Derived categories and rationality of conic bundles joint work with M.Bernardara June 30, 2011
Fano 3-folds with κ(v ) = V smooth irreducible projective 3-fold over C. V is birational (Reid-Mori, Miyaoka) to X with at most terminal singularities of the following types : X is a minimal Q-Fano variety X is a minimal Del Pezzo fibration X is a minimal conic bundle π : X S, S normal. Up to birational equivalence X and S are nonsingular and projective.
Rationality of conic bundles π : X S standard conic bundle over S smooth, rational, projective surface. C S degeneracy curve, C C associated double cover and J(X ) the intermediate Jacobian of X. Necessary : S rational, C connected [A-M] and J(X ) direct sum of Jacobians of smooth projective curves [C-G]. Theorem (Beauville) J(X ) is isomorphic to the Prym variety P( C/C). Theorem (Shokurov, Beauville) If S is minimal, X is rational if and only if J(X ) splits.
Rationality of conic bundles Possible cases : S = P 2 and C is a cubic or a quartic. S = P 2, C is a quintic and C C is induced by an even θ-characteristic. S P 1 is ruled, C is trigonal or hyperelliptic and the gr 1 induced by the ruling. is
Derived categories and rationality Question : Can we relate the derived category D b (X ) and the rationality of X? The most promising way is by looking at semi-orthogonal decompositions of D b (X ). If T is a linear triangulated category, a semiorthogonal decomposition T = T 1,..., T n is an ordered collection of orthogonal (from right to left, Hom T (T >i, T i ) = 0) subcategories generating the whole category.
Semi-orthogonal decompositions Example : [BMMS] If X is a smooth cubic threefold D b (X ) = T, O X, O X (1), and the equivalence class of the category T corresponds to the isomorphism class of J(X ). Both T and J(X ) are indecomposable. Idea : The rationality of conic bundles may be equivalent to admitting a certain semi-orthogonal decomposition.
Derived categories of conic bundles Theorem (Kuznetsov) If π : X S is a conic bundle, let B 0 be the sheaf of even parts of the Clifford algebra associated to it and D b (S, B 0 ) the derived category of B 0 -algebras. D b (X ) = ΦD b (S, B 0 ), π D b (S), where Φ and π are fully faithful. If S is rational, D b (S) is generated by exceptional objects and then : D b (X ) = ΦD b (S, B 0 ), E 1,..., E s,
Main Theorems Theorem (-,Bernardara) If there are smooth projective curves Γ i with fully faithful functors Ψ i : D b (Γ i ) D b (S, B 0 ) and a semiorthogonal decomposition D b (S, B 0 ) = Ψ 1 D b (Γ 1 ),... Ψ k D b (Γ k ), E 1,..., E l ; (1) with E j exceptional objects in D b (S, B 0 ), then J(X ) = J(Γ i ). Theorem (-,Bernardara) If S is minimal, then X is rational and J(X ) = J(Γ i ) if and only if D b (S, B 0 ) decomposes like (1).
From semiorthogonal decomposition to rationality FIRST PART : Assume that D b (S, B 0 ) = Ψ 1 D b (Γ 1 ),... Ψ k D b (Γ k ), E 1,..., E l, then J(X ) = J(Γ i ).
From semiorthogonal decomposition to rationality We suppose that there exists Ψ : D b (Γ) D b (S, B 0 ) D b (X ) and study the map that Ψ induces on the motive h 1 (Γ), for Γ a smooth, projective curve and g(γ) > 0. If Ψ : D b (Γ) D b (S, B 0 ) is fully faithful, then it is a FM. Ψ admits a right adjoint Ψ R, which is also FM. In D b (Γ X ) we have E := Ker(Ψ), (1) F := Ker(Ψ R ), (2) Ψ Ψ R = Id D b (Γ). (3) Define e := ch(e).td(γ) and f := ch(f).td(x ), mixed cycles in CHQ (X Γ).
From semiorthogonal decomposition to rationality e := ch([e]).td(γ) CHQ (X Γ) D b (Γ) Φ E D b (X ) (4) CHQ (Γ) e CHQ (X ), The vertical arrows are obtained by taking the Chern character. This (plus E = D b (pt)) induces the decomposition : CH Q (X ) = k CHQ (Γ i) Q r. i=1
The motive of a projective curve If Γ is a smooth projective curve h(γ) = h 0 (Γ) h 1 (γ) h 2 (Γ), where h 0 (Γ) = Q, h 2 (Γ) = Q( 1) and h 1 (Γ) corresponds to J(Γ) up to isogenies, i.e. Hom(h 1 (Γ), h 1 (Γ )) = Hom(J(Γ) Q, J(Γ ) Q ). No nontrivial map h 1 (Γ) h 1 (Γ) factors through Q( j). The cycles e and f induce e i : h(γ) h(x )(i 3) and f i : h(x )(i 3) h(γ). By GRR we have f e = Id h(γ).
From semiorthogonal decomposition to rationality Proposition (Nagel,Saito) π : X S standard conic bundle. there is a submotive Prym h 1 ( C), corresponding to the Prym variety, and Prym( 1) h 3 (X )( 1). If S is rational, h(x ) is the direct sum of Prym( 1) and a finite number of copies of Q( j) (with different twists). Hence h 1 (Γ) is a direct summand of h(x )( 1), notably of Prym( 1), and we have an isogeny ψ Q between J(Γ) and a subvariety of J(X ). Now : f e = i (f i e 4 i ). By the decomposition of h(x ), (f i e 4 i ) h 1 (Γ) is zero unless i = 2. The isogeny ψ Q is the algebraic morphism ψ : J(Γ) J(X ) given by the cycle ch 2 (E).
From semiorthogonal decomposition to rationality P( C/C) = J(X ) is the algebraic representative of the algebraically trivial part A 2 (X ) CH 2 (X ). The polarization θ P is the incidence polarization with respect to X. In particular ψ θ J(X ) = θ J(Γ) and then ψ is an isomorphism between J(Γ) and a principally polarized abelian subvariety of J(X ). Consider the semiorthogonal decomposition of D b (S, B 0 ). D b (X ) = Ψ 1 D b (Γ 1 ),... Ψ k D b (Γ k ), E 1,..., E r. Each Ψ i gives a morphism ψ i : J(Γ i ) J(X ). Let ψ = ψ i. CH Q (X ) = k CHQ (Γ i) Q r = i=1 k Pic Q (Γ i ) Q r+k. i=1
From semiorthogonal decomposition to rationality 0 k i=0 Pic0 Q (Γ i) ψ Q k i=0 Pic Q(Γ) Q k+r 0 A 2 Q (X ) CHQ 2 (X ) The coker of ψ Q is a finite dimensional Q-vector space. ψ : J(Γ i ) J(X ) is a morphism of abelian varieties. Hence coker is trivial. Then ψ is an isomorphism. Corollary If S is minimal and D b (S, B 0 ) admits a decomposition like (1), then X is rational and J(X ) = J(Γ i ).
From rationality to semiorthogonal decomposition SECOND PART : Assume X is rational (i.e. S = P 2 and C cubic, quartic, quintic etc...), then D b (S, B 0 ) = Ψ 1 D b (Γ 1 ),... Ψ k D b (Γ k ), E 1,..., E l.
From rationality to semiorthogonal decomposition π : X S a rational, standard CB and S a minimal rational surface. 0 Br(S) Br(K(S)) α H 1 β et(k(d), Q/Z) µ 1, D S x S Br(S) is trivial, and B 0 is a 2-torsion point of Br(K(S)), hence B 0 and D b (S, B 0 ) are fixed once it is fixed the data of the discriminant curve C and its double cover C C.
From rationality to semiorthogonal decomposition Given a degeneration curve and its double cover, we provide an explicit construction as follows : π X χ S Z. π Z is a smooth projective rational threefold with known semiorthogonal decomposition, π : X S is induced by an explicit linear system on Z, and χ is the blow up of the smooth curve Γ in the base locus.
From rationality to semiorthogonal decomposition The decompositions are obtained comparing (via mutations) the decompositions induced respectively by the blow-up and by the conic bundle structure : (A) (B) D b (X ) = ΨD b (Γ), χ D b (Z), D b (X ) = ΦD b (S, B 0 ), π D b (S). Here is a table summarizing the five different cases C S Z Γ D b (S, B 0 ) quintic in P 2 P 3 genus 5 D b (Γ), 1 exc. quartic in P 2 Quadric genus 2 D b (Γ), 1 exc. cubic in P 2 P 1 -bd. over P 2 3 exc. trigonal in F n P 2 -bd. over P 1 tetragonal D b (Γ), 2 exc. hyperell. in F n Quadr. bd. over P 1 hyperell. D b (Γ), D b (Γ )
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