Non-emptiness of Brill-Noether Loci over very general quintic hypersurface
aa r X i v : . [ m a t h . AG ] O c t NON-EMPTINESS OF BRILL-NOETHER LOCI OVERVERY GENERAL QUINTIC HYPERSURFACE
KRISHANU DAN AND SARBESWAR PAL
Abstract.
In this article we study Brill-Noether loci of moduli space ofstable bundles over smooth surfaces. We define Petri map as an analogywith the case of curves. We show the non-emptiness of certain Brill-Noether loci over very general quintic hypersurface in P , and use thePetri map to produce components of expected dimension. Introduction
Let X be a smooth, irreducible, projective variety of dimension n over C , H be an ample divisor on X , and let M := M X,H ( r ; c , · · · , c s ) bethe moduli space of rank r , H -stable vector bundles E over X with Chernclasses c i ( E ) = c i , where s := min { r, n } . A Brill-Noether locus B kr,X,H is aclosed subscheme of M whose support consists of points E ∈ M such that h ( X, E ) ≥ k + 1. G¨ottsche et al ([6]) and M. He ([7]) studied the Brill-Noether loci of stable bundles over P , and Yoshioka ([15], [16]), Markman([4]), Leyenson ([10], [11]) studied it for K C over C , the Brill-Noether loci of the modulispace, Pic d ( C ), of degree d line bundles on C is well-studied. The questionslike non-emptiness, connectedness, irreducibility, singular locus etc of Brill-Noether loci are known when C is a general curve in the sense of moduli(see e.g. [1]). This concept was generalized for vector bundles over curvesby Newstead, Teixidor and others. For an account of the results and historyin this case see [5] and the references therein.Recently, in [2], authors have constructed a Brill-Noether loci over higherdimensional varieties under the additional cohomology vanishing assump-tions: H i ( X, E ) = 0 , ∀ i ≥ E ∈ M . This is a natural general-ization of the Brill-Noether loci over the curves for higher dimensional vari-eties. In [2], [3], authors gave several examples of non-empty Brill-Noetherlocus, and examples of Brill-Noether locus where “expected dimension” isnot same as the exact dimension. In all these examples, the surfaces and/orvarieties chosen have the canonical bundle has no non-zero sections. In thisarticle, we defined “Petri map” over a smooth projective variety with canon-ical bundle ample, as an analogue of that for curves. Similar to the case ofcurves, the injectivity of the “Petri map” implies the existence of smooth Mathematics Subject Classification.
Key words and phrases.
Vector Bundles, Brill-Noether loci, Moduli space, Surfaces. oints in Brill-Noether loci. We then use this fact to prove the existenceof a smooth point and hence a component of expected dimension in theBrill-Noether loci over a very general quintic hypersurface in P where thecanonical bundle is ample and globally generated. Notation:
We work throughout over the field C of complex numbers. If X is a smooth, projective variety, we denote by K X the canonical bundle on X .For a coherent sheaf F on X , we denote by H i ( X, F ) the i -th cohomologygroup of F and by h i ( X, F ) its (complex) dimension. If V is a vector bundleon X , we denote by V ∗ the dual of V .2. Brill-Noether Loci
In this section we will briefly recall the construction of Brill-Noether lociover higher dimensional varieties, following [2].Let X be an irreducible, smooth, projective variety of dimension n , andlet H be an ample divisor on X . For a torsion free sheaf F over X , let c i ( F )denotes the i -th Chern class of F . Set µ ( F ) = µ H ( F ) := c ( F ) · H n − rank( F ) . Definition 2.1.
A torsion-free sheaf F over X of rank r is called H - semistable if for all non-zero subsheaf G of F with rank( G ) < rank( F ),we have µ ( G ) ≤ µ ( F ) . We say F is H - stable if the above inequality is strict.Let M := M X,H ( r ; c , · · · , c s ) be the moduli space of rank r , H -stablevector bundles E over X with Chern classes c i ( E ) = c i , where s := min { r, n } .Assume that M is a fine moduli space, and let E → M × X be an universalfamily such that for any t ∈ M , E t := E| t × X is a rank r, H -stable bundleover X with Chern classes c i ( E t ) = c i . Choose an effective divisor D on X such that H i ( X, E t ( D )) = 0 , ∀ i ≥ ∀ t ∈ M . Let D := M × D be theproduct divisor, and φ : M × X → M be the projection map. From theexact sequence 0 → E → E ( D ) → E ( D ) / E →
M × X , we get an exact sequence on M :0 → φ ∗ ( E ) → φ ∗ ( E ( D )) γ −→ φ ∗ ( E ( D ) / E ) → R φ ∗ ( E ) → . Note that, γ is a map between two locally free sheaves of ranks χ ( E t ( D ))and χ ( E t ( D )) − χ ( E t ) respectively, on M . For an integer k ≥ −
1, let B kr,X,H ⊂ M be the ( χ ( E t ( D )) − ( k + 1))-th determinantal variety associatedto the map γ . Now assume H i ( X, E t ) = 0 , ∀ i ≥ ∀ t ∈ M . Then wehave Support( B kr,X,H ) = { E ∈ M : h ( X, E ) ≥ k + 1 } . When M is not a fine moduli space, it is possible to carry out this construc-tion locally and then can be glued together to get a global algebraic object.We summarize the above construction as heorem 2.2. ( [2] , Theorem 2.3) Let X be a smooth, irreducible, projec-tive variety of dimension n , H be a fixed ample divisor on X , and M := M X,H ( r ; c , · · · , c s ) be a moduli space of rank r, H -stable vector bundles E on X with fixed Chern classes c i ( E ) = c i , s = min { r, n } . Assume that forany E ∈ M , H i ( X, E ) = 0 for i ≥ . Then for any k ≥ − , there exists adeterminantal variety B kr,X,H ⊂ M such thatSupport ( B kr,X,H ) = { E ∈ M : h ( X, E ) ≥ k + 1 } . Moreover, each non-empty irreducible component of B kr,X,H has dimensionat least dim ( M ) − ( k + 1)( k + 1 − χ ( r ; c , · · · , c s )) where χ ( r ; c , · · · , c s ) := χ ( E t ) for any t ∈ M and B k +1 r,X,H ⊂ Sing ( B kr,X,H ) whenever B kr,X,H = M . Definition 2.3.
The variety B kr,X,H is called the k -th Brill-Noether locus ofthe moduli space M and the number ρ kr,X,H := dim( M ) − ( k + 1)( k + 1 − χ ( r ; c , · · · , c s ))is called the generalized Brill-Noether number. By the above theorem, the dimension of B kr,X,H is at least ρ kr,X,H . Wecall ρ kr,X,H , the expected dimension of the Brill-Noether locus B kr,X,H . Whenthere is no confusion about X and H , we will simply denote these by B kr and ρ kr . 3. Petri Map
In this section we will define “Petri Map” for higher dimensional varieties,as an analogue of the one defined for curves. We note that the description ofPetri map over curves as given in [5] works for higher dimensional varietiesalso. For convenience, we recall this description.Let X be an irreducible, smooth, projective variety of dimension n , H be an ample divisor on X , and let M := M X,H ( r ; c , · · · , c s ) be the mod-uli space of rank r , H -stable vector bundles E over X with Chern classes c i ( E ) = c i , where s := min { r, n } . The tangent space to M at a point E ∈ M is given by H ( X, E ⊗ E ∗ ), where E ∗ is the dual of E . A tangent vector to M at E can be identified with a vector bundle E ǫ on X ǫ := X × Spec( k [ ǫ ] /ǫ )whose restriction on X is E and it fits into the exact sequence0 → E → E ǫ → E → . We call it a first order deformation of E .One can give an explicit description of the bundle E ǫ as follows: Let { U i } be an open cover of X such that E i := E | U i is the trivial bundle. Set U ij := U i ∩ U j , and let φ ij ∈ H ( U ij , E ⊗ E ∗ ) be the co-boundary map orresponding to φ ∈ H ( X, E ⊗ E ∗ ). Consider the trivial extension of E i to U i × Spec( k [ ǫ ] /ǫ ) given by E i ⊕ ǫE i . Then the matrix (cid:20) Id 0 φ ij Id (cid:21) will give the gluing data for the bundle E ǫ .Assume that a section s of E can be extended to a section of E ǫ . Thenwe have local sections ( s ′ i ) ∈ H ( U i , E i ) such that ( s | U i , s ′ i ) defines a sectionof E ǫ . If this is the case, then we have (cid:20) Id 0 φ ij Id (cid:21) (cid:20) s | U i s ′ i (cid:21) = (cid:20) s | U j s ′ j (cid:21) . This gives two conditions: ( s | U i ) | U ij = ( s | U j ) | U ij and φ ij ( s ) = s ′ j − s ′ i . Thefirst condition is automatically satisfied, since s is a global section and fromthe second condition we see that, in this case, ( φ ij ( s )) satisfies the co-cyclecondition. In other words, ( φ ij ( s )) is in the kernel of the map H ( X, E ⊗ E ∗ ) −→ H ( X, E ) , ( ν ij ) ( ν ij ( s )) . Let E ∈ B kr , k ≥ s ∈ H ( X, E ). Then the first order deformationof E , as an element of B kr , is the subset { E ǫ : the section s can be extendedto a section of E ǫ } of H ( X, E ⊗ E ∗ ), i.e. E ǫ ∈ Ker (cid:0) H ( X, E ⊗ E ∗ ) −→ H ( X, E ) , ( ν ij ) ( ν ij ( s )) (cid:1) . Now assume E ∈ B kr − B k +1 r and let T be the tangent space to B kr at thepoint E . From the discussion above, we have a map α : H ( X, E ) ⊗ H ( X, E ⊗ E ∗ ) −→ H ( X, E ) . This induces the map µ : H ( X, E ) ⊗ H n − ( X, K X ⊗ E ∗ ) −→ H n − ( X, K X ⊗ E ⊗ E ∗ ) . Note that, T can be identified with (Im( µ )) ⊥ . We call the map µ , the Petrimap.
Remark 3.1.
Let X be an irreducible, smooth, projective surface and H be an ample divisor on X . In this case, Petri map, as defined above, is thecup product map(1) µ : H ( X, E ) ⊗ H ( X, K X ⊗ E ∗ ) −→ H ( X, K X ⊗ E ⊗ E ∗ ) . If E is a smooth point in the moduli space M , we havedim( T ) = dim( M ) − h ( X, E ) h ( X, E ) + dim Ker( µ ) . Thus, if the Petri map is injective, then E is a smooth point of B kr and andthe component of B kr through E has the expected dimension. Remark 3.2.
The Petri map can also be derived from [7]. Indeed, by takingΛ = Λ ′ = ( H ( X, E ) , Id, E ) in [7, Corollary 1.6], we see that the above Petrimap is dual of the map Ext ( E, E ) → Hom ( H ( X, E ) , H ( X, E )) in thegiven exact sequence. . Brill-Noether loci over quintic hypersurface
Let X be a very general quintic hypersurface in P . Then we havePic( X ) ≃ Pic( P ) ≃ Z , K X ≃ O X (1), χ ( X, O X ) = 5. Let H be a hy-perplane class, and M ( c ) := M X,H (2; 3
H, c ) be the moduli space of ranktwo H -stable bundles on X with first Chern class 3 H , and second Chernclass c . It is known ([13]) that M ( c ) is irreducible for c ≥
14, genericallysmooth for c ≥
21, and has the expected dimension 4 c −
60. Also notethat H ( X, E ) = 0 , ∀ E ∈ M . Thus the hypothesis of the Theorem 2.2 issatisfied.Let C be a smooth, irreducible, projective curve in the complete linearsystem | H | . Then the genus of C, g := 31.
Proposition 4.1.
With the notations as above, assume that there is a basepoint free line bundle of degree ≥ on C with exactly two sections, then B ⊂ M ( c ) is non-empty.Proof. Let A be a base point free line bundle on C with h ( C, A ) = 2.Consider the elementary transformation(2) 0 → F → H ( C, A ) ⊗ O X → A → . Then F is a rank two vector bundle on X with c ( F ) = − H , c ( F ) =deg( A ), h ( X, F ) = 0 = h ( X, F ) ([8, Chapter 5, Proposition 5.2.2]). Dual-izing the above exact sequence we get(3) 0 → H ( C, A ) ∗ ⊗ O X → F ∗ → O C ( C ) ⊗ A ∗ → . Thus h ( X, F ∗ ) = 2 + h ( C, O C ( C ) ⊗ A ∗ ) ≥ Claim: F ∗ ∈ B .It is sufficient to show that F is H -stable. Let O X ( m ) destabilizes F . Then m ≥ −
1. On the other hand, from (2), we have m ≤
0. Since h ( X, F ) = 0, m = 0. Thus we are reduced to show that h ( X, F ⊗ O X (1)) = 0. Note that F ⊗ O X (1) ≃ F ∗ ⊗ O X ( − O X ( − → O X ( − ⊕ → F ∗ ⊗ O X ( − → O C ( H ) ⊗ A ∗ → . Since deg( O C ( H ) ⊗ A ∗ ) < h ( C, O C ( H ) ⊗ A ∗ ) = 0 and consequently h ( X, F ∗ ⊗ O X ( − . (cid:3) Since C ∈ | H | , C is a smooth complete intersection of two smoothhypersurfaces of P of degrees 5 and 3. Thus by [1, Page 139, C-4], C doesnot have a g . In particular, C is not hyperelliptic, trigonal or tetragonal(i.e. C does not have a g , g , g respectively). Also note that K C = O C (4).Now for D = O C (3), K C ⊗ O C ( − D ) = O C (1), and this gives an embeddingof C ֒ → P . So by [1, Page 221, B-4], C is not a bi-elliptic. Proposition 4.2.
With the notations as above, for ≤ d ≤ , there existsa base point free line bundle of degree d on C . roof. Let us denote by W rd ( C ) the Brill-Noether loci of degree d line bun-dles L on C with h ( C, L ) ≥ r + 1. Case I: d = 32.In this case, dim W ( C ) = 31. If W ( C ) − W ( C )( = ∅ ) does not con-tain any base point free line bundle, then tensoring by the ideal sheaf ofthe base locus, we obtain a family of base-point free line bundles with ex-actly two sections of dimension 31 and is contained in S e ≤ W e ( C ). Nowdim W e ( C ) = e ≤
31 = g , and W e ( C ) is a proper closed subset of W e ( C ).So dim W e ( C ) ≤ e − ≤
30. Thus dim( S e ≤ W e ( C )) <
31, a contradiction.
Case II: d = 31.In this case, dim W ( C ) ≥
29. Applying Martens’ theorem [1, ChapterIV, Theorem 5.1], we see that dim( S ≤ e ≤ W e ( C )) <
29. Note that, forany L ∈ Pic( C ) with h ( C, L ) = 2, if B is the base locus of | L | , thendeg( B ) ≤ deg( L ) −
2. Now arguing as above, we get a base point free linebundle on C of degree 31 with exactly two sections. Case III: d = 30.This follows from the existence of base point free complete g on C [9,Theorem 1.4]. Case IV: d = 29.We have dim W ( C ) ≥
25. By Mumford’s theorem [1, Chapter IV, Theo-rem 5.2], we get dim( S ≤ e ≤ W e ( C )) <
25. Now we argue as in Case I toconclude.
Case V: d = 28.Here dim W ( C ) ≥
23. Then using Keem’s theorem [1, Page 200], we getdim( S e ≤ W e ( C )) <
23. Arguing as in Case I, we conclude that there ex-ists a base point free line bundle of degree 28 on C with exactly two globalsections. (cid:3) Now we are ready to prove
Theorem 4.3.
With the notations as in the beginning of this section, B − B ⊂ M ( c ) is non-empty for ≤ c ≤ .Proof. Combining Propositions 4.1 and 4.2, we see that, for 28 ≤ c ≤ B ⊂ M ( c ) is non-empty. Moreover, for 28 ≤ d ≤
32, we can find basepoint free line bundles A on C with h ( C, A ) = 2 and h ( C, O C ( C ) ⊗ A ∗ ) = 0.Now using (3), we conclude. (cid:3) Proposition 4.4.
With the notations as in the beginning of this section, let E ∈ B − B ⊂ M ( c ) , ≤ c ≤ be a general element constructed as inProposition 4.1. Then E is a smooth point in M ( c ) .Proof. Since the morphism M X,H (2; 3
H, c ) ⊗O X ( − −−−−−−→ M X,H (2;
H, c − E ⊗ O X ( −
1) is a smoothpoint of M X,H (2;
H, c − F := E ⊗ O X ( − F is not a smoothpoint of M X,H (2;
H, c − H ( X, AdF ⊗ K X ) = 0. Thus thereis a non-zero map φ : F → F ⊗ K X . Two cases can occur: ase I: φ drops rank every where.Case II: φ does not drop rank every where.Following [12], we call the Case I as singularity of first kind, and the CaseII as singularity of second kind. If Case I occurs, we will have h ( X, F ) = 0(see [12, Proposition 5.1]), which is impossible. Also, from [13, Lemma 10.1],we see that the dimension of the singular locus of the second kind is at most8 . But the dimension of the family of globally generated line bundles A on C with 28 ≤ deg( A ) ≤ , h ( C, A ) = 2 and h ( C, O C ( C ) ⊗ A ∗ ) = 0 isgreater than 8. Hence a general E , as constructed in Proposition 4.1, willbe a smooth point in M X,H (2; 3
H, c ). (cid:3) Let E ∈ B − B be a smooth point in M ( c ) , ≤ c ≤ , as con-structed in Proposition 4.1. Existence of such a smooth point is assured byProposition 4.4. Then E fits into the following exact sequence(4) 0 → O ⊕ X → E → O C ( C ) ⊗ A ∗ → E ∗ ⊗ K X we get(5) 0 → ( E ∗ ⊗ K X ) ⊕ → E nd ( E ) ⊗ K X → O C ( C ) ⊗ A ∗ ⊗ ( E ∗ ⊗ K X ) | C → . Note that h ( X, E ∗ ⊗ K X ) = h ( X, E ) = 0, by our assumption, and since E is a smooth point in the moduli space, H ( X, E nd ( E ) ⊗ K X ) ≃ H ( X, K X ).Thus, by taking cohomology long exact sequence corresponding to the exactsequence (5), we get0 → H ( X, K X ) → H ( C, O C ( C ) ⊗ A ∗ ⊗ ( E ∗ ⊗ K X ) | C ) → H ( X, ( E ∗ ⊗ K X ) ⊕ ) η −→ H ( X, E nd ( E ) ⊗ K X ) → . . . (6)Note that the Petri map µ in (1) is same as the map η above. Now we willshow that the map η is injective, and this will in turn imply that the bundle E is a smooth point in B .Dualizing the exact sequence (4) and restricting to the curve C , we obtainthe exact sequence (on C )0 → O C ( − C ) ⊗ A → E ∗ | C → O ⊕ C → A → C )(7) 0 → O C ( − C ) ⊗ A → E ∗ | C → A ∗ → . Tensoring the sequence (7) by O C ( C ) ⊗ A ∗ ⊗ K X | C and taking cohomologylong exact sequence we get0 → H ( C, K X | C ) → H ( C, O C ( C ) ⊗ A ∗ ⊗ ( E ∗ ⊗ K X ) | C ) → H ( C, O C ( C ) ⊗ A ∗⊗ ⊗ K X | C ) → . . . By our assumption on A , H ( C, O C ( C ) ⊗ A ∗ ) = 0. Since degree( A ∗ ⊗ K X | C ) <
0, we have H ( C, O C ( C ) ⊗ A ∗⊗ ⊗ K X | C ) = 0. Thus from the In [12, Corollary 5.1], the authors have estimated the bound for singularity of secondkind as ≤
13. But in [13, Lemma 10.1], the authors have improved the above bound andshown that it is ≤ bove exact sequence H ( C, K X | C ) ≃ H ( C, O C ( C ) ⊗ A ∗ ⊗ ( E ∗ ⊗ K X ) | C ) . It is easy to see that H ( X, K X ) ≃ H ( C, K X | C ). Consequently, the map η in (6) is injective.We summarize the above discussion as Theorem 4.5. B ⊂ M ( c ) , ≤ c ≤ , contains a smooth point, andhence the irreducible component containing it has the expected dimension. Non-emptiness of B Let X and M ( c ) be as in the previous section. Let Hilb c ( X ) lci denotesthe open subscheme of the Hilbert scheme Hilb c ( X ) consisting of length c subschemes of X which are locally complete intersections. Given any point Z ∈ Hilb c ( X ) lci , we have an exact sequence0 → O X → E → I Z ⊗ O X (3) → I Z is the ideal sheaf of Z and E is a rank two torsion-free sheaf on X . The space of such (isomorphic classes of) extensions is parametrized by P Ext ( I Z ⊗ O X (3) , O X ). By duality, Ext ( I Z ⊗ O X (3) , O X ) = H ( X, I Z ⊗O X (4)) ∗ . Since h ( X, O X (4)) = 35, for a general element of length c ≥ c ( X ) lci , we have h ( X, I Z ⊗ O X (4)) = 0. Thus a general elementin Hilb c ( X ) lci , c ≥
36, satisfies the Cayley-Bacharach property for O X (4),and consequently, we get that the corresponding extension E is locally free.Also any such vector bundle E is H -stable. Thus for c ≥ , B ⊂ M ( c )is non-empty.The following two results give a bound for dim B . Lemma 5.1. ( [14, Proposition 1.1] ) With the notations as above, for c ≥ , dim B ≤ c − . Lemma 5.2. ( [12, Corollary 3.1] ) With the notations as above, every irre-ducible component of B has dimension ≥ c − h ( X, O X (3) ⊗ K X ) − . Since h ( X, O X (3) ⊗ K X ) = 35 , we see that, for c ≥
36, every irreduciblecomponent of B has dimension exactly 3 c −
36. On the other hand, theBrill-Noether number ρ = 3 c −
36. We summarize the above discussion as
Proposition 5.3.
With the notations as above, B ⊂ M ( c ) is non-emptyfor c ≥ , and every irreducible component of B has the expected dimen-sion.Acknowledgement: We would like to thank Prof. P. Newstead for his val-ueable comments and pointing out the gap in the earlier version. We alsowould like to thank D.S. Nagaraj, V. Balaji, P. Sastry for their encourage-ment and helpful discussion. First named author would like to thank IISERTrivandrum for their hospitality during the stay where this work started. In [14], Proposition 1 . c ( E ) = H , and thebound author got there is dim B ≤ c −
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Chennai Mathematical Institute, H1 Sipcot IT Park, Siruseri, Kelambakkam- 603103, INDIA.
E-mail address : [email protected] IISER - Thiruvananthapuram, Computer Science Building, College of En-gineering Trivandrum Campus, Trivandrum - 695016, Kerala, India
E-mail address : [email protected]@iisertvm.ac.in