Ulrich bundles on intersections of two 4-dimensional quadrics
aa r X i v : . [ m a t h . AG ] A p r ULRICH BUNDLES ONINTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS
YONGHWA CHO, YEONGRAK KIM, AND KYOUNG-SEOG LEE
Abstract.
In this paper, we investigate the existence of Ulrich bundles on a smooth completeintersection of two 4-dimensional quadrics in P by two completely different methods. First, wefind good ACM curves and use Serre correspondence in order to construct Ulrich bundles, which isanalogous to the construction on a cubic threefold by Casanellas-Hartshorne-Geiss-Schreyer. Next,we use Bondal-Orlov’s semiorthogonal decomposition of the derived category of coherent sheavesto analyze Ulrich bundles. Using these methods, we prove that any smooth intersection of two 4-dimensional quadrics in P carries an Ulrich bundle of rank r for every r ≥
2. Moreover, we providea description of the moduli space of stable Ulrich bundles. Introduction
Let P n be the n -dimensional projective space over the field of complex numbers C . A famoustheorem by Horrocks states that a vector bundle E on P n splits as the direct sum of line bundles ifand only if E has no intermediate cohomology, i.e. , h i ( E ( j )) = 0 for all 0 < i < n and j ∈ Z . It isnatural to ask the algebro-geometric meaning of these vanishing conditions for the other varieties. Let X ⊂ P N be an n -dimensional smooth projective variety with a fixed polarization O X (1) = O P N (1) | X .We call that a vector bundle E on X is ACM (arithmetically Cohen-Macaulay) if E has no intermediatecohomology with respect to the given polarization O X (1). Roughly speaking, the presence of nontrivialACM bundles measures how X is apart from the projective space P n . Due to their interestingproperties, ACM bundles have played a significant role in the study of vector bundles.In commutative algebra, ACM bundles correspond to MCM (maximal Cohen-Macaulay) moduleswhich are Cohen-Macaulay modules achieving the maximal dimension. A particularly interesting casehappens when the minimal free resolution of an MCM module becomes completely linear. Such anMCM module has the maximal possible number of minimal generators which are concentrated on asingle degree [37]. Eisenbud and Schreyer made a comprehensive study on the geometric analogueof these linear MCM modules, and named them Ulrich sheaves [14]. Thanks to foundational worksby Beauville [5] and Eisenbud-Schreyer [14], Ulrich sheaves provide a number of fruitful applications;for example, linear determinantal representations of hypersurfaces, matrix factorizations by linearmatrices, the cone of cohomology tables, and Cayley-Chow forms. Eisenbud and Schreyer conjecturedthat every projective variety carries an Ulrich sheaf [14], and verified it for a few simple cases. Theconjecture is still wildly open even for smooth surfaces. In very recent years, there were severalprogresses on the conjecture for surfaces; for instance, general K3 surfaces [1], abelian surfaces [6],and nonspecial surfaces of p g = q = 0 [13]. Much less is known for ACM and Ulrich bundles on threefolds. On a smooth quadric Q ⊂ P ,there is only one nontrivial indecomposable ACM bundle, namely, the spinor bundle [10]. Arrondoand Costa studied ACM bundles on Fano 3-folds of index 2 of degree d = 3 , , P [26]. He also classified all the possibleChern classes of rank 2 ACM bundles on prime Fano 3-folds and complete intersection Calabi-Yau3-folds [27]. Their results expected the existence of Ulrich bundles on 3-folds of small degree, however,constructions were not complete except for a very few cases. On the other hand, Beauville showedthat a general hypersurface of degree ≤ P is linearly Pfaffian. In other words, such a hypersurfacecarries a rank 2 Ulrich bundle [5]. He also checked that every Fano 3-fold of index 2 carries a rank2 Ulrich bundle [7]. In particular, a general smooth cubic 3-fold carries Ulrich bundles of rank r forevery r ≥
2, proved first by Casanellas, Hartshorne, Geiss, and Schreyer [11]. Recently, Lahoz, Macr`ı,and Stellari extended this result to every smooth cubic 3-fold using the derived category of coherentsheaves and also described the moduli space of stable Ulrich bundles [24].It is quite natural to ask for the next case, a del Pezzo threefold X = Q ∩ Q ∞ of degree fourwhich is the complete intersection of two quadric 4-folds. Indeed, X is very attractive since thereare several ways to understand vector bundles on X . Since X is a 3-fold, we may construct vectorbundles on X by observing curves lying on X via Serre correspondence. On the other hand, it is alsowell-known that the geometry of the intersection of 2 even dimensional quadrics is closely related toa hyperelliptic curve. Bondal and Orlov showed that the derived category of coherent sheaves on theintersection of 2 even dimensional quadrics has a semiorthogonal decomposition whose components arethe derived category of the hyperelliptic curve associated to the 2 given quadrics and the exceptionalcollection [8]. Recently, there were several attempts to understand vector bundles on a variety usingthe semiorthogonal decomposition of its derived category. For instance, Kuznetsov studied instantonbundles on some index 2 Fano 3-folds via semiorthogonal decompositions [23]. Lahoz, Macr`ı, andStellari studied ACM bundles on cubic 3-folds and 4-folds via semiorthogonal decomposition [24, 25].Therefore, it is reasonable to apply the semiorthogonal decomposition to understand vector bundleson the intersection of two even dimensional quadrics.Being motivated by earlier works mentioned above, we investigate the existence and the modulispace of Ulrich bundles on the intersection of two 4-dimensional quadrics by two completely differentmethods: classical Serre corrseponence and Bondal-Orlov theorem. The main result is the followingtheorem: Theorem 1.1 (see Theorem 3.8 and 4.14) . The moduli space of stable Ulrich bundles of rank r ≥ on X = Q ∩ Q ∞ is isomorphic to a nonempty open subscheme of U s C ( r, r ) , where U s C ( r, r ) is themoduli space of stable vector bundles of rank r and degree r on a curve C of genus . Our approach using Serre correspondence closely follows the works of Arrondo and Costa [4] andof Casanellas, Hartshorne, Geiss, and Schreyer [11], and our approach using derived categories isstrongly influenced by the works of Kuznetsov [23] and of Lahoz, Macr`ı, and Stellari [24, 25]. Thestructure of this paper is as follows. In Section 2, we recall a few useful facts related to ACM andUlrich bundles. In Section 3, we construct Ulrich bundles of any rank r ≥ X = Q ∩ Q ∞ using Serre correspondence and Macaulay2 . In Section 4, we
LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 3 prove the existence of Ulrich bundles of any rank r ≥ X = Q ∩ Q ∞ using Bondal-Orlov theorem. We also analyze the moduli of stableUlrich bundles of rank r on X and provide a description in terms of vector bundles on C .2. Preliminaries on ACM and Ulrich bundles
In this section, we briefly review the definition of ACM and Ulrich bundles and their basic prop-erties.
Definition 2.1.
Let X ⊂ P N be an n -dimensional smooth projective variety embedded by a veryample line bundle O X (1).(1) A coherent sheaf E on X is ACM if H i ( E ( j )) = 0 for all 0 < i < n and j ∈ Z .(2) An ACM sheaf E on X is Ulrich if H ( E ( − h ( E ) = deg( X ) rank( E ). Remark 2.2.
Since the underlying space X is smooth, E being ACM implies that E is locally free.Hence it is natural to call ACM (Ulrich) bundles for the objects occurring in the above definition.We recall the following proposition by Eisenbud and Schreyer. We refer to [7, 14] for more details. Proposition 2.3 ([7, Theorem 1], [14, Proposition 2.1]) . Let X ⊂ P N and E as above. The followingare equivalent: (1) E is Ulrich; (2) H i ( E ( − i )) = 0 for all i > and H j ( E ( − j − for j < n . (3) H i ( E ( − j )) = 0 for all i and ≤ j ≤ n . (4) For some (all) finite linear projections π : X → P n , the sheaf π ∗ E is isomorphic to the trivialsheaf O ⊕ t P n for some t . (5) The section module M := ⊕ j H ( E ( j )) is a linear MCM module, that is, the minimal S = C [ x , . . . , x N ] -free resolution of M F : 0 → F N − n → · · · → F → F → M → is linear in the sense that F i is generated in degree i for every i . In particular, by Serre duality, we immediately have the following proposition as a consequence:
Proposition 2.4.
Let X n ⊂ P N be as above, and let H := O X (1) be a very ample line bundle. (1) If E is an ACM bundle on X , then E ∗ ( K X ) is also an ACM bundle. (2) When X is subcanonical, that is, K X = O X ( k ) for some k ∈ Z , E is ACM if and only if E ∗ isACM. (3) If E is an Ulrich bundle on X , then E ∗ ( K X + ( n + 1) H ) is an Ulrich bundle. The following proposition about the stability is very useful in later sections.
Proposition 2.5 ([11, Theorem 2.9]) . Let X be a smooth projective variety, and let E be an Ulrichbundle on X . Then (1) E is semistable and µ -semistable. YONGHWA CHO, YEONGRAK KIM, AND KYOUNG-SEOG LEE (2) If → E ′ → E → E ′′ → is an exact sequence of coherent sheaves with E ′′ torsion-free, and µ ( E ′ ) = µ ( E ) , then both E ′ and E ′′ are Ulrich. (3) If E is stable, then it is also µ -stable. Geometric approach via Serre correspondence
In this section, we show the existence of Ulrich bundles using Serre correspondence.3.1.
Serre correspondence.
We briefly recall Serre correspondence which enables us to constructa vector bundle as an extension from a codimension 2 subscheme. To obtain a vector bundle, sucha subscheme has to satisfy certain generating conditions. For instance, it is well-known that a 0-dimensional subscheme on a smooth surface should satisfy Cayley-Bacharach condition to provide alocally free extension. For higher dimensional cases, the situation gets much more complicated. Forexample, a curve in P occurs as the zero locus of a rank 2 vector bundle on P if and only if it isa local complete intersection and subcanonical [18]. It is clear that not all curves come from vectorbundles. When it happens, we cannot construct a vector bundle as an extension. However, still inmany cases, it is a powerful tool providing constructions of vector bundles. We refer to [3] for theproof and more details. Theorem 3.1 (Serre correspondence) . Let X be a smooth variety and let Y ⊂ X be a local completeintersection subscheme of codimension in X . Let N be the normal bundle of Y in X and let L bea line bundle on X such that H ( L ∗ ) = 0 . Assume that ( ∧ N ⊗ L ∗ ) | Y has ( r − generating globalsections s , . . . , s r − . Then there is a rank r vector bundle E as an extension → O r − X ( α ,...,α r − ) −−−−−−−−→ E −→ I Y/X ( L ) → such that the dependency locus of ( r − global sections α , . . . , α r − of E is Y with P r − i =1 s i α i | Y = 0 .Moreover, if H ( L ∗ ) = 0 , such an E is unique up to isomorphism. ACM bundles of rank 2 via Serre correspondence.
From now on, let Q , Q ∞ be twosmooth quadric hypersurfaces in P meeting transversally and let X = Q ∩ Q ∞ ⊂ P be a smoothFano 3-fold of degree 4 and index 2, i.e. , ω X = O X ( − H X ], [ L X ], [ P X ] be the class of a hyperplane section, a line, and a point in X respectively.Then, H ( X, Z ) ≃ Z · [ H X ] , H ( X, Z ) ≃ Z · [ L X ] , and H ( X, Z ) ≃ Z · [ P X ] . (3.1)The ring structure is given as follows: H X = 4 L X , H X · L X = P X . For a vector bundle F on X , wedefine its slope µ with respect to H by µ H ( F ) := deg H F rank F By virtue of (3.1), we fix our convention as follows.
Notation 3.2.
Via the isomorphisms Z · [ H X ] ≃ Z , Z · [ L X ] ≃ Z , and Z · [ P X ] ≃ Z , we may regard c i ( F ) as an integer, by omitting the cyclic generators of H i ( X, Z ). Under this convention, one caneasily see that µ H ( F ) = c ( F ) deg X rank F = 4 · c ( F )rank F LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 5
We also omit the redundant coefficient 4 in the formula and redefine the slope of F as follows: µ ( F ) := c ( F )rank F . The following proposition is useful in later sections.
Proposition 3.3 ([21, Proposition 1.2.7]) . Let E and E ′ be µ -stable bundles with µ ( E ) > µ ( E ′ ) . Then Hom( E , E ′ ) = 0 . Applying Proposition 2.4 to X = Q ∩ Q ∞ , we get the following: Proposition 3.4.
Let E be an Ulrich bundle of rank r on X = Q ∩ Q ∞ . Then, (1) µ ( E ) = 1 , and (2) E ∗ (2) is an Ulrich bundle. In [4], Arrondo and Costa made a comprehensive study of ACM bundles on X extending [36].They classified the possible Chern classes for ACM bundles under a mild assumption. In particular,they classified all the rank 2 ACM bundles on X . Theorem 3.5 ([4, Theorem 3.4]) . An indecomposable rank ACM vector bundle on X is a twist ofone of the following; (1) A line type: a semistable vector bundle E l fitting in an exact sequnce → O X → E l → I l → where l ⊂ X is a line contained in X ; (2) A conic type: a stable vector bundle E λ fitting in an exact sequence → O X → E λ → I λ (1) → where λ ⊂ X is a conic contained in X ; (3) An elliptic curve type: a stable vector bundle E e fitting in an exact sequence → O X → E e → I e (2) → where e ⊂ X is an elliptic curve of degree . It is classically well-known that the Fano scheme F ( X ) of lines l ⊂ X is isomorphic to the Jacobian J ( C ) of the hyperelliptic curve C of genus 2 associated to X (see [28, Theorem 5], [29, Theorem 2] or[34]). Since E l has the unique global section up to constants, the space also coincides with the spaceof line type ACM bundles.Conic type ACM bundles are also well understood as in the following way. Given a conic λ ⊂ X ,note that there is only one quadric Q ∈ d := | Q + tQ ∞ | t ∈ P in a pencil containing the plane Λ = h λ i .It is clear that Λ ∩ X = λ . Since Q is a 4-dimensional quadric, there is a spinor bundle whose globalsections sweep out a family of planes in Q containing Λ. The bundle E λ is the restriction of this spinorbundle. Hence, the moduli of conic type ACM bundle can be naturally identified with the space ofspinor bundles associated to the pencil d .The last case is particularly interesting. When e ⊂ X ⊂ P is an elliptic normal curve of degree6, we have h ( I e (1)) = 0 and h ( I e (2)) = h ( I e/ P (2)) − h ( I X/ P (2)) = 9 − E e is an YONGHWA CHO, YEONGRAK KIM, AND KYOUNG-SEOG LEE initialized ACM bundle with h ( E e ) = 8 = (deg X ) · (rank E e ), in other words, it is an Ulrich bundleof rank 2. We refer to [7] for an explicit construction of such curves. Proposition 3.6 ([7, Proposition 8]) . There exists an Ulrich bundle of rank on X . Ulrich bundles of higher ranks via Serre correspondence and
Macaulay2 . Similar asthe case of cubic 3-folds in P , the existence of rank 3 Ulrich bundles on X was expected earlier in [4,Example 4.4]. However, as Casanellas and Hartshorne pointed out [11, Remark 5.5], the constructionwas incorrect not only for cubic 3-folds but also for our X . Arrondo and Costa constructed anarithmetically Cohen-Macaulay curve D of degree 15 and genus 12 using a Gorenstein liaison, however,2 sections of H ( ω D ( − H ∗ ( ω D ). Indeed, in loc. cit., the authorsstarted with a twisted cubic curve D ′ , and then found an arithmetically Gorenstein curve B ′ of degree18 containing D ′ where the residual curve is D . Hence we have a short exact sequence0 → I B ′ → I D ′ → ω D ( − → . Since B ′ is arithmetically Gorenstein, we have a short exact sequence of graded S = H ∗ ( O P )-modules0 → H ∗ ( I B ′ ) → H ∗ ( I D ′ ) → H ∗ ( ω D ( − → . (3.2)Note that B ′ is the zero locus of a section of E e (1), so I ′ B fits into the short exact sequence 0 → O X →E e (1) → I B ′ (4) →
0. Hence, the first 2 nonzero terms in the sequence (3.2) are H ( I D ′ (1)) ≃ H ( ω D ( − H ( I D ′ (2)) ≃ H ( ω D ) . Via the exact sequence 0 → I X/ P → I D ′ / P → I D ′ →
0, we may lift the sections in H ( I D ′ ( j )) asthe homogeneous form of degree j in S . It is clear that a twisted cubic curve D ′ ⊂ P is generated by2 linear forms and 3 quadratic forms in S , and hence H ( ω D ( − l and l , namely.However, sections in the image of H ( ω D ( − ⊗ H ( O P (1)) → H ( ω D ) can only span 11 quadrics,since two sections of ω D ( −
1) admit a linear Koszul relation l l − l l = 0 in S . Hence we concludethat H ( ω D ( − ⊗ H ( O P (1)) → H ( ω D ) cannot be surjective.We need to construct a curve satisfying the generating condition to construct a rank 3 Ulrichbundle E on X . If it exists, then two independent global sections of E will degenerate along a curve D of degree 15 since E is globally generated always. It is easy to see that the numerical conditionssuggested in [4, Example 4.4] are valid. Hence, we need to construct an ACM curve D ⊂ X of giveninvariants such that H ( ω D ( − H ( ω D ( − ⊗ H ( O P ( j )) → H ( ω D ( j − j ≥ ω ∗ D (1 + j ) is nonspecial for j ≥
2, Castelnuovo pencil trick implies that the map is automat-ically surjective for j ≥
2. Hence it is sufficient to check only for the j = 1 case. The constructionfollows from Macaulay2 [17] computations, which is analogous to [11, Appendix] or [16]. Although
LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 7 the proof goes into the same strategy, in particular, the
Macaulay2 scripts are almost same, it isworthwhile to write down since the difference between the cubic 3-fold case is not that much straight-forward.
Proposition 3.7 (See also [11, Theorem A.3]) . The space of pairs D ⊂ X ⊂ P of smooth ACMcurves of degree and genus on a complete intersection of quadrics X has a component whichdominates the Hilbert scheme of intersections of quadrics in P . Moreover, the module H ∗ ( ω D ) isgenerated by its sections in degree − as S P = H ∗ ( O P ) -modules for a general pair D ⊂ X . Inparticular, a general intersection of two quadrics in P carries a desired curve we discussed above.Proof. We prove by constructing a family of such curves as in the following strategy. First, we take afamily of smooth curves of genus 12 in P × P . Next, we observe that a general (precisely, a randomlychosen) curve in this family admits an embedding to P in a natural way. Finally, we check that sucha curve in P satisfies the desired properties. Then the whole statement will follow by the deformationtheory and the semicontinuity.Let D be a smooth projective curve of genus 12 together with line bundles L and L with | L | a g and | L | a g . Let D ′ be the image of the map D | L | , | L | −−−−−→ P × P . Suppose that the maps H ( P × P , O ( n, m )) → H ( D, L n ⊗ L m ) are of maximal rank for all n, m ≥ D is isomorphic to its image D ′ and we may compute the Hilbert series ofthe truncated ideal I trunc = M n,m ≥ H ( I D ′ ( n, m ))in the Cox ring S P × P = k [ x , x ; y , y , y ] of P × P , namely, H I trunc ( s, t ) = 5 s t − s t − s t + 3 s t + 10 s t (1 − s ) (1 − t ) . Hence, by reading off the Hilbert series, we may expect that I trunc admits a bigraded free resolutionof type 0 → F → F → F → I trunc → F = S P × P ( − , − ⊕ S P × P ( − , − , F = S P × P ( − , − ⊕ S P × P ( − , − ,and F = S P × P ( − , − .We will construct a curve D ′ ⊂ P × P in a converse direction. First, we take a free resolution ofthe above form, and then observe that the module represented by such a resolution is indeed an idealof a curve D ′ . Let M : F → F be a general map chosen randomly, and let K be the cokernel of thedual map M ∗ : F ∗ → F ∗ . The first terms of a minimal free resolution of K are: · · · → G N ′ → F ∗ M ∗ → F ∗ → K → G be the module generated by syzygies of M ∗ . Composing N ′ with a general map F ∗ → G and dualizing again, we get a map N : F → F . The following script shows that the kernel of N ∗ is S P × P so that the entries of the matrix S P × P → F ∗ generate an ideal. YONGHWA CHO, YEONGRAK KIM, AND KYOUNG-SEOG LEE i1 : setRandomSeed "RandomCurves";p=997;Fp=ZZ/p;S=Fp[x_0,x_1,y_0..y_2, Degrees=>{2:{1,0},3:{0,1}}]; -- Cox ringm=ideal basis({1,1},S); -- irrelevant ideali2 : randomCurveGenus12Withg17=(S)->(M:=random(S^{6:{-3,-5},11:{-4,-4}},S^{5:{-4,-5}}); -- random map MN’:=syz transpose M; -- syzygy matrix of the dual of MN:=transpose(N’*random(source N’,S^{3:{4,3},10:{3,4}}));ideal syz transpose N) -- the vanishing ideal of the curvei3 : ID’=saturate(randomCurveGenus12Withg17(S),m); -- ideal of D’
Since the maximal rank assumption is an open condition, the above example provides that thereis a component
H ⊂
Hilb (7 , , ( P × P ) in the Hilbert scheme of curves of bidegree (7 ,
10) andgenus 12 defined by free resolutions of the above form. Also note that D ′ ∈ H admits both g and g induced by the natural projections.We want to verify that a general D ∈ H ′ equipped with two natural projections acts like a generalcurve D ∈ M , L , and L in order to show that H ′ dominates M . Recall from Brill-Noethertheory that for a general curve D of genus g , the Brill-Noether locus W rd ( D ) = { L ∈ Pic( D ) | deg( L ) = d, h ( L ) ≥ r + 1 } is nonempty and smooth away from W r +1 d ( D ) of dimension ρ if and only if ρ = ρ ( g, r, d ) = g − ( r + 1)( g − d + r ) ≥ . Also note that the tangent space at L ∈ W rd ( D ) \ W r +1 d ( D ) is the dual of the cokernel of Petri map H ( D, L ) ⊗ H ( D, ω D ⊗ L − ) → H ( D, ω D ) . We expect that both L and L are smooth isolated points of dimension ρ = ρ = 0, equivalently,both Petri maps are injective. We refer to [2, Chapter IV] for details on Brill-Noether theory.Now let η : D → D ′ be a normalization of a given point D ′ ∈ H , since we do not know that D ′ is smooth yet. We check that L i are smooth points in the associated Brill-Noether loci as follows,where L i is a line bundle on D obtained by pulling back natural g and g on D ′ for i = 1 , L ; we take the plane model Γ ⊂ P of D ′ . i4 : Sel=Fp[x_0,x_1,y_0..y_2,MonomialOrder=>Eliminate 2];R=Fp[y_0..y_2]; -- coordinate ringIGammaD=sub(ideal selectInSubring(1,gens gb sub(ID’,Sel)),R);-- ideal of the plane model We observe that Γ is a curve of desired degree and genus, and its singular locus ∆ consists only ofordinary double points as follows.
LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 9 i5 : distinctPoints=(J)->(singJ:=minors(2,jacobian J)+J;codim singJ==3)i6 : IDelta=ideal jacobian IGammaD + IGammaD; -- singular locusdistinctPoints(IDelta)o6 = truei7 : delta=degree IDelta;d=degree IGammaD;g=binomial(d-1,2)-delta;(d,g,delta)==(10,12,24)o7 = true
We can also compute the minimal free resolution of I ∆ : i8 : IDelta=saturate IDelta;betti res IDelta0 1 2o8 = total: 1 4 30: 1 . .1: . . .2: . . .3: . . .4: . . .5: . 4 .6: . . 3 Thanks to the above Betti table, we immediately check that Γ is irreducible since ∆ is not a completeintersection (4,6). Indeed, there is no way to write a degree 10 curve Γ ⊂ P with 24 nodes as a unionof 2 curves. In particular, the normalization of Γ is isomorphic to a smooth irreducible curve of genus g = 12, and thus D ′ is smooth since 12 = g ≤ p a ( D ′ ) ≤
12. Hence from now on, we do not distinguish D and D ′ since they coincide.By Riemann-Roch, we have h ( D, L ) = 3 since h ( D, L ) = h ( D, ω D ⊗ L − ) = h ( P , I ∆ (6)) = 4by the adjunction formula applied to D ⊂ Bl ∆ P . Hence | L | is complete and the Petri map for L is identified with the muiltiplication H ( P , O P (1)) ⊗ H ( P , I ∆ (6)) → H ( P , I ∆ (7)) . Note that the map is injective since there is no linear relation among the 4 sextic generators of I ∆ .In fact, the Petri map becomes an isomorphism, and L ∈ W ( D ) is a smooth isolated point ofdimension ρ = 0. To check that L is Petri generic, we first compute the embedding D → P H ( ω D ⊗ L − ) = P andits minimal free resolution by choosing sections of H ( ω D ) ≃ H ( P , I ∆ (7)) which vanish on a fiberof D → P induced by | L | : i9 : LK=(mingens IDelta)*random(source mingens IDelta, R^{12:{-7}});-- compute a basisPt=random(Fp^1,Fp^2); -- a random point in a lineL1=substitute(ID’,Pt|vars R); -- fiber over the pointKD=LK*(syz(LK % gens L1))_{0..5};-- compute a basis for elements in LK vanish in L1T=Fp[z_0..z_5]; -- coordinate ringphiKD=map(R,T,KD); -- embeddingID=preimage_phiKD(IGammaD);degree ID==15 and genus ID==12o9 = truei10 : betti(FD=res ID)0 1 2 3 4o10 = total: 1 12 25 16 20: 1 . . . .1: . 2 . . .2: . 10 25 16 .3: . . . . 2 We observe that the curve D ⊂ P verifies the desired properties. Since the length of the minimalfree resolution of I D equals to the codimension, D ⊂ P becomes ACM. Note that the dual complexHom • S P ( F D , S P ( − ⊕ n ∈ Z H ( ω D ( n )) where F D is the minimal free resolutionof D . The Betti table also tells us that this module is generated by its 2 global sections in degree − h ( L ) = h ( ω D ( − | L | is also complete and the Petri map for L is identifiedwith H ( D, ω D ( − ⊗ H ( P , O P (1)) → H ( D, ω D ) . This map is also injective since there is no linear relation between the 2 generators in H ( ω D ( − L ∈ W ( D ) is a smooth isolated point ofdimension ρ = 0. As consequences, H dominates Z = W × M W and M thanks to Brill-Noether theory.It remains to check the existence of a dominating family of desired curves in P over the space ofintersections of two quadrics in P . Since a random curve D ∈ H provides an embedding D ⊂ P given by a Petri generic line bundle O D (1) := ω D ⊗ L − , the above construction provides a nonemptycomponent H ′ ⊂ Hilb t +1 − ( P ) together with a dominant rational map H ′ //Aut ( P ) → M .Note that choosing an intersection of 2 quadrics X ⊂ P containing D is equivalent to choosing a LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 11 H ( P , I D/ P (2)). Consider the incidence variety V = { ( D, X ) | D ∈ H ′ ACM and X ∈ Gr(2 , H ( P , I D/ P (2))) smooth } . Since the graded Betti numbers are upper semicontinuous in a flat family having the same Hilbertfunction, we observe that V is birational to H ′ since H ( P , I D/ P (2)) is spanned by 2 quadrics for arandomly chosen D .We compute the normal sheaf N D/X for a random pair (
D, X ) ∈ V as follows: i11 : IX=ideal((mingens ID)*random(source mingens ID,T^{2:-2}));ID2=saturate(ID^2+IX);cNDX=image gens ID / image gens ID2; -- conormal sheafNDX=sheaf Hom(cNDX,T^1/ID); -- normal sheafHH^0 NDX(-1)==0 and HH^1 NDX(-1)==0o11 = truei12 : HH^0 NDX==Fp^30 and HH^1 NDX==0o12 = true In particular, the Hilbert scheme of X is smooth of dimension 30 at [ D ⊂ X ], and h i ( N D/X ( − i = 0 ,
1. We do a similar computation for N D/ P : i13 : cNDP=prune(image (gens ID)/ image gens saturate(ID^2));NDP=sheaf Hom(cNDP,T^1/ID);HH^0 NDP==Fp^68 and HH^1 NDP==0o13 = true Hence H ′ ⊂ Hilb t +1 − is smooth of expected dimension 68 at a general smooth point [ D ⊂ P ] ∈ H ′ .Consider the natural projections V π (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ π ' ' PPPPPPPPPPPPP H ′ Gr (2 , H ( P , O P (2))) . We observe that V is irreducible of dimension 68 since the fiber of π over D is exactly a singlepoint. Also note that the map π is smooth of dimension h ( D, N D/X ) = 30 at (
D, X ). Sincedim Gr(2 , H ( P , O P (2))) = 38, we conclude that π is dominant. In particular, a general X ∈ Gr (2 , H ( O P (2))) contains a curve D ∈ H ′ . By the semicontinuity, we conclude that a general( D, X ) also satisfies the desired properties. (cid:3)
Existence of such a curve D on X provides a construction of a rank 3 Ulrich bundle on X viaSerre correspondence. The idea by Casanellas and Hartshorne also makes sense in our case, andconsequently, we have the following theorem: Theorem 3.8 (See also [11, Proposition 5.4 and Theorem 5.7]) . Let X ⊂ P be the intersection of general quadrics in P . Then X carries an ( r + 1) -dimensional family of stable Ulrich bundles ofrank for every r ≥ . Proof.
Since the strategy is almost same as in [11], we only provide a shorter proof here. Note firstthat there is a rank 2 Ulrich bundle on any smooth complete intersection X [4, 7], namely, an ellipticcurve type ACM bundle. Since there is no Ulrich line bundle, any rank 2 Ulrich bundle must bestable by Proposition 2.5. Because of the same reason, if there is a rank 3 Ulrich bundle, then it isalso stable.Proposition 3.7 implies that a general X contains a smooth ACM curve D of degree 15 and genus12 such that ω D ( −
1) has two sections which generate the graded module H ∗ ( ω D ) as S P -modules.By Serre correspondence, those two generators define a rank 3 vector bundle E as an extension0 → O X → E → I D (3) → . Since D is ACM, we immediately check that H ( E ( j )) = 0 for every j ∈ Z . Furthermore, we alsohave H ( E ∗ ( j )) = H ( E ( − j − j ∈ Z from the dual sequence0 → O X ( − → E ∗ → O X → ω D ( − → . Hence E is an ACM bundle. Applying Riemann-Roch on D , we have h ( O D (2)) = 19 = h ( O X (2))and thus h ( E ( − h ( I D (2)) = 0. Similarly, we have h ( I D (3)) = h ( O X (3)) − h ( O D (3)) = 10,and thus h ( E ) = 12 = (deg X ) · (rank E ). Indeed, E is a rank 3 Ulrich bundle. As consequences, weshow the existence of Ulrich bundles on X of every rank r ≥ E of rank r for every r ≥
2. By Riemann-Roch,we have χ ( E ⊗ E ∗ ) = − r . Since the computations in [11, Proposition 5.6] also holds for our X ,we have h ( E ⊗ E ∗ ) = h ( E ⊗ E ∗ ) = 0. Since E is simple, we conclude that h ( E ⊗ E ∗ ) = 1 and h ( E ⊗ E ∗ ) = r + 1 as desired. Hence the moduli space of stable Ulrich bundles is smooth of expecteddimension if it is nonempty.It only remains to show the existence of stable Ulrich bundles of rank bigger than 3. Let r ≥ E ′ and E ′′
6≃ E ′ be stable Ulrich bundles of rank 2 and r −
2, respectively. By Riemann-Rochand [11, Proposition 5.6], we have h ( E ′ ⊗ E ′′∗ ) = − χ ( E ′ ⊗ E ′′∗ ) = 2 r − >
0. Hence the space P Ext X ( E ′′ , E ′ ) is nonempty and each element gives a nonsplit extension0 → E ′ → E → E ′′ → E is a simple and strictly semistable Ulrich bundle of rank r . Such extensions form a family ofdimension dim {E ′ } + dim {E ′′ } + dim P Ext X ( E ′′ , E ′ ) = r − r + 5 < r + 1 . Since all the other extensions by different ranks form smaller families, we conclude that a generalUlrich bundle of rank r is stable. This completes the proof. (cid:3) Remark 3.9.
We finish this section by a few remarks.(1) In fact, the proof of Proposition 3.7 implies much stronger results. For instance, one can checkthat H is a unirational family which dominates the moduli space M of smooth curves of genus12 as in [11, Appendix].(2) Because we made a computer-based computation over a finite field, we cannot remove theassumption X being general. It is also mysterious that “how general” X should be. LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 13 (3) As we mentioned, the above approach closely follows [11]. In loc. cit., the authors also checkedthat any smooth cubic 3-fold contains an elliptic normal curve of degree 5. Similarly, any smoothcomplete intersection of two quadrics in P contains an elliptic normal curve of degree 6, as in[7, Proposition 8]. It is an interesting task to construct smooth ACM curves of degree 15 andgenus 12 on any smooth complete intersection of two 4-dimensional quadrics.4. Derived categorical approaches
The notion of semiorthogonal decomposition enables us to reduce problems about Ulrich bundleson X to problems about vector bundles on the associated curve C . Let us recall some necessary factsabout the moduli space of vector bundles on curves and the derived category of coherent sheaves on X .4.1. Stable vector bundles on curves.
Let C be a smooth projective curve of genus g , U C ( r, d )be the moduli space of S-equivalence classes of rank r semistable vector bundles of degree d on C ,and SU C ( r, L ) be the moduli space of S-equivalence classes of rank r semistable vector bundles ofdeterminant L on C . We use the superscript (–) s to describe the sub-moduli space parametrizingstable objects. It is well-known that U C ( r, d ) and SU C ( r, L ) are normal projective varieties (see[28, 35]).The lemma below is one of the well-known results for (semi-)stable bundles on curves. Lemma 4.1.
Let F be a stable vector bundle of rank ≥ on C . Then, (1) µ ( F ) ≥ g − implies h ( F ) = 0 , and (2) µ ( F ) ≥ g − implies that F is globally generated.If the inequalities on µ are strict, then the same results are valid for F semistable.Proof. Assume that h ( F ) = 0. Then h ( F ∗ ⊗ ω C ) = 0, which is imposible unless F = ω C since F is stable and deg( F ∗ ⊗ ω C ) ≤
0. This proves (1). If µ ( F ) ≥ g −
1, then H ( F ( − P )) = 0 forany P ∈ C , hence H ( F ) → F ⊗ κ ( P ) is surjective. Using Nakayama’s lemma, we conclude that H ( F ) ⊗ O C → F is surjective. (cid:3) Lemma 4.2 ([32, Exercise 2.8]) . Let
F, G be vector bundles on C such that H p ( F ⊗ G ) = 0 for p = 0 , . Then both F and G are semistable.Proof. By Riemann-Roch, µ ( F ⊗ G ) = g −
1. Assume that there exists 0 = F ′ ⊂ F such thatrank F ′ < rank F and µ ( F ′ ) > µ ( F ). Then, µ ( F ′ ⊗ G ) > µ ( F ⊗ G ) = g −
1. This shows that χ ( F ′ ⊗ G ) >
0, in particular, h ( F ′ ⊗ G ) >
0. This contradicts to F ′ ⊗ G ⊂ F ⊗ G and h ( F ⊗ G ) = 0.It follows that F is semistable, and the same argument applies to G . (cid:3) Similar as in the case of line bundles, we may define the Brill-Noether locus as follows: W k − r,d ( C ) := { [ F ] ∈ U s C ( r, d ) | h ( C, F ) ≥ k } which is a subscheme of U s C ( r, d ) of expected dimension ρ k − r,d = r ( g −
1) + 1 − k ( k − d + r ( g − Theorem 4.3 ([9, Theorem B]) . The locus W k − r,d ( C ) is nonempty if and only if d > , r ≤ d + ( r − k ) g and ( r, d, k ) = ( r, r, r ) . Derived categories of X . Let Q n ⊂ P n +1 be a smooth quadric hypersurface. We unify allthe notations which involve spinor bundles in accordance with [8]. Hence, the spinor bundles on thequadric Q n give the semiorthogonal decomposition [22]D b ( Q n ) = ( (cid:10) O ( − n + 1) , . . . , O , S (cid:11) if n is odd (cid:10) O ( − n + 1) , . . . , O , S + , S − (cid:11) if n is evenEspecially in the case n = 4, S ± correspond to the universal quotient bundle and the dual of theuniversal subbundle under the isomorphism Q ≃ Gr(2 , C ).Let Q , Q ∞ ⊂ P be two nonsingular 4-dimensional quadrics whose intersection defines X . Withoutloss of generalities, we may assume Q = ( x + . . . + x = 0) and Q ∞ = ( λ x + . . . + λ x = 0)for some λ , . . . , λ ∈ C . We define X := Q ∩ Q ∞ a smooth threefold of degree 4. One wellknown approach to X is to associate the quadric pencil d := | Q + tQ ∞ | t ∈ P on P . Let us assumethat the pencil d is nonsingular in the sense of [34], namely, each singular quadric Q λ i ( i = 0 , . . . , Q ⊂ P over a point. Note that this condition isequivalent to saying that λ , . . . , λ are pairwise distinct. Also note that none of λ , . . . , λ is zerosince Q ∞ is smooth.The resolution of indeterminacy of ϕ d : P P gives the relative quadric Q → P . Let σ : C → P be the double cover ramified over [1 : λ ] , . . . , [1 : λ ] ∈ P , and let Q C := Q × P C be the fiberproduct. Bondal and Orlov [8] showed that C is the fine moduli space of spinor bundles on thequadrics in d , i.e. there exists a vector bundle S Q C on Q C such that for each c ∈ C , the restriction S Q C (cid:12)(cid:12) Q×{ c } is one of the spinor bundles on the quadric Q σ ( c ) . When Q σ ( c ) is a singular quadric, thenit is a cone C ( Q ) of a 3-dimensional quadric over a point v ∈ P . In this case S σ ( c ) is the pullback ofthe unique spinor bundle on Q by C ( Q ) \ { v } → Q . We define the vector bundle S := S Q C (cid:12)(cid:12) X × C . Theorem 4.4 (Bondal–Orlov [8]) . The Fourier–Mukai transform Φ S : D b ( C ) → D b ( X ) , F • Rp X ∗ ( Lp ∗ C F • L ⊗ S ) is fully faithful, and induces a semiorthogonal decomposition D b ( X ) = (cid:10) O X ( − , O X , Φ S (cid:0) D b ( C ) (cid:1) (cid:11) . Furthermore, X can be regarded as the fine moduli space of stable vector bundles of rank 2 withfixed determinant of odd degree [29], and S is the universal bundle of this moduli problem. There arisesan ambiguity of the choice of this fixed determinant (the theorem of Bondal and Orlov is independentof the replacement S 7→ S ⊗ p ∗ C L for any line bundle L ∈ Pic C ). Definition 4.5.
We choose ξ a line bundle of degree 1, and assume that S is the universal familyof the fine moduli space SU C (2 , ξ ∗ ) ≃ X which parametrizes the stable vector bundles of rank 2 anddeterminant ξ ∗ . Equivalently, S is determined by imposing the condition det S = O X (1) ⊠ ξ ∗ . LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 15
This choice of S is precisely dual to the same symbol in Section 5 of [23]. We remark that someparts of the next subsection are following the arguments in [23]. This may cause confusions, so werephrase the details which are necessary for the rest part of the paper.4.3. Ulrich bundles via derived categories.
Let Coh( X ) be the category of coherent sheaves on X . There is a natural functor Coh( X ) → D b ( X ) which maps a coherent sheaf E to the complexconcentrated at degree zero: . . . → → E → → . . . . This identifies Coh( X ) to a full (but not triangulated) subcategory of D b ( X ), hence we may regarda coherent sheaf on X as an object in D b ( X ). Conversely, we call an object E • ∈ D b ( X ) a coherentsheaf (resp. a vector bundle) if E • is isomorphic to an object (resp. a locally free sheaf) in D b ( X ) ∩ Coh( X ).We use derived categories to classify Ulrich bundles on X . We first assume that there existsan Ulrich bundle E of rank r ≥ X (the existence will be proved later). By Proposition 2.3, H p ( E ( − i )) = Hom D b ( X ) ( E ∗ (1) , O ( − i +1)[ p ]) = 0 for all p and i = 1 , ,
3. Using the semiorthogonal de-composition in Theorem 4.4, one immediately sees that E ∗ (1) ∈ Φ S D b ( C ). Since D b ( C ) → Φ S (D b ( C ))is an equivalence of categories, the study of Ulrich bundles on X boils down to the study of cer-tain objects in D b ( C ). Such objects are obtained by mapping E ∗ (1) along the projection functorΦ ! S : D b ( X ) → D b ( C ). Before to proceed, let us note that the projection Φ ! S is right adjoint to Φ S .Since the functor Φ S is given by F Rp X ∗ ( Lp ∗ C F L ⊗ S ) where p X : X × C → X and p C : X × C → C are the natural projections, its right adjoint has the following form ( cf. [20, Proposition 5.9]):Φ ! S : D b ( X ) → D b ( C ) , E 7→ Rp C ∗ (cid:0) Lp ∗ X E L ⊗ S ∗ ) ⊗ ω C [1] . Meanwhile, the Ulrich conditions in Proposition 2.3-(3) impose an extra condition on E ∗ (1) otherthan E ∗ (1) ∈ Φ S D b ( C ). Indeed, the condition H • ( E ( − E ∗ (1) ∈ Φ S D b ( C ).It can be expressed as follows:Hom D b ( X ) ( E ∗ (1) , O X ( − p ]) = 0 ⇔ Hom D b ( X ) (Φ S Φ ! S ( E ∗ (1)) , O X ( − p ]) = 0 ⇔ Hom D b ( C ) (Φ ! S ( E ∗ (1)) , Φ ! S ( O X ( − p ]) = 0 . (4.1) Lemma 4.6.
We have Φ ! S ( O X ( − ≃ R ∗ ⊗ ω ⊗ C , where R is the second Raynaud bundle whichappears in [23, Section 5.4] .Proof. By [23],
R ≃ Φ ! S ∗ O X [ −
1] = p C ∗ (cid:0) S ⊗ p ∗ X O X ) ⊗ ω C . Thus, R ∗ ≃ H om D b ( C ) ( p C ∗ S ⊗ ω C , O C ) ≃ p C ∗ H om D b ( X × C ) ( S ⊗ p ∗ C ω C , p ∗ X ω X [3]) ≃ p C ∗ (cid:0) S ∗ ⊗ p ∗ X O X ( − (cid:1) ⊗ ω ∗ C [3] ≃ Φ ! S ( O X ( − ⊗ ω ⊗ ( − C [2] , where the second isomorphism is given by Grothendieck-Verdier duality. (cid:3) Together with the orthogonality condition (4.1), we have to understand how the object Φ ! S ( E ∗ (1))looks like. One standard way is to analyze the restriction to the point Φ ! S E ∗ (1) ⊗ κ ( c ) ∈ D b ( { c } ). Wefix the notations to avoid confusion as follows. Notation 4.7.
For x ∈ X , we denote by S x the vector bundle over C determined by the restrictionof S to { x } × C ≃ C . Similarly, the vector bundle S c ( c ∈ C ) over X is defined to be the restrictionof S to X × { c } ≃ X .The proof of the following proposition is essentially due to [23, Theorem 5.10], but we write downthe proof to prevent the confusions arising from the choice of a convention. Proposition 4.8.
Suppose there exists an Ulrich bundle E of rank r on X . Then, F := Φ ! S ( E ∗ (1)) ∈ D b ( C ) is a semistable vector bundle over C of rank r and degree r . Furthermore, F satisfies (1) Ext pC ( R , F ∗ ⊗ ω ⊗ C ) = 0 for p = 0 , and (2) H ( F ⊗ S x ) = 0 for each x ∈ X .Conversely, if F is a semistable vector bundle over C of rank r and degree r satisfying the conditions(1) and (2) above, then Φ S F = E ∗ (1) for some Ulrich bundle E over X .Proof. Let c ∈ C be a point. Then F ⊗ κ ( c ) ∈ D b ( { c } ) is the complex of C -vector spaces whosecohomology sheaves are controlled by H p +1 ( X, E ∗ (1) ⊗ S ∗ c ) ≃ Ext p +1 X ( E ( − , S ∗ c ) . (4.2)By [30, p. 310], µ ( S ∗ c ) = − /
2, regardless whether c is a ramification point or not. Hence µ ( E ( − X ( E ( − , S ∗ c ) ≃ Hom X ( E ( − , S ∗ c ) = 0 . Consider the following short exact sequence ( cf. [30, Theorem 2.8])0 → S ∗ τc → O ⊕ X → S ∗ c (1) → τ : C → C is the hyperelliptic involution arising from the double cover C → P . Note that evenfor the ramification points c ∈ C , one can compose the sequence (4.3) in a natural way. Tensoring(4.3) with E ∗ ( j ) for j = − , ,
1, we have H p +1 ( E ∗ (1) ⊗ S ∗ c ) ≃ H p +2 ( E ∗ ⊗ S ∗ τc ) ≃ H p +3 ( E ∗ ( − ⊗ S ∗ c )and the latter one vanishes for p ≥
1. This proves that (4.2) is zero unless p = 0, in other words, F is a coherent sheaf concentrated at degree 0. Furthermore, since p ∗ X ( E ∗ (1)) ⊗ S is flat over C , c χ ( E ∗ (1) ⊗ S ∗ c ) is a constant function and thus F is a vector bundle on C .To compute rank F and deg F , we use Grothendieck-Riemann-Roch which readsch(Φ S F ) = ch( Rp X ∗ ( p ∗ C F ⊗ S )) = p X ∗ (cid:0) ch( p ∗ C F ) ch( S ) td( T p X ) (cid:1) = (2 d − s ) + 13 (2 s − d ) P X − sL X + ( d − s ) H X , (4.4)where d = deg F and s = rank F . The computation method is identical to the one introduced in[23, Lemma 5.2] except that the Fourier-Mukai kernels are dual to each other. LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 17
Since Φ S F = E ∗ (1) is of rank r and of degree zero, we find 2 d − s = r and d − s = 0. It followsthat s = r and d = 2 r . By (4.1) and Lemma 4.6,Hom D b ( C ) (Φ ! S ( E ∗ (1)) , Φ ! S ( O X ( − p ]) ≃ Hom D b ( C ) ( F, R ∗ ⊗ ω ⊗ C [ p − ≃ Ext p − C ( R , F ∗ ⊗ ω ⊗ C ) . Since both R and F are vector bundles, it suffices to require Ext pC ( R , F ∗ ⊗ ω ⊗ C ) = 0 for p = 0 , F follows from Lemma 4.2. Finally, H ( F ⊗ S x ) = 0 follows fromthe fact that Φ S F = E ∗ (1) is a vector bundle on X ; indeed, Φ S F = Rp X ∗ ( p ∗ C F ⊗ S ) is the complexconcentrated at zero, hence R p X ∗ ( p ∗ C F ⊗ S ) = 0. By the cohomology base change, H ( F ⊗ S x ) = 0for each x ∈ X .Conversely, assume that F is a semistable vector bundle on C satisfying all the prescribed condi-tions. The condition (2) implies that Φ S F ∈ D b ( X ) is a vector bundle on X . Then Φ S F ∈ Φ S D b ( C )together with (1) can be interpreted as Ext pX (Φ S F, O X ( − j )) = 0 for j = 0 , ,
2, showing that E := (Φ S F ) ∗ ⊗ O X (1) is an Ulrich bundle over X . (cid:3) Using (4.4) and Φ S F = E ∗ (1), we can immediately check that( c i ( E ) ) i = ( 1 , r, r − r, r ( r − r + 1) ) . Proposition 4.8 gives a bijection between the set of Ulrich bundles on X with the set of certainsemistable vector bundles on C . From now on, we bring our focus into the semistable vector bundleson C satisfying the conditions described in Proposition 4.8. First of all, we prove that a general stablebundle in U C ( r, r ) ( r ≥
2) satisfies the condition (2) of Proposition 4.8.
Proposition 4.9.
For r ≥ , let U s C ( r, r ) be the moduli space of stable vector bundles on C of rank r and degree r . The subset (cid:8) [ F ] ∈ U s C ( r, r ) : h ( F ⊗ S x ) = 0 for every x ∈ X (cid:9) is open and nonempty.Proof. First of all, we claim that the set { [ F ] ∈ U s C ( r, r ) : h ( F ⊗ S x ) = 0 for every x ∈ X } is openin U s C ( r, r ). Consider the closed subset Z ⊂ X × U s C ( r, r ) defined by { ( x, [ F ]) : h ( F ⊗ S x ) ≥ } . Since the projection morphism pr : X × U s C ( r, r ) → U s C ( r, r ) is proper, V := U s C ( r, r ) \ pr ( Z ) isopen in U s C ( r, r ). Writing down the locus V set-theoretically, we can easily find that V = { [ F ] ∈ U s C ( r, r ) : h ( F ⊗ S x ) = 0 for every x ∈ X } . For r = 2, we know that any smooth X carries an Ulrich bundle E of rank 2 as in Proposition 3.6.Note that its projection image F := Φ ! S ( E ∗ (1)) is a rank 2 vector bundle of degree 4 on C satisfyingthe desired property. Assume that r ≥
3. Let F be a stable vector bundle of rank r and degree2 r , and let x ∈ X . Suppose that H ( F ⊗ S x ) ≃ Hom C ( F, S ∗ x ⊗ ω C ) ∗ is nonzero. By the stabilitycondition, any nonzero morphism F → S ∗ x ⊗ ω C must be surjective, so we have a short exact seqeunce0 → F ′ → F → S ∗ x ⊗ ω C → where F ′ is a semistable vector bundle of rank ( r −
2) and degree (2 r − ext C ( S ∗ x ⊗ ω C , F ′ ) = 3 r −
4. Hence, for each x ∈ X , the locus of vector bundles F fit into theabove exact sequence has dimension at most ( r − + 1 + (3 r −
5) = r − r . As varying x ∈ X , thebad locus can sweep out a set of dimension at most r − r + 3 < r + 1. Hence we conclude that ageneral F ∈ U s C ( r, r ) does not admit a surjection to S ∗ x ⊗ ω C for any x ∈ X . (cid:3) Remark 4.10.
The formula (4.4) tells us that there is no line bundle F of degree 2 such that Φ S F is locally free. Indeed, there is no line bundle E on X such that ch( E ) = 1 − L X . In particular, thereis no Ulrich line bundle on X .Our aim is to find a semistable vector bundle F of rank r and degree 2 r such that Ext pC ( R , F ∗ ⊗ ω ⊗ C ) = 0 for p = 0 ,
1. Since G := F ∗ ⊗ ω ⊗ C is also a semistable vector bundle of rank r and degree2 r , the following proposition guarantees the existence of Ulrich bundles at least when r = 3: Proposition 4.11.
Hom C ( R , G ) = 0 for a generic stable vector bundle G of rank and degree .Proof. Suppose that there is a nontrivial morphism
R → G. Note that R is a stable vector bundle [19,Corollary 6.2]. By the stability condition, we observe that the image of R → G is either a rank 2vector bundle of degree 3, or a rank 3 vector bundle of degree 4, 5, 6. We show by cases that theseconditions are not generic.(1) Suppose that the image of R → G is a rank 2 vector bundle of degree 3. There are two shortexact sequences 0 → G ′′ → R → G ′ → → G ′ → G → L → G ′ is the image of R , G ′′ is a rank 2 vector bundle of degree 1. Note that both G ′ and G ′′ are stable. Also, L is locally free: indeed, if ¯ G ′ is the kernel of the morphism G → L/ Tors L ,then the stability argument forces that G ′ = ¯ G ′ , hence L = L/ Tors L showing that L is locallyfree. Since h ( C, G ′ ) >
0, a nonzero section s ∈ H ( C, G ′ ) defines the following exact sequence0 → O C ( D ) s → G ′ → M → D is the zero locus V ( s ) of s and M = det G ′ ⊗ O C ( − D ) is a line bundle. By the stability,we have either deg D = 0 or 1. Tensoring by G ′′∗ , we have0 → G ′′∗ ( D ) → G ′ ⊗ G ′′∗ → G ′′∗ ⊗ M → . When deg D = 0, that is, D = 0, the stability of G ′′ assures thatdim Hom C ( G ′′ , G ′ ) ≤ h ( C, G ′′∗ ) + h ( C, G ′′∗ ⊗ M )= 0 + 3 = 3 . When deg D = 1,dim Hom C ( G ′′ , G ′ ) ≤ h ( C, G ′′∗ ( D )) + h ( C, G ′′∗ ⊗ M )= 1 + 2 = 3 LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 19 since both the Brill-Noether loci W , ( C ) and W , ( C ) are empty by Theorem 4.3. In any cases,we observe that the Quot scheme [ R → G ′ ] ∈ Quot , ( R ) has the local dimension at most 3for any stable quotient G ′ ∈ U s C (2 , G ∈ U s C (3 ,
6) which is anextension of L by G ′ has the dimension at mostdim { G } ≤ dim Quot , ( R ) + dim Pic ( C ) + dim P Ext C ( L, G ) ≤ <
10 = dim U s C (3 , . (2) Suppose that the image of R is a rank 3 vector bundle of degree 4. We have two short exactsequences 0 → L → R → G ′ → → G ′ → G → T → L is a line bundle of degree 0, G ′ is the image of R , and T is a torsion sheaf of length2. Since dim Hom C ( L, R ) = 1 ( cf. the proof of [23, Lemma 5.9]), the dimension of the familyof stable vector bundles G ′ ∈ U C (3 ,
4) which fit into the first short exact sequence is at mostdim Pic ( C ) = 2. Hence the dimension of the family of stable vector bundles G which fit intothe second short exact sequence is at most dim { T } + dim { G ′ } + dim P Ext C ( T, G ′ ) = 9.(3) Suppose that the image of R is a rank 3 vector bundle of degree 5. We have two short exactsequences 0 → L → R → G ′ → → G ′ → G → T → L is a line bundle of degree − G ′ is the image of R , and T is a torsion sheaf of length 1.Since R is stable, we have dim Ext C ( L, R ) = dim Hom C ( R , L ⊗ ω C ) = 0. By Riemann-Roch, wehave dim Hom C ( L, R ) = 4, and thus the dimension of the family of stable vector bundles G ′ ∈U C (3 ,
5) which fit into the first exact sequence is at most dim Pic − ( C )+dim P Hom C ( L, R ) = 5.Therefore, the dimension of the family of stable vector bundles G which fit into the second exactsequence is at most dim { T } + dim { G ′ } + dim P Ext C ( T, G ′ ) = 8.(4) Suppose that the image of R is a rank 3 vector bundle of degree 6, in other words, it coincideswith G . We have the following short exact sequence0 → L → R → G → L is a line bundle of degree −
2. By the stability and Riemann-Roch formula, we havedim Hom C ( L, R ) = χ ( L, R ) = 8. Hence the dimension of the family of stable vector bundles G which fits into the above exact sequence is at most dim Pic − ( C ) + dim P Hom C ( L, R ) = 9.To sum up, we conclude that a generic stable vector bundle G ∈ U C (3 ,
6) yields Hom C ( R , G ) = 0. (cid:3) Corollary 4.12.
For each r ≥ , a generic stable vector bundle G ∈ U C ( r, r ) satisfies Ext pC ( R , G ) = 0 , p = 0 , . Proof.
Assume that G i ∈ U C ( r i , r i ) ( i = 1 ,
2) are stable vector bundles satisfying Ext pC ( R , G i ) = 0.Then G := G ⊕ G is a semistable vector bundle satisfying Ext pC ( R , G ) = 0. By the semicontinuity,we see that Ext pC ( R , G ) = 0 for a general G ∈ U C ( r + r , r + r )). By [7, Proposition 9],Proposition 4.8, and Proposition 4.11, there are vector bundles G ∈ U C (2 ,
4) and G ∈ U C (3 ,
6) suchthat Ext pC ( R , G i ) = 0. Since direct sums of G and G can produce all the ranks ≥
4, we get thedesired result. (cid:3)
Recall that the projection image F = Φ ! S ( E ∗ (1)) is always a semistable vector bundle. It is easy tosee that both the stability and the strict semistability are preserved by this Fourier-Mukai projection. Proposition 4.13.
Let E be an Ulrich vector bundle of rank r ≥ , and let F := Φ ! S E ∗ (1) be asemistable vector bundle on C . If E is stable (resp. strictly semistable), then so is F .Proof. First assume that E is strictly semistable. There is a destabilizing sequence0 → E ′ → E → E ′′ → E ′ and E ′′ are Ulrich bundle of smaller ranks by Proposition 2.5. This gives the following shortexact sequence 0 → F ′′ := Φ ! S E ′′∗ (1) → F = Φ ! S E ∗ (1) → F ′ := Φ ! S E ′∗ (1) → . Since E ′′ is Ulrich, we see that F ′′ ⊂ F is a vector bundle of slope 2 on C , so F cannot be stable.Now assume that E is stable, but F is strictly semistable. Consider the destabilizing sequence0 → F ′′ → F → F ′ → . Since F comes from an Ulrich bundle, the conditions in Proposition 4.8 ensures that h ( F ′ ⊗ S x ) = 0and Ext pC ( R , F ′∗ ⊗ ω ⊗ C ) = 0. It follows that E ′ is Ulrich where E ′∗ (1) := Φ S ( F ′ ). The existence ofthe nonzero map E ∗ (1) → E ′∗ (1) leads to a contradiction; indeed, E ∗ (1) is stable of µ = 0 and E ′∗ (1)is semistable of µ = 0, thus there is no nonzero map from E ∗ (1) to E ′∗ (1). (cid:3) To sum up the above discussions, we have the following theorem.
Theorem 4.14.
Let M ( r ) ( r ≥ ) be the moduli space of S-equivalence classes of Ulrich bundles ofrank r over X . The projection functor Φ ! S : D b ( X ) → D b ( C ) induces the morphism ϕ : M ( r ) → U C ( r, r ) , [ E ] ϕ ( E ) := [Φ ! S ( E ∗ (1))] of moduli spaces. Moreover, ϕ satisfies the following properties: (1) set-theoretically, ϕ is an injection; (2) ϕ maps stable(resp. semistable) objects to stable(resp. semistable) objects; (3) let M s ( r ) be the stable locus. Then ϕ induces an isomorphism of M s ( r ) onto ϕ ( M s ( r )) = ( [ F ] ∈ U s C ( r, r ) : Ext pC ( R , F ∗ ⊗ ω ⊗ C ) = 0 , p = 0 , ,h ( F ⊗ S x ) = 0 for each x ∈ X. ) , which is a nonempty open subscheme of U C ( r, r ) . LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 21
Proof.
First of all, to be well defined, ϕ has to preserve S-equivalence classes. Assume that E and E are Ulrich bundles which are S-equivalent, i.e. there are Jordan-H¨older filtrations0 = E (0) i ⊂ E (1) i ⊂ . . . ⊂ E ( m ) i = E ∗ i (1)such that E ( j )1 / E ( j − =: gr j ( E ∗ (1)) ≃ gr j ( E ∗ (1)) := E ( j )2 / E ( j − . For each j ,0 → E ( j − i → E ( j ) i → gr j ( E ∗ i (1)) → ϕ preserves both the stabilityand the strict semistability by Proposition 4.13, so it immediately follows that0 = Φ ! S ( E (0) i ) ⊂ Φ ! S ( E (1) i ) ⊂ . . . ⊂ Φ ! S ( E ( m ) i ) = ϕ ( E i )is a Jordan-H¨older filtration with gr j ( ϕ ( E i )) ≃ Φ ! S (gr j ( E ∗ i (1))). This shows that ϕ ( E ) and ϕ ( E ) areS-equivalent.The statement (1) follows from the fact that Φ S : D b ( C ) → Φ S (D b ( C )) is an equivalence of cate-gories, and E ∗ (1) ∈ Φ S (D b ( C )) for each Ulrich bundle E over X . The statement (2) is already provedin Proposition 4.13, so it only remains to prove (3). For any stable Ulrich bundle [ E ] ∈ M s ( r ), thefunctor Φ ! S induces T [ E ] M s ( r ) ≃ Ext X ( E , E ) ≃ Ext C ( ϕ ( E ) , ϕ ( E )) ≃ T [ ϕ ( E )] U s C ( r, r ) . Hence together with (1), ϕ is an isomorphism near [ E ]. Finally, by Proposition 4.9 and Corollary 4.12, ϕ ( M s ( r )) is open and nonempty. (cid:3) Remark 4.15.
It is not true in general that ϕ ( M s ( r )) = U s C ( r, r ). For example, choose a point P ∈ C and consider a stable bundle F := R ∗ ⊗ O C ( − P ) ⊗ ω ⊗ C of rank 4 and degree 8. Then F ∗ ⊗ ω ⊗ C = R ⊗ O C ( P ), hence we see thatHom C ( R , F ∗ ⊗ ω ⊗ C ) = 0 . This shows that ϕ ( M s (4)) is a proper subset of U s C (4 , ℓ ⊂ X is a jumping line for E if the direct sum decomposition of E (cid:12)(cid:12) ℓ contains at least two non-isomorphic direct summands. Proposition 4.16.
Let E be a stable Ulrich bundle of rank r over X . For a generic line ℓ ⊂ X , E (cid:12)(cid:12) ℓ ≃ O X (1) ⊕ r .Proof. We may assume that ξ = O C ( P ) for a point P ∈ C . Indeed, if we choose a suitable L ∈ Pic ( C )and make a replacement S ′ := S ⊗ p ∗ C L , then all the arguments in this section are still valid for thenew Fourier-Mukai transform Φ S ′ : D b ( C ) → D b ( X ) and its right adjoint Φ ! S ′ . In particular, theRaynaud bundle R ′ obtained from Φ ! S ′ O X ( −
2) as in Lemma 4.6 satisfies R ′ = R ⊗ L . Let F := Φ ! S ( E ∗ (1)) and G := F ∗ ⊗ ω ⊗ C . We have G ⊗ ξ ∗ = R ; otherwise0 = Hom C ( R , G ) = Hom C ( R , R ⊗ O C ( P )) = 0gives a contradiction. Since G is stable and G ⊗ ξ ∗ = R , we have Hom C ( R , G ⊗ ξ ∗ ) = 0. By[19, Lemma 2.4 and Theorem 2.5], H ( C, L ⊗ G ⊗ ξ ∗ ) = 0 for a general L ∈ Pic ( C ). On the otherhand, H p ( C, L ⊗ G ⊗ ξ ∗ ) = Ext − pC ( G, L ∗ ⊗ ξ ⊗ ω C ) ∗ = Ext − pC ( L ⊗ ξ ∗ ⊗ ω C , F ) ∗ = Ext − pX (Φ S ( L ⊗ ξ ∗ ⊗ ω C ) , E ∗ (1)) ∗ . We have Φ S ( L ⊗ ξ ∗ ⊗ ω C ) = I ℓ (1)[ −
1] for a line ℓ ⊂ X and its ideal sheaf I ℓ ( cf. [23, Lemma 5.5]).This establishes a bijection between Pic ( C ) and the Fano variety F ( X ) of lines in X . Thus, H p ( C, L ⊗ G ⊗ ξ ∗ ) ≃ Ext − pX ( I ℓ , E ∗ ) ∗ ≃ H p +1 ( E ⊗ I ℓ ( − . In the short exact sequence 0 → E ⊗ I ℓ ( − → E ( − → E ( − ⊗ O ℓ →
0, we easily find that H p +1 ( E ⊗I ℓ ( − ≃ H p ( E ( − ⊗O ℓ ). In particular, h p ( E ( − ⊗O ℓ ) = 0 which implies E (cid:12)(cid:12) ℓ ≃ O X (1) ⊕ r for a general ℓ ∈ F ( X ). (cid:3) We finish this paper by some important remarks.
Remark 4.17 (Arrondo–Costa revisited) . (1) Arrondo–Costa’s classification (Theorem 3.5) also can be interpreted via derived categories ofcoherent sheaves on X . The moduli space of rank 2 ACM bundles of line type is isomorphic tothe abelian surface J ( C ), and the interpretation in terms of categorical language is explainedin [23, Lemma 5.5]. The moduli space of rank 2 ACM bundles of conic type is isomorphic to C and this can be explained by the result of [8] because the image of a conic type ACM bundlealong the projection functor is a skyscraper sheaf. Finally, rank 2 ACM bundles of elliptic curvetype are Ulrich, hence E 7→ Φ ! S ( E ∗ (1)) shows that the moduli space of ACM bundles of ellipticcurve type is isomorphic to an open subset of U C (2 , E constructed in [4, Example 4.4] is not Ulrich.Indeed, two global sections of ω D ( −
1) has a nontrivial linear relation, that is, H ( E ∗ (1)) ≃ ker[ H ( ω D ( − ⊗ H ( P , O P (1)) → H ( ω D )] ≃ C . Hence h ( E ( − h ( E ( − E ( −
1) and E ( −
2) have no cohomology, we see that E ∗ (1) isa semistable vector bundle of rank 3 contained in Φ S D b ( C ). Indeed, the nonzero section of H ( E ∗ (1)) ≃ H ( E ( − ∗ induces a short exact sequence0 → ¯ E → E ( − → O X → , where ¯ E is a rank 2 vector bundle so called an “instanton bundle” of charge 3 (see [15] and[23, Definition 1.1 and Theorem 3.10]). Note that rank 2 Ulrich bundles are instanton bundlesof charge 2, which are minimal. Arrondo–Costa construction shows the existence of a non-minimal instanton bundle. LRICH BUNDLES ON INTERSECTIONS OF TWO 4-DIMENSIONAL QUADRICS 23
Remark 4.18.
The second Raynaud bundle R has an interesting property. Note that a (semi-)stablevector bundle F of rank r and slope g − C defines the theta locusΘ F := { L ∈ Pic ( C ) | h ( C, F ⊗ L ) = 0 } , which is a natural generalization of the theta divisor. The locus is either a divisor linearly equivalentto r Θ where Θ ⊂ Pic ( C ) is the usual theta divisor, or the whole Picard group Pic ( C ). Indeed, thetheta map θ : SU C ( r, det F ) | r Θ | gives a rational map, which is a morphism when r ≤
3. However, when r = 4, R does not have atheta divisor since h ( R ⊗ L ) = 1 for every L ∈ Pic ( C ) as treated above (see also [23, Lemma 5.9]).We refer interested readers to [19, 31, 32] for more details on generalized theta divisors and R .The strange duality provides a following geometric interpretation in terms of generalized thetadivisors. Denote L by the ample generator of Pic SU C (4 , det R ), we see that R is a base point of |L k | if and only if H ( C, R ⊗ G ) = 0 , for all G ∈ U C ( k, . By Serre duality, the above condition is equivalent toHom C ( R , G ∗ ⊗ ω C ) = Ext C ( R , G ∗ ⊗ ω C ) = 0 . Note that G ∗ ⊗ ω C is a vector bundle of rank k and degree 2 k .Corollary 4.12 actually implies that R is not a base point of |L k | for k ≥
2. Since Proposition 4.11holds not only for R but for any stable rank 4 vector bundle of degree 4, we conclude that(1) R 6∈ Bs |L | , i.e. , Bs |L | is a proper subset of Bs |L| = { the set of 16 Raynaud type bundles on C } which correspond to 16 theta characteristics of C ;(2) The linear system |L k | is base-point-free for k = 3.Since |L k | is base-point-free for k ≥ g = 2 and r = 4 (cf. [33, Section 8]).Even though our argument do not assure that a generic vector bundle F ∈ U C (2 ,
4) is orthogonal toall the 16 Raynaud type bundles, however, it sounds very promising that |L | is also base-point-free. Remark 4.19.
The strategy in Proposition 4.8 is also useful to classify Ulrich bundles for smoothcomplete intersection varieties of two even dimensional quadrics of higher dimensions. In higherdimensional cases, we also observe that every Ulrich bundle is a image of Fourier-Mukai transform ofa semistable vector bundle on the associated hyperelliptic curve from Bondal-Orlov’s semiorthogonaldecomposition. Moreover, the moduli space of stable Ulrich bundles is a smooth Zariski open subsetof the moduli space of stable vector bundles on the associated hyperelliptic curve. However, showingthe existence becomes more complicated for higher dimensional cases. For instance, there is no Ulrichbundle of rank 2 on such an n -dimensional del Pezzo variety of degree 4 when n ≥ g − might exist. Acknowledgement.
The authors thank Fabrizio Catanese and Universit¨at Bayreuth for kind hos-pitality during their visit. Yonghwa Cho would like to express his gratitude to JongHae Keum andKorea Institute for Advanced Study for hospitality when he was visiting there. Yeongrak Kim thanksGeorge Harry Hitching, Mihnea Popa, and Frank-Olaf Schreyer for helpful discussion and suggestions.Kyoung-Seog Lee is grateful to Mudumbai Seshachalu Narasimhan for many invaluable teachings, en-couragements and kind hospitality. He thanks Alexander Kuznetsov, Carlo Madonna, and PaoloStellari for kind explanations and motivating discussions. Part of this work was done while he wasa research fellow of Korea Institute for Advanced Study and was visiting Indian Institute of Science.He thanks Korea Institute for Advanced Study and Indian Institute of Science for wonderful workingconditions and kind hospitality. He thanks Gadadhar Misra for kind hospitality during his stay inIndian Institute of Science.Yonghwa Cho was partially supported by Basic Science Research Program through the NRF ofKorea funded by the Ministry of Education (2016930170). Yeongrak Kim was supported by BasicScience Research Program through the National Research Foundation of Korea funded by the Ministryof Education (NRF-2016R1A6A3A03008745). Kyoung-Seog Lee was supported by IBS-R003-Y1.
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