O

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Similar Documents

Description

ON STABLE RATIONALITY OF FANO THREEFOLDS AND DEL PEZZO FIBRATIONS BRENDAN HASSETT AND YURI TSCHINKEL 1. Introduction Recent breakthroughs of Voisin [Voi15], developed and amplified by Colliot-Thélène Pirutka

Transcript

ON STABLE RATIONALITY OF FANO THREEFOLDS AND DEL PEZZO FIBRATIONS BRENDAN HASSETT AND YURI TSCHINKEL 1. Introduction Recent breakthroughs of Voisin [Voi15], developed and amplified by Colliot-Thélène Pirutka [CTP14, CTP15], Beauville [Bea14], and Totaro [Tot15], have reshaped the classical study of rationality questions for higher-dimensional varieties. Failure of stable rationality is now known for large classes of rationally-connected threefolds. The key tool is (Chow-theoretic) integral decompositions of the diagonal, which necessarily exist for stably rational varieties. Integral decompositions of the diagonal specialize well, even to mildly singular varieties, connecting logically the stable rationality of various classes of varieties. This puts a premium on discovering appropriate degenerations linking different classes of rationally connected varieties. Using these techniques, we prove: Theorem 1. Let X be a very general smooth non-rational Fano threefold over C. Assume that X is not birational to a cubic threefold. Then X is not stably rational. While smooth cubic threefolds are all known to be non-rational, determining whether or not they are stably rational remains an open problem. No smooth cubic threefolds are known to be stably rational. However, Voisin [Voi14] has shown that the cubic threefolds where her techniques fail to apply, i.e., those admitting an integral decomposition of the diagonal, are dense in moduli. Several common geometric threads, developed in collaboration with Kresch, unify our approach to Theorem 1. In [HKT15b], we showed that very general conic bundles over rational surfaces with sufficiently large discriminant fail to be stably rational. The conic bundle structures on cubic threefolds arising from projection from a line have quintic plane curves as their discriminants too small for our techniques Date: January 25, 2 BRENDAN HASSETT AND YURI TSCHINKEL to apply. Nevertheless, conic bundles are a useful tool for analyzing stable rationality of Fano threefolds. Second, in [HKT15a] we classified quartic del Pezzo surfaces with mild singular fibers and maximal monodromy; previously [HT14] we showed that a number of these arise as specializations of Fano threefolds of index one. Together, these facilitate a streamlined approach to most families of non-rational Fano threefolds. Acknowledgments: The first author was supported by NSF grant We are grateful to Andrew Kresch for his foundational contributions that made this research possible. We also benefitted from conversations with Alena Pirutka. Kresch and Pirutka also offered helpful feedback on early drafts of this paper. 2. Organization of the cases The tables in [IP99] enumerate non-rational Fano threefolds; see also the summary in [Bea15, Section 2.3], which includes references to methods used to establish non-rationality. In the results that follow, very general refers to the complement to a countable union of Zariski-closed proper subsets of the families enumerated in this section. Let V be a Fano threefold of Picard rank one. Write Pic(V ) = Zh, for some ample h, and express the anti-canonical class K V = rh. Let h 1,2 = dim H 1 (V, Ω 2 V ), which equals the dimension of the intermediate Jacobian IJ(V ). We enumerate non-rational Fano threefolds V of Picard rank one, using the invariants (r, KV 3, h1,2 ). (1, 2, 52): double cover of P 3 ramified in a surface of degree 6, unirationality is unknown, very general V are not stably rational [Bea14] (1, 4, 30): quartic in P 4, unirationality is unknown, very general V are not stably rational [CTP14] (1, 6, 20): intersection of a quadric and a cubic, unirational (1, 8, 14): intersection of three quadrics in P 6, unirational (1, 10, 10): section of Gr(2, 5) by a subspace of codimension 2 and a quadric, general such V are non-rational, all are unirational (1, 14, 5): section of Gr(2, 5) by a subspace of codimension 5, unirational (2, 8, 21): V 1, unirationality is unknown (2, 8 2, 10): V 2, double cover of P 3, ramified in a smooth quartic, unirational, very general V 2 are not stably rational [Voi15, Bea15] ON STABLE RATIONALITY 3 (2, 8 3, 5): V 3, cubic in P 4, unirational We next list non-rational minimal Fano threefolds of Picard rank 2, using the invariants ( K 3 V, h1,2 ). All of these admit conic bundle structures, induced by projection onto a rational surface. (6, 20): double cover of P 1 P 2 branched in a divisor of bidegree (2, 4), unirational (12, 9): divisor in P 2 P 2 of bidegree (2, 2), or a double cover of F(1, 2) P 2 P 2 of bidegree (1, 1), ramified in B K F(1,2), unirational (14, 9): double cover of V 7 Bl p (P 3 ), branched in B K V7, unirational (12, 8): double cover of P 1 P 1 P 1 branched in a divisor of degree (2, 2, 2), unirational 3. Conic bundles over rational surfaces We recall the set-up for the results of [HKT15b]: Let S be a smooth projective rational surface over C. Fix a linear system L of effective divisors on S such that the generic member is smooth and irreducible. Consider the space of pairs { D L nodal and reduced, D D étale of degree two } L and let M be one of its irreducible components. Assume it contains a point {D D} with the following properties: the nodes of D are disjoint from the base locus of L; D is reducible and for each irreducible component D 1 D the induced cover D D D 1 D 1 is non-trivial. Results of Artin and Mumford [AM72] and Sarkisov [Sar82] allow us to assign to each point of M a conic bundle X S, unique up to birational equivalences over S. Essentially, M parametrizes ramification data for the associated Brauer elements in the function field of S, which determine them as S is rational. The condition on the distinguished point implies that the corresponding conic bundle has non-trivial Brauer group. Using Voisin s decomposition of the diagonal technique, we proved in [HKT15b] that a very general point [X] M parametrizes a threefold that fails to be stably rational. We first observe an obvious strengthening of the main theorem of [HKT15b]: M need not dominate the linear series L but can be any smooth irreducible parameter space of reduced nodal curves D L 4 BRENDAN HASSETT AND YURI TSCHINKEL with étale double covers D D. Let K denote the image of M in L, so we have M ϕ K L where ϕ is étale and a covering space over the open subset parametrizing smooth curves. We still insist that there is a reducible member whose nodes are disjoint from the base locus of K, such that the cover over each component is non-trivial. Second, our result is easiest to apply in cases where the monodromy action is large, e.g., when M parametrizes all non-trivial double covers of the generic point [D] K, or equivalently, when the monodromy representation on H 1 (D, Z/2Z)\{0} is transitive. This is the case when S = P 2 and L parametrizes plane curves of even degree; in odd degree there are two such orbits [Bea86]. Large monodromy actions make it easier to decide which component contains a given distinguished point {D D}. 4. Classification of quartic del Pezzo fibrations and stable rationality Consider quartic del Pezzo surface fibrations π : X P 1 satisfying two non-degeneracy conditions: the discriminant is square-free, i.e., X is regular and the degenerate fibers are complete intersections of two quadrics with at most one ordinary singularity; the monodromy action on the Picard groups of the fibers is the full Weyl group W (D 5 ). The fundamental invariant of such fibrations is the height h(x ) = deg(c 1 (ω π ) 3 ) = 2 deg(π ω 1 π ), an even integer (see [HKT15a, HT14] for more background). The principal results we require are [HKT15a, Th. 10.2]: under the non-degeneracy conditions we have h(x ) 8; when h(x ) = 8 or 10, the moduli space of these fibrations has two irreducible components; when h(x ) 12 the moduli space is irreducible. When h(x ) = 8, 10, 12 the total space X is either rational or birational to a cubic threefold; see [HKT15a, 11] and [HT14, 8-10] for details. Thus we will focus on fibrations with heights at least fourteen. Note that Alexeev [Ale87] established non-rationality in these cases by ON STABLE RATIONALITY 5 relating the del Pezzo fibrations to conic bundles. (We will review this below.) Theorem 2. Let X P 1 be a fibration in quartic del Pezzo surfaces satisfying our non-degeneracy conditions, with h(x ) 14, and very general in moduli. Then X fails to admit an integral decomposition of the diagonal and thus is not stably rational. Proof. We first reduce to the conic bundle case, following Alexeev. Choose a section σ : P 1 X, which we may assume is not contained in any line of the generic fiber. Blowing up this section gives a cubic surface fibration with a distinguished line and projecting from this line gives a conic fibration: L X π S P 1 Here S P 1 is a rational ruled surface. The conic bundle structure over S yields a discriminant curve D S and an étale double cover D D. Note that D D coincides with the spectral data introduced in [HKT15a, 2,8] and S is the natural ruled surface containing D described in [HKT15a, 10]. Using [HKT15a, 6] we pin down the numerical invariants: Suppose first that h(x ) = 4n + 2 for n 3. Here the surface S F 1, the Hirzebruch surface. Let ξ denote the ( 1)-curve and f the class of a fiber. Then [D] = 5ξ +(n+3)f which has genus h(x ) 4. If h(x ) = 4n for n 4 then S F 0 P 1 P 1. Here D has bidegree (n, 5), also of genus h(x ) 4. The fundamental dictionary between del Pezzo fibrations and spectral data [HKT15a, Th. 10.1] implies that the D S arising from del Pezzo fibrations are generic in the linear series L = D. The analysis of [HKT15a, 3] shows that the monodromy acts on H 1 (D, Z/2Z) via the full symplectic group, hence transitively on the non-trivial elements. To apply the main result of Section 3, it suffices to exhibit a distinguished point in D, i.e., a reducible curve D = D 1 D 2 with D 1 and D 2 smooth of positive genus, intersecting transversally. Note that any étale double cover on D 1 D 2 deforms to the parameter space M. Indeed, such double covers correspond to homomorphisms H 1 (D 1 D 2, Z) Z/2Z; 6 BRENDAN HASSETT AND YURI TSCHINKEL a specialization D D 1 D 2 induces a homomorphism H 1 (D, Z) H 1 (D 1 D 2, Z) collapsing vanishing cycles, thus a double cover of D via composition. Producing the reducible curve is simple: For S = F 1 take D 1 2ξ + 3f, the proper transform of a cubic plane curve, and D 2 3ξ + nf, a smooth curve of genus 2n 5 1. For S = F 0 P 1 P 1 take D 1 of bidegree (2, 2), an elliptic curve, and D 2 of bidegree (n 2, 3), of genus 2n Quartic del Pezzo surfaces of height 22 A quartic del Pezzo surface of height 22 may be constructed as follows [HT14, 4, Case 5]: Let V = O P 1 O P 1(1) 4 and consider the injection V O 9 P 1 associated with the global sections of V. Then we have morphisms P(V ) P 1 P 8 π 2 P 8, where the composition collapses the distinguished section σ : P 1 P(V ) arising from the O P 1 summand. We use π = π 1 for the fibration over P 1. Let ξ = c 1 (O P(V )(1)) and h = π (c 1 (O P 1(1))) so that ξ 5 = 4ξ 4 h. A generic height 22 quartic del Pezzo X P 1 admits an embedding X P(V ) P 1 as a complete intersection of divisors of degrees 2ξ h and 2ξ. Let Q P 1 denote the former divisor, which is canonically determined. It necessarily contains the section σ. The second divisor Q is a pull-back of a quadric hypersurface via π 2 ; it is typically disjoint from σ. Consider projection from the section σ: inducing a birational map ϖ : P(V ) P(O P 1( 1) 4 ) P 1 P 3 Q P 1 P 3. Restricting to X yields a generically finite morphism φ : X P 3. ON STABLE RATIONALITY 7 We compute its invariants via intersections in P(V ). The pullback of the hyperplane class on P 3 via φ is ξ h. First, we have deg(φ) = (ξ h) 3 (2ξ)(2ξ h) = 2 which means φ is a double cover. Its ramification divisor R = K X φ K P 4 = ξ + h + 4(ξ h) = 3(ξ h) maps to the branch surface B P 3 of degree six. We interpret when φ fails to be finite. Points p P 3 correspond to line subbundles σ(p 1 ) L(p) P(O P 1 O P 1( 1)) F 1 P(V ) where F 1 is the blowup of the projective plane at a point. Thus Q L(p) is the union of the ( 1)-curve and the proper transform of a line l and Q L(p) is the proper transform of a conic disjoint from the ( 1)- curve. These typically meet at two points but the conic might contain the line l, i.e., φ 1 (p) = l; this is a codimension-three condition on p and corresponds to singular points of B. How many singularities do we expect on B? For the moment, assume these are ordinary double points for generic X. We have [HT14, Cor. 3.5]: h 1 (Ω 2 X ) = 22 5 = 17 but for a generic sextic double solid V we have h 1 (Ω 2 V ) = 52 [IP99]. If V admits n ordinary singularities and rank r class group then we have 52 = n r h 1 (Ω 2 Ṽ ), where Ṽ V is the blowup of the singularities. This can be seen by comparing the topological Euler characteristics and class groups of V and Ṽ. We must have r = 2 if V is a contraction of a del Pezzo fibration X P 1. Thus we expect n = 36. Proposition 3. Let y 0, y 1, y 2, y 3 denote coordinates on P 3. The equation for B takes the form det(m) = 0 where M = L2 Q 0 Q 1 Q 0 Q 00 Q 01, Q 0 Q 01 Q 11 with L linear in the y i and the remaining entries quadratic. The generic such matrix arises from a height 22 fibration in quartic del Pezzo surfaces. 8 BRENDAN HASSETT AND YURI TSCHINKEL The singularities of B are of two types. The first type corresponds to the vanishing of the 2 2 minors of M; there are 32 such singularities. The second type corresponds to the locus L = Q 0 = Q 1 = 0; there are four such singularities. Proof. Let x 0 and x 1 denote homogeneous coordinates on P 1 and their pullbacks to P(V ). Designate generating global sections y 0, y 1, y 2, y 3 Γ(O P(V )(ξ h)) Γ(O P 3(1)) and z, x 0 y 0, x 1 y 0,..., x 0 y 3, x 1 y 3 Γ(O P(V )(ξ)). After completing the square to eliminate the term linear in z, the defining equation Q may be written in the form z 2 = Q 00x Q 01x 0 x 1 + Q 11x 2 1, where the Q ij are quadratic in the y i. The defining equation for Q takes the form zl(y 0, y 1, y 2, y 3 ) + Q 0 x 0 + Q 1 x 1 = 0, where L is linear and Q 0 and Q 1 are quadratic in the y i. Eliminating z we obtain x 2 0(Q 00L 2 Q 2 0) + 2x 0 x 1 (Q 01L 2 Q 0 Q 1 ) + x 2 1(Q 11L 2 Q 2 1) = 0, which is the defining equation for the image of X in P 1 P 3. The discriminant of this polynomial regarded as a binary quadratic form in x 0 and x 1 can be written as L 4 ((Q 01) 2 Q 00Q 11) + L 2 ( 2Q 01Q 0 Q 1 + Q 00Q Q 11Q 2 0). After dividing out by L 2 we obtain det(m). This proves the first assertion. Reversing the algebra gives the second assertion. We analyze the singularities of the hypersurface det(m) = 0. In general, the singularities of the determinant of a symmetric 3 3 matrix of forms is given by the vanishing of the 2 2 minors. In geometric terms, this is the Veronese surface Ver P 5 which has degree four. If the entries are quadratic forms in y 0,..., y 3 then the image of the associated morphism P 3 P 5 has degree eight. Bezout s Theorem gives 32 points of intersection. However, we also have to take into account singularities of the entries. Given the form of the upper-left entry of M, these occur precisely when L = 0. (The other entries are generic.) The determinantal hypersurface ON STABLE RATIONALITY 9 thus has additional singularities along the locus L = Q 0 = Q 1 = 0. Our genericity assumption implies this is a complete intersection, thus we obtain four additional ordinary double points. We have the following corollary: Corollary 4. The generic sextic double solid arises as a deformation of a nodal birational model of a generic height 22 fibration X P 1 in quartic del Pezzo surfaces. 6. Index one Fano threefolds Let V be a smooth Fano threefold with Pic(V ) = ZK V, i.e., with rank one and index one. Its degree d(v ) = KV 3 takes the following values [IP99]: d(v ) = 2, 4, 6, 8, 10, 12, 14, 16, 18, 22. For each d(v ) there is an irreducible parameter space for the corresponding Fano threefolds. The cases d(v ) = 12, 16, 18, and 22 are rational. When d(v ) = 14 the generic X P 9 arises as a linear section of the Grassmannian Gr(2, 6). Projective duality gives a codimension ten section of the Pfaffian cubic hypersurface in P 14, a cubic threefold V. There is a birational map V V ; see [IM00, 1], for example, for additional details. This example is related to quartic del Pezzo fibrations: One of the two species of quartic del Pezzo fibrations of height ten X P 1 admits a natural morphism [HKT15a, 11] X V P 9 ; the image is a nodal Fano threefold of degree 14. However, stable rationality of cubic threefolds (and birationally equivalent varieties) remains an open problem d(v ) = 2: Sextic double solids. Failure of stable rationality in this case has been established by Beauville [Bea14] and by Colliot- Thélène Pirutka [CTP15]. It also follows naturally from our formalism: Apply Corollary 4 to reduce to the corresponding del Pezzo fibration of height 22. This realizes a generic height 22 fibration in quartic del Pezzo surfaces as a nodal sextic double solid. If a smooth threefold admits an integral decomposition of the diagonal the same holds true for a specialization with nodes [Voi15, Th. 1.1]. Since a very general del Pezzo fibration of height 22 lacks such a decomposition (by Theorem 2), the same holds for a very general sextic double solid. 10 BRENDAN HASSETT AND YURI TSCHINKEL 6.2. d(v ) = 4: Quartic threefolds. Failure of stable rationality in this case has been established by Colliot-Thélène and Pirutka [CTP14]. To see this from the perspective of fibrations in quartic del Pezzo surfaces, it suffices to recall that a generic such fibration of height 20 admits a birational model as a determinantal quartic threefold with sixteen nodes [HT14, 11 ] (cf. [Che06, Th. 11]). This lacks an integral decomposition of the diagonal (Theorem 2), so very general quartic threefolds have the same property, hence fail to be stably rational d(v ) = 6: Complete intersections of a quadric and a cubic in P 5. We proceed as before, using the fact that a generic quartic del Pezzo fibration of height 18 admits a birational model as a complete intersection Y P 5 with eight nodes. Indeed, realize X P(O 4 P O 1 P 1( 1)) P 1 P 5 as a complete intersection of forms of bidegree (1, 1), (0, 2), and (1, 2), as in Case 3 of [HT14, 4]. (Here we are using the irreducibility of the moduli space of quartic del Pezzo fibrations of height 18.) Let Y P 5 denote the image of projection onto the second factor. Consider first the image in P 5 [x 0,...,x 5 ] of the locus cut out by the forms of bidegree (1, 1) and (1, 2): sl 0 + tl 1 = sq 0 + tq 1 = 0, with L 0, L 1 C[x 0,..., x 5 ] 1, Q 0, Q 1 C[x 0,..., x 5 ] 2. Its equation is obtained by eliminating s and t, which yields ( ) L0 L L 1 Q 0 L 0 Q 1 = det 1 = 0. Q 0 Q 1 This is a cubic fourfold W singular along the elliptic quartic curve C = {L 0 = L 1 = Q 0 = Q 1 = 0}. Let Q C[x 0,..., x 5 ] 2 be the form of bidegree (0, 2), so that Y = W {Q = 0}. This is a complete intersection of Q with W, having eight nodes at the intersection C {Q = 0} = {p 1,..., p 8 }. The preimages of these nodes in X are distinguished sections of X P 1. We establish failure of stable rationality for very general complete intersections as in the previous cases: Use Theorem 2 to deduce the failure of integral decomposition of the diagonal for very general quartic ON STABLE RATIONALITY 11 del Pezzo fibrations of height eighteen. It follows that very general complete intersections V P 5 of a quadric and a cubic also lack such decompositions, so stable rationality fails d(v ) = 8: Complete intersections of three quadrics in P 6. Let V P 3 denote a complete intersection of three quadrics. Beauville [Bea77, 6.4, 6.23] has shown that V is birational to a conic fibration X P 2, with discriminant D P 2 of degree seven, and a generic plane curve of degree seven arises in this way. Thus the results in 3 a

Related Search

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks