A categorical invariant for geometrically rational surfaces with a conic bundle structure
aa r X i v : . [ m a t h . AG ] S e p A CATEGORICAL INVARIANT FOR GEOMETRICALLY RATIONALSURFACES WITH A CONIC BUNDLE STRUCTURE
MARCELLO BERNARDARA AND SARA DURIGHETTO
Abstract.
We define a categorical birational invariant for minimal geometrically ratio-nal surfaces with a conic bundle structure over a perfect field via components of a naturalsemiorthogonal decomposition. Together with the similar known result on del Pezzo sur-faces, this provide a categorical birational invariant for geometrically rational surfaces. Introduction
In recent years the study of the derived category of an algebraic variety has been widelydeveloped. It is clear now that semiorthogonal decompositions can provide a useful toolin order to detect the geometrical structure of a variety. In particular its interest focus infinding a birational invariant to be used, for example, to study the rationality of the variety.In this context, the first author and M. Bolognesi [6] introduced the concept of cate-gorical representability and formulated the following question: is a rational variety alwayscategorically representable in codimension 2? Analogously, is it possible to characterizeobstruction to rationality via natural components of some semiorthogonal decompositionwhich cannot be realized in codimension 2? On the complex field, for example, if we con-sider a V Fano threefold X , its derived category admits a semiorthogonal decompositionwith only one nontrivial component A X . For a smooth cubic threefold Y we can also finda decomposition with only one nontrivial component A Y , and and Kuznetsov showed that A X is equivalent to A Y if Y is the unique cubic threefold birational to X [12]. This suggeststhat one could consider A X as a birational invariant.In the case of complex conic bundles π : X → S over a minimal rational surface thesituation is quite well-known. A necessary condition to rationality of X is that the inter-mediate Jacobian J ( X ), as principally polarized Abelian variety, is the direct sum of theintermediate Jacobian of smooth projective curves. It follows for example that smooth cubicthreefolds are not rational [7]. From a categorical point of view it is possible to characterizethe rationality of the conic bundle from the semiorthogonal decomposition of the derivedcategory. By Kuznetsov [11] we have(1.1) D b ( X ) = h Φ D b ( D, B ) , π ∗ D b ( S ) i , where B is the sheaf of even parts of Clifford algebras associated to the quadratic formdefining the fibration and Φ : D b ( S, B ) → D b ( S ) is a fully faithful functor from the derivedcategory of B -algebras over S . If S is rational the only nontrivial part for this semiorthogonaldecomposition must then be contained in the component D b ( S, B ). If S is minimal, the firstauthor and Bolognesi proved that X is rational if and only if D b ( S, B ) has a decompositionwhose components are derived categories of smooth curves or exceptional objects [5]. All those results holds on the complex field C , but we want to study the problem overan arbitrary perfect field k . Auel and the first author worked out the case of del Pezzosurfaces [2] . Given a minimal del Pezzo surface S of degree d and Picard rank 1, a naturalsubcategory A S ⊂ D b ( S ) is defined by the orthogonal complement to the structure sheaf.In [2], a category GK S can be defined, roughly speaking, as the product of all componentsof A S which are not representable in dimension 0, and it is a birational invariant. SuchGriffiths-Kuznetsov component, where it is defined, is then the suitable birational invariantto detect the rationality of the given variety. Our aim is to extend this approach to theother class of geometrically rational minimal surfaces,that is, conic bundles.The precise definition of such an invariant is given in Definition 10. Roughly speaking, wedefine the Griffiths-Kuznetsov component to be the direct sum of subcategories of D b ( S )which are not representable in dimension 0. However, unlikely in the case of del Pezzosurfaces, there is no, to the best of the authors’ knowledge, argument to prove that the(natural) decomposition we choose to define GK S is unique up to mutations. This motivatesthe involved case-by-case definition, and a fundamental part of this work is to prove thatGK S is indeed well-defined. Our main result is the following. Theorem 1.
Let k be a perfect field and S be a geometrically rational surface birationalto a conic bundle over k . The Griffiths-Kuznetsov component GK S is well defined and is abirational invariant. Recall the classification of minimal geometrically rational surfaces over an arbitrary field(see, e.g., [9]): minimal conic bundles are one of the two possible classes of such surfaces,namely the ones with Picard rank two, the other being del Pezzo surfaces with Picard rankone. Combining Theorem 1 with the results from [2], we obtain the following result.
Theorem 2.
Let S be a geometrically rational surface over a perfect field k . Then theGriffiths-Kuznetsov component GK S is well-defined and it is a birational invariant. Notations.
Functors of geometric origin between derived categories will be denoted un-derived (i.e. f ∗ instead of Lf ∗ for the pull-back via a morphism). Given a k -algebra A , thenotation D b ( k, A ) stands for the k -linear bounded derived category of coherent A -modules. Acknowledgments.
The authors are grateful to St´ephane Lamy for providing and dis-cussing the example in remark 4, and to J´er´emy Blanc and Michele Bolognesi for fruitfulconversations. 2.
Basics on geometrically rational surfaces
In this section we will introduce some useful and known results. Let k be a perfect fieldand k an algebraic closure. Let us consider S , a smooth projective geometrically integralsurface over k . We say that S is geometrically rational if S := S × k k is k -rational. A fieldextension l of k is a splitting field for S if S × k l is birational to P l through a sequence ofmonoidal transformations centered at closed l -points.A smooth projective surface S is minimal over k if every birational morphism φ : S → Y ,defined over k , to a smooth variety Y is an isomorphism. If k is algebraically closed, the the results in [2] are claimed to hold over general fields, but perfection is required, as we will show inRemark 4, to ensure that evey birational map can be factored into Sarkisov links as in [10]. ATEGORICAL INVARIANT FOR CONIC BUNDLES 3 only minimal rational surfaces are the projective plane and projective bundles over P . Overa general field, we have the following classification (see, e.g., [9]). Proposition 3.
Let S be a minimal geometrically rational surface over k . Then S is oneof the following:(i) S = P k is the projective plane, so Pic( S ) = Z , generated by the hyperplane O (1) ;(ii) S ⊂ P k is a smooth quadric and Pic( S ) = Z , generated by the hyperplane section O (1) ;(iii) S is a del Pezzo surface with Pic( S ) = Z , generated by the canonical class ω S ;(iv) S is a conic bundle f : S → C over a geometrically rational curve, with Pic( S ) ≃ Z ⊕ Z . Elementary links.
We recall some elements of the Sarkisov program which describesthe factorization of a birational map between minimal rational surfaces in elementary links[10]. Let π : S → Y be a minimal geometrically rational surface with an extremal contrac-tion. Then either Y is a point and S is a minimal surface with Picard rank 1 or Y is acurve and S is a conic bundle with Picard rank 2. If π : S → Y and π ′ : S ′ → Y ′ are twoextremal contractions, an elementary link is a birational map φ : S S ′ of one of thefollowing types:Type I) There is a commutative diagram S (cid:15) (cid:15) S ′ σ o o (cid:15) (cid:15) Y Y ′ ψ o o where φ = σ − , σ : S ′ → S is a Mori divisorial elementary contraction and ψ : Y ′ → Y is a morphism. In this case, Y = Spec( k ), ρ ( S ) = 1, S is a minimal delPezzo, and S ′ → Y ′ is a conic bundle over a geometrically rational curve.Type II) There is a commutative diagram S (cid:15) (cid:15) X σ o o τ / / S ′ (cid:15) (cid:15) Y Y ′∼ = o o where φ = τ ◦ σ − , σ : X → S and τ : X → S ′ are Mori divisorial elementarycontractions. In this case, S and S ′ have the same Picard number, and are henceeither both del Pezzo surfaces (and Y is a point) or both conic bundles (and Y isa geometrically rational curve).Type III) There is a commutative diagram S (cid:15) (cid:15) σ / / S ′ (cid:15) (cid:15) Y ψ / / Y ′ where φ = σ , σ : S → S ′ is a Mori divisorial elementary contraction and ψ : Y → Y ′ is a morphism. Links of type III are nothing but inverse of links of type I. MARCELLO BERNARDARA AND SARA DURIGHETTO
Type IV) There is a commutative diagram S (cid:15) (cid:15) φ / / ❴❴❴❴❴❴❴❴❴❴ S ′ (cid:15) (cid:15) Y ψ ●●●●●●●●● Y ′ ψ ′ { { ✇✇✇✇✇✇✇✇✇ Spec( k )where S ≃ S ′ are isomorphic, Y and Y ′ are geometrically rational curves and ψ and ψ ′ are the structural morphisms. Then both S and S ′ are conic bundles andthe link amounts to a change of conic bundle structure on S .Any birational map φ : S S ′ between minimal geometrically rational surfaces canbe factored through elementary links, and Iskovskikh gives the complete list of all possiblesuch links [10]. We note that the Picard rank is invariant under links of type II and IV,while it changes under links of type I and III. Moreover, if we suppose that S is not rationalthe list of links of type I (and hence of their inverses of type III) is very limited: either S isof degree 8 with a point of degree 2, and S ′ is of degree 6 and the curve C can be rational(according to S being a quadric or not), or S is of degree 4, has a rational point and S ′ isof degree 3 and C is a rational curve. Remark . If k is not perfect, than a birational map may not be decomposable in a finitesequence of elementary links centered at closed points. An example of such map was givenin [13, Rmk. 1.3]: if k = ( Z / Z )[ t ], the birational map φ of P k given by[ x : x : x ] [ x x : x x : x + tx ]has [ √ t : 1 : 0] as a base point, and such a point is never defined over a separable fieldextension of k . 3. Basics on derived categories
Categorical representability.
Using semiorthogonal decompositions, one can definea notion of categorical representability for triangulated categories. In the case of smoothprojective varieties, this is inspired by the classical notions of representability of cycles, see[6]. We refrain here to recall standard notions of semiorthogonal decompositions, excep-tional objects, and mutations, the interested reader can refer to [1]. Let us just recall anonstandard definition of exceptional object.
Definition 5.
Let A be a division (not necessarily central) simple k -algebra (i.e., the centerof A could be a field extension of k ), and A a k -linear triangulated category. An object V of A is called A -exceptional ifHom( V, V ) = A and Hom( V, V [ r ]) = 0 for r = 0 . An exceptional object in the classical sense of the term [8, Def. 3.2] is a k -exceptional object.By exceptional object, we mean A -exceptional for some division k -algebra A . Example . Let A be a central simple algebra over k and X := SB ( A ) the Severi-Brauervariety associated to it, and let n = dim X . The Quillen vector bundle V is a rank n + 1 ATEGORICAL INVARIANT FOR CONIC BUNDLES 5 indecomposable vector bundle whose base change to a splitting field is O (1) ⊕ n +1 , and is inparticular an A -exceptional object [14]. Definition 7.
A triangulated category A is representable in dimension m if it admits asemiorthogonal decomposition A = hA , . . . , A r i , and for each i = 1 , . . . , r there exists a smooth projective k -variety Y i with dim Y i ≤ m ,such that A i is equivalent to an admissible subcategory of D b ( Y i ).The motivation for the above definition is the possibility to formulate the following ques-tion: Question 8.
Let X be a smooth projective k -variety of dimension n . Does X rational imply D b ( X ) categorically representable in dimension n − ? In this work, we consider the above question for surfaces, and we are hence interested incharacterizing categories which are representable in dimension 0. This was done in [2].
Lemma 9.
A triangulated category A is representable in dimension if and only if thereexists a semiorthogonal decomposition A = hA , . . . , A r i , such that for each i , there is a k -linear equivalence A i ≃ D b ( K i /k ) for a separable fieldextension K i /k Conic bundles.
We recall a natural semiorthogonal decomposition of the derivedcategory of a surface with conic bundle structure, following the work of Kuznetsov [11] andits generalization to general fields [3]. Let S be a surface over the field k with a structure ofconic bundle π : S → C over a geometrically rational curve. Such conic bundle is associatedto a quadratic form q : E → L on a locally free O C -module E of rank 3. Denote by O S/C (1)the restriction to S of the line bundle O P E/C (1), and let B be the even Clifford algebraassociated to the form q , which is a locally free O C -algebra whose isomorphism class isinvariant for π : S → C .Under these conditions, we have that π ∗ : D b ( C ) → D b ( S ) is fully faithful, and thereexist a fully faithful functor Φ : D b ( C, B ) → D b ( S ) such that D b ( S ) = h π ∗ D b ( C ) , Φ D b ( C, B ) i . Moreover, since C is a geometrically rational curve, there is a simple k -algebra A (trivialif and only if C ≃ P k ) such that C = SB ( A ). In particular, there is an A -exceptional object V , which is either O (1) if C ≃ P k or the Quillen bundle V as in example 6 if C is notrational, such that D b ( C ) = hO C , V i = h V ∗ , O C i . It follows that we can refine the semiorthogonal decomposition of S (abusing of notation,setting V := π ∗ V ) as follows:(3.1) D b ( S ) = hO S , V, Φ D b ( S, B ) i . Now, let π : S → P k be the base change of the conic bundle to the algebraic closure.Such a conic bundle is not necessarily a Hirzebruch surface and can indeed be not minimal,and have a finite number, say r , of singular fibers which are given by two lines meeting in MARCELLO BERNARDARA AND SARA DURIGHETTO a point. We can pick one line in each fiber and denote such set of lines by F , . . . , F r . ThePicard rank of S is then 2 + r , and there is a semiorthogonal decomposition obtained byconsidering the k -minimal model S → S , which is a Hirzebruch surface:(3.2) D b ( S ) = hO S , O S ( F ) , O S (Σ) , O S (Σ + F ) , O F , . . . , O F r i , where F is the general fiber of π , Σ is a section of π .We finally notice that the base change of the semiorthogonal decomposition (3.1) isexactly the semiorthogonal decomposition (3.2): indeed, either C is rational and we alreadyhave V = O S ( F ), or C is not rational, V has rank 2 and we have V = O S ( F ) ⊕ . The lattergenerates the same category as O S ( F ) since we are considering thick subcategories.It follows that the base change of Φ D b ( C, B ) to S is the subcategory(3.3) hO S , O S ( F ) i ⊥ = hO S (Σ) , O S (Σ + F ) , O F , . . . , O F r i . Links of type I/III and the definition of the Griffiths-Kuznetsovcomponent
We are going to construct a birational invariant for geometrically rational surfaces witha conic bundle π : S → C as the collection of subcategories in the semiorthogonal decom-position (3.1) which are not representable in dimension 0. Such an invariant will matchthe one constructed in [2] in the case where S is birational to a minimal del Pezzo surface.Hence, we first have to deal with links of type I/III in the non-rational cases to give a properdefinition. Indeed, the subcategory Φ D b ( C, B ) can admit semiorthogonal decompositionsand even exceptional object if S is birational to a quadric or to a del Pezzo surface of degree4. These cases were already treated in [2], and we quickly recall them.Let S ′ be a minimal non-rational del Pezzo surface of degree 8 with a point of degree2. Then, S ′ is an involution surface in a Severi-Brauer threefold SB ( B ), and there is anassociated even Clifford algebra C , which is a simple algebra whose center is a degree twofield extension of k , and a semiorthogonal decomposition D b ( S ′ ) = h D b ( k ) , D b ( k, B ) , D b ( k, C ) i , where the first category is generated by O S ′ and the second one either by O S ′ (1) (in whichcase B = 0 and S ′ is a quadric) or by the restriction of the Quillen bundle of SB ( B ) to S ′ (in the case where S ′ is not a quadric). It follows that the Griffiths-Kuznetsov componentfor S ′ should be GK S ′ := D b ( k, C ) if S ′ is a quadric and GK S ′ := D b ( k, C ) ⊕ D b ( k, B ) if S ′ isnot a quadric. In this case, such a category is not shown to be a birational invariant in [2],and this is due to the existence of a link of type I, from which follows that the birationalclass of S ′ contains minimal conic bundles of degree 6. Indeed, the blow-up of a degree 2point S → S ′ is a conic bundle π : S → C , with C either rational if S ′ is a quadric ornon-rational if S ′ is not a quadric.In [2, § B], it is proved that the component we want to construct is indeed related to thestandard semiorthogonal decomposition of the conic bundle as follows: writing C = SB ( A )we have that A and B are Morita-equivalent (note that B has order dividing 2 since it hasan involution defining S ′ ), and that there is a degree 2 extension l/k and a semiorthogonaldecomposition: D b ( C, B ) = h D b ( l ) , D b ( k, C ) i . ATEGORICAL INVARIANT FOR CONIC BUNDLES 7
It follows that the components which are (potentially) not representable in dimension 0 inthe standard decomposition (3.1) are exactly the ones we considered above for S ′ .Let S ′ be a minimal del Pezzo surface of degree 4 with a rational point. Note that S ′ isnot rational. In particular, there is semiorthogonal decomposition D b ( S ′ ) = hO S ′ , A S ′ i , and GK S ′ := A S ′ is expected to be the good candidate for the birational invariant we arelooking for. This case neither was treated in [2], since, again, the existence of a link of typeI implies that the birational class of S ′ contains minimal conic bundles of degree 3. Indeed,the blow-up of a rational point S → S ′ is a conic bundle π : S → P .In [2, A.2] (see also [3]), it is proved that the component we want to construct is indeedrelated to the standard semiorthogonal decomposition of the conic bundle since there isan equivalence D b ( C, B ) ≃ A S ′ . It follows that the component which is (potentially) notrepresentable in dimension 0 in the standard decomposition (3.1) is exactly the one weconsidered above for S ′ .Now that we have analyzed all the possible links of type I/III between non-rationalsurfaces, we can give the definition of the Griffiths-Kuznetsov component of a conic bundle. Definition 10.
Let S be a surface with a structure of conic bundle π : S → C over ageometrically rational smooth curve C . If S is minimal, the Griffiths-Kuznetsov componentGK S of S is defined as follows: (i) if S is rational, GK S = 0; (ii) if S = C × C where C i is a geometrically rational curve with associated Azu-maya algebra A i , then GK S is the sum of those between D b ( k, A ), D b ( k, A ) and D b ( k, A ⊗ A ) which are not equivalent to D b ( k ) (equivalently, the algebra is notBrauer-trivial); (iii) if C ≃ P and S is birational to a non-rational quadric with associated even Cliffordalgebra C , then GK S = D b ( k, C ); (iv) if C ≃ P , and S is neither rational nor birational to a quadric, GK S = D b ( P , B ); (v) if C is not rational with associated Azumaya algebra A , and S is birational to aquadric with associated even Clifford algebra C , then GK S = D b ( k, A ) ⊕ D b ( k, C ); (vi) if C is not rational with associated Azumaya algebra A , and S is not birational toa quadric, then GK S = D b ( k, A ) ⊕ D b ( C, B );If S is not minimal, the Griffiths-Kuznetsov component is GK S = GK S for a minimalmodel S → S .Note that the term component is slightly abused here, since there are cases where GK S is not a component of D b ( S ) but rather the direct sum of some components. Since we canoperate mutations on semiorthogonal decompositions (and we will indeed do to proof themain theorem), we cannot in general give any canonical gluing of components contributingto GK S .The rest of the paper is dedicated to the proof of Theorem 1. Note that we can restrictto minimal models. If π : S → C and π ′ : S ′ → C ′ be are minimal conic bundle structuresand φ : S S ′ is a birational morphism, then φ can be decomposed in a finite numberof links. The invariance of GK S under links of type I and III has been studied above and MARCELLO BERNARDARA AND SARA DURIGHETTO follows by results from [2]. To complete the proof, we will prove the invariance under linksof type II (Theorem 11) and type IV (Corollary 14).5.
Links of type II
Links of type II between conic bundles are the most common birational transformation,that is given by an elementary transformation along a closed fiber of the conic bundlestructure. Let π : S → C be a conic bundle. To define a link of type II, pick a closedpoint x ∈ S of degree d , and denote by S x the fiber of π containing it. Then perform theblow-up of x followed by the subsequent contraction of the fiber S x . This gives a conicbundle π ′ : S ′ → C and a commutative diagram: Z p (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ q ❅❅❅❅❅❅❅❅ S π (cid:15) (cid:15) φ / / ❴❴❴❴❴❴❴ S ′ π ′ (cid:15) (cid:15) C id / / C Theorem 11.
In the above setting, we have GK S ≃ GK S ′ .Proof. Let E be the exceptional divisor of p and E ′ the exceptional of q . We denote by f and f ′ the fibers of π and π ′ in Z respectively. Recall [10] that p ∗ ( − K X ) = q ∗ ( − K Y ) + df − E ′ f = f ′ E = df ′ − E ′ = p ∗ ( − K X ) − q ∗ ( − K Y ) + E ′ Let V be the Quillen bundle on C . Since the isomorphism class of C is preserved underthis link, so is the algebra A , and therefore the category D b ( k, A ).We are left to prove that the equivalence class of the category A S := D b ( C, B ) is alsopreserved. Over the algebraic closure, the category A S ⊂ D b ( S ) admits the followingsemiorthogonal decomposition:(5.1) A S = hO S (Σ) , O S (Σ + F ) , O F , . . . , O F r i , where F i are given a choice of a line in each singular fibers of the fibration S → P k , and F and Σ are respectively the fiber and the section of the conic bundle. Similarly, we have asemiorthogonal decomposition(5.2) A S ′ = hO S ′ (Σ ′ ) , O S ′ (Σ ′ + F ′ ) , O F ′ , . . . , O F ′ r i . Over k , we have that S is the blow up of r points on a Hirzebruch surface F n , for some n , so that K S = 2Σ + ( n + 2) F . Similarly, S ′ is the blow-up of r points on a Hirzebruchsurface F m and one can see that the map φ is obtained by lifting to S the composition of d elementary transformations on F n along fibers that do not contain the points blown-up by S → F n . In particular, m = n − d and K S ′ = 2Σ ′ + ( n − d + 2) F ′ − E ′ . It follows that inour transformation: Σ = Σ ′ − E ′ .Then we have the following equivalence of subcategories of D b ( Z ): p ∗ A S ⊗ O ( E ′ ) = q ∗ A S ′ . ATEGORICAL INVARIANT FOR CONIC BUNDLES 9
Indeed, first note the singular fibers are preserved under the birational transformation φ ,we can choose F ′ i such that p ∗ F i = q ∗ F ′ i for i = 1 , . . . , r . Moreover O q ∗ F i does not changeunder tensor with O ( E ′ ), since the exceptional divisor is not supported on singular fibers.Secondly, using the above relation Σ = Σ ′ − E ′ , it is not difficult to see that hO (Σ) , O (Σ + F ) i ⊗ O ( E ′ ) = hO (Σ ′ ) , O (Σ ′ + F ′ ) i We can now conclude since the autoequivalence ⊗ ( E ′ ) descends to an autoequivalence of D b ( Z ), since E ′ is defined over k . It follows, that A S is equivalent to A S ′ and the proof iscomplete. (cid:3) Links of type IV
A link of type IV is a birational self-transformation of a minimal surface S exchangingtwo conic bundle structures π i : S → C i , for i = 1 ,
2. We will than denote by A i :=Φ i D b ( C i , B i ) ⊂ D b ( S ) to keep simple notations. In particular, the birational map φ : S S fits a commutative diagram S π (cid:15) (cid:15) φ / / ❴❴❴❴❴❴❴❴❴❴ S π (cid:15) (cid:15) C ❍❍❍❍❍❍❍❍❍ C { { ✇✇✇✇✇✇✇✇✇ Spec( k )As proved in [10], S must have degree 8,4,2 or 1, and the list of birational maps is quitelimited. In this case, we need to prove that the Griffiths-Kuznetsov component is welldefined, namely that it does not depend on the choice of semiorthogonal decompositiongiven by the different conic bundle structures. We proceed by a case by case analysis. Proposition 12.
Let S be a degree surface and φ a link of type IV. Then GK S is welldefined.Proof. In this case, S = C × C . The fact that GK S is well-defined is proved in [2, § C] (cid:3) Proposition 13.
Let S be a surface of degree , , or , and φ a link of type IV. Then GK S is well defined.Proof. For i = 1 ,
2, consider the semiorthogonal decomposition D b ( S ) = hO S , V i , A i i , and recall that h V i i = hO S ( F i ) i , for F i a geometric fiber of π i , since either C i is rationaland V i is such a line bundle, or C i is not rational and V i = O S ( F i ) ⊕ . Now we proceed bycase by case analysis following the possibilities given by [10]. Degree 4.
Assume S has degree 4. We have F = − K S − F , so that V = V ∗ ⊗ ω ∗ . Itfollows that h V i ≃ h V ∗ i ≃ h V i , since A op and A are Brauer equivalent.Now consider D b ( S ) = h V ∗ , O S , A i = h V , ω ∗ S , A ⊗ ω ∗ S i = hO S , A , V i , where the first equality is given by the autoequivalence ⊗ ω S on D b ( S ) and the second isthe mutation of h ω ∗ S , A ′ i to the left with respect to V . We can then mutate A to theright with respect to V and obtain then A , so that we have shown that A ≃ A and wefinished the proof. Degree 2.
Assume S has degree 2. We have F = − K S − F , so that V = V ∗ ⊗ ( ω ∗ ) ⊗ .It follows that h V i ≃ h V ∗ i ≃ h V i , since A op and A are Brauer equivalent.Now consider the first conic bundle structure and the semiorthogonal decompositions D b ( S ) = hO S , V , A i = h V , A , ω ∗ S i = hA ′ , V , ω ∗ S i , where the first equality is the mutation of O S to the right with respect to its orthogonalcomplement and the second one is the mutation of A to the left with respect to V , so that A ′ = ⊥ h V , ω ∗ S i is equivalent to A .Now consider the second conic bundle structure and the semiorthogonal decompositions D b ( S ) = h V ∗ , O S , A i = h V , ( ω ∗ S ) ⊗ , A ⊗ ( ω ∗ S ) ⊗ i == h ω ∗ S , A ⊗ ( ω ∗ S ) , V i = hA ′ , ω ∗ S , V i , where we first tensor with ( ω ∗ S ) ⊗ , then mutate h ( ω ∗ S ) ⊗ , A ⊗ ( ω ∗ S ) ⊗ i to the left with respectto its orthogonal complement, then mutate A ⊗ ω ∗ S to the left with respect to ω ∗ S . It followsin particular that A ′ = ⊥ h ω ∗ S , V i is equivalent to A .Finally, the two semiorthogonal decompositions give the full orthogonality between V and ω ∗ S , so that h ω ∗ S , V i = h V , ω ∗ S i . This implies that A ′ = A ′ and the proof follows. Degree 1.
Assume S has degree 1. We have then that C i are rational and F = − K S − F . In particular, we only need to prove that the categories A and A areequivalent.Let us consider the first conic bundle structure and the semiorthogonal decompositions: D b ( S ) = hO ( − F ) , O S , A i = hO ( F ) , ( ω ∗ S ) ⊗ , A ⊗ ( ω ∗ S ) ⊗ i == h ( ω ∗ S ) ⊗ , A ⊗ ( ω ∗ S ) ⊗ , O ( F ) i = h ( ω ∗ S ) ⊗ , O ( F ) , A ′ i , where we first tensor by ( ω ∗ S ) ⊗ , then mutate h ( ω ∗ S ) ⊗ , A ⊗ ( ω ∗ S ) ⊗ i to the left withrespect to its orthogonal complement, then mutate A ⊗ ( ω ∗ S ) ⊗ to the right with respectto O ( F ).Now we need to mutate O ( F ) to the left with respect to ( ω ∗ S ) ⊗ . To this end, let uscalculate: Hom i (( ω ∗ S ) ⊗ , O ( F ) = H i ( S, O (3 K S + F )) . First of all, note that (3 K S + F ) .F <
0, which implies that H ( S, O (3 K S + F )) = 0.Similarly, by Serre duality we have that H ( S, O (3 K S + F )) = H ( S, O (2 K S + F )) = 0since (2 K S + F ) .F < H ( S, O (3 K S + F )) = − χ ( O S , O (3 K S + F )). The lattercan be calculated by Riemann-Roch: χ ( O S , O (3 K S + F )) = 12 (3 K S + F ) · (2 K S + F ) + 1 = − , since K S · F = − S has degree 1. It follows, that there is a unique extension0 −→ O ( F ) −→ F −→ ( ω ∗ S ) ⊗ −→ , ATEGORICAL INVARIANT FOR CONIC BUNDLES 11 which has rank 2 and first Chern class F − K S . Moreover, F is the result of the mutationof O ( F ) to the left with respect to ( ω ∗ S ) ⊗ , so that we end up with the decomposition(6.1) D b ( S ) = hF , ( ω ∗ S ) ⊗ , A ′ i . Now consider the second conic bundle structure and the semiorthogonal decompositions: D b ( S ) = hO S , O ( F ) , A i = hO ( F ) , A , ω ∗ S i = hO ( F ) , ω ∗ S , A ′ i , where the first equality is the mutation of O S to the right with respect to its right orthogonal,and A ′ is the mutation of A to the left with respect to ω ∗ S and is therefore equivalent to A .We mutate now O ( F ) to the right with respect to ω ∗ S . A calculation similar to the aboveone shows that there is exactly one nontrivial extension0 −→ ω ∗ S −→ G −→ O ( F ) −→ , which has rank 2 and first Chern class F − K S . Moreover, G is the result of the mutation of O ( F ) to the right with respect to ω ∗ S , and is an exceptional object. Thanks to Gorodentsev[8], exceptional bundles on S are characterized by their rank and their first Chern class.Note that the F and G have both rank 2, while the first Chern class of G is the first Chernclass of F ⊗ ω S . It follows that G ≃ F ⊗ ω S is the mutation of O ( F ) to the right withrespect to ω ∗ S . We hence end up with the decompositions D b ( S ) = h ω ∗ S , F ⊗ ω S , A ′ i = h ( ω ∗ S ) ⊗ , F , A ′ ⊗ ω ∗ S i == hF , A ′ ⊗ ω ∗ S , ( ω ∗ S ) ⊗ i = hF , ( ω ∗ S ) ⊗ , A ′′ i , where first we tensor by ω ∗ S , then mutate ( ω ∗ S ) ⊗ to the right with respect to its rightorthogonal, and A ′′ is the left mutation of A ′ ⊗ ( ω ∗ S ) ⊗ to the right with respect to ( ω ∗ S ) ⊗ and is therefore equivalent to A . The proof follows then by comparison with 6.1. (cid:3) Corollary 14.
The Griffiths-Kuznetsov component is well-defined for minimal conic bun-dles and hence birational invariant under links of type IV.
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