree sheaf with v ( π ∗ ( F )) = v . By Lemma 3.5, det E = L . Therefore M H ( r, L, a ) = ∅ . (cid:3) Proposition 3.7.
Assume that r ∈ Z ≥ , s ∈ Z and L ∈ NS( X ) satisfy r ≡ s mod 2 and ( c ( L ) )+ rs = − .If r = 0 , then we further assume that ( c ( L ) , H ′ ) > for an ample divisor H ′ on X . Then M H ( r, L, − s ) = ∅ for a general H if and only if L = D + 2 A + r K X , where D is a nodal cycle and A ∈ NS( X ) .Proof. If r >
0, then the claim is a consequence of Lemma 3.6 and [5, Thm. 3.4]. If r = 0, then the claim isa consequence of Remark 2.19 (see also Corollary 4.5). (cid:3) . Appendix
Let X be any Enriques surface and H be an ample divisor on X . For ω = tH , t >
0, let Z (0 ,ω ) : D ( X ) → C be a stability function defined by(4.1) Z (0 ,ω ) ( E ) := h e ω √− , v ( E ) i , E ∈ D ( E ) . Let T (0 ,ω ) be the full subcategory of Coh( X ) generated by torsion sheaves and torsion free stable sheaves E with Z (0 ,ω ) ( E ) ∈ H ∪ R < . Let F (0 ,ω ) be the full subcategory of Coh( X ) generated by torsion free stablesheaves E with − Z (0 ,ω ) ( E ) ∈ H ∪ R < . Let A (0 ,ω ) ( ⊂ D ( X )) be the category generated by T (0 ,ω ) and F (0 ,ω ) [1].If( ω ) = 1, then σ (0 , ω ) := ( A (0 ,ω ) , Z (0 ,ω ) ) is a stability condition. A (0 ,ω ) is constant on ( ω ) = 1. We set Definition 4.1. (1) For ( ω ) >
1, we set T µ := T (0 ,ω ) , F µ := F (0 ,ω ) and A µ := A (0 ,ω ) .(2) For ( ω ) <
1, we set T := T (0 ,ω ) , F := F (0 ,ω ) and A := A (0 ,ω ) .For E ∈ F µ , we have an exact sequence(4.2) 0 → E → E → E → E is generated by O X and K X , and(2) E ∈ F µ satisfies Hom( O X , E ) = Hom( O X ( K X ) , E ) = 0, i.e., E ∈ F .Since H ( O X ( K X )) = H ( O X ) = 0, E ∼ = O ⊕ nX ⊕ O X ( K X ) ⊕ m . We also have E = Hom( O X , E ) ⊗ O X ⊕ Hom( O X ( K X ) , E ) ⊗ O X ( K X ). For E ∈ T , the natural homomor-phism φ : E → Hom( E, O X ) ∨ ⊗ O X ⊕ Hom( E, O X ( K X )) ∨ ⊗ O X ( K X )is surjective and ker φ ∈ T µ .We set E := ker( O X ⊠ O X ⊕ O X ( K X ) ∨ ⊠ O X ( K X ) → O ∆ ) . As in [11], Φ E ∨ [1] X → X : D ( X ) → D ( X ) induces an isomorphism A → A µ and we have a commutative diagram(4.3) A Φ E∨ [1] X → X −−−−→ A µZ (0 ,ω ) y y Z (0 ,ω ′ ) C ←−−−− × ( ω ) C where ω ′ = ω/ ( ω ). In particular, we get the following. Proposition 4.2. Φ E ∨ [1] X → X induces an isomorphism (4.4) M (0 ,ω ) (cid:16) r, η + r K X , − s (cid:17) ∼ = M (0 ,ω ′ ) (cid:16) s, η + s K X , − r (cid:17) . Applying Toda’s argument to the wall crossing along the line ω = tH , t >
0, we get the following result(see also the argument in [11]).
Proposition 4.3 (cf. Toda [16]) . (1) If ( ω ) ≫ and ( η, ω ) > , then M (0 ,ω ) (cid:16) r, η + r K X , − s (cid:17) = M ω (cid:16) r, η + r K X , − s (cid:17) . (2) e (cid:0) M (0 ,ω ) (cid:0) r, η + r K X , − s (cid:1)(cid:1) is independent of a general choice of ω .Remark . Wall crossing along the line ω = tH is very similar to the classical wall crossing of Giesekersemi-stability, since A (0 ,ω ) is almost the same. Corollary 4.5. e (cid:16) M H (cid:16) r, η + r K X , − s (cid:17)(cid:17) = e (cid:16) M H (cid:16) s, η + s K X , − r (cid:17)(cid:17) for a general H . We have another proof of Proposition 3.7.
Proposition 4.6.
Assume that ( η ) + rs = − . M H ( r, η + r K X , − s ) = ∅ for a general H if and only if η ≡ D mod 2 , where D is a nodal cycle.Proof. By the proof of Theorem 2.6, we have e ( M H ( r, η + r K X , − s )) = e ( M H (2 , η ′ + K X , − s ′ )), where η ≡ η ′ mod 2. By [5], M H (2 , η ′ + K X , − s ′ ) = ∅ if and only if η ≡ D mod 2, where D is a nodal cycle.Therefore the claim holds. (cid:3) eferences [1] T. Bridgeland, Fourier-Mukai transforms for elliptic surfaces,
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E-mail address : [email protected]@math.kobe-u.ac.jp