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Math.AG] 7 Jun 2018 Épijournal de Géométrie Algébrique epiga.episciences.org Volume 2 (2018), Article Nr. 5 Stable rationality of higher dimensional conic bundles Hamid Ahmadinezhad and Takuzo Okada n 1 Abstract. We prove that a very general nonsingular conic bundle X P − embedded in a n 1 ! projective vector bundle of rank 3 over P − is not stably rational if the anti-canonical divisor of X is not ample and n 3. ≥ Keywords. Stable Rationality; conic Bundles 2010 Mathematics Subject Classification. 14E08 [Français] Titre. Rationalité stable des fibrés en coniques de grande dimension n 1 Résumé. Nous démontrons qu’un fibré en coniques non-singulier très général X P − plongé n 1 ! dans le projectivisé d’un fibré vectoriel de rang 3 au dessus de P − n’est pas stablement rationnel lorsque le diviseur anti-canonique de X n’est pas ample et n 3. ≥ arXiv:1612.04206v5 [math.AG] 7 Jun 2018 Received by the Editors on February 7, 2018, and in final form on April 25, 2018. Accepted on May 21, 2018. Hamid Ahmadinezhad Department of Mathematical Sciences, Loughborough University, LE11 3TU, United Kingdom e-mail: [email protected] Takuzo Okada Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502 Japan e-mail: [email protected] © by the author(s) This work is licensed under http://creativecommons.org/licenses/by-sa/4.0/ 2 1. Introduction Contents 1. Introduction ................................................ 2 2. Embedded conic bundles ........................................ 3 3. Stable non-rationality .......................................... 7 1. Introduction An important question in algebraic geometry is to determine whether an algebraic variety is rational; that is, birational to projective space. Two algebraic varieties are said to be birational if they become isomorphic after removing finitely many lower-dimensional subvarieties from both sides. The closest varieties to being rational are those that admit a fibration into a projective space with all fibres rational curves; so-called conic bundles. In this article, we study stable (non-)rationality of conic bundles over a projective space of arbitrary dimension (greater than one). A non-rational variety X may become rational after being multiplied by a suitable projective space, i.e., X Pm is birational to Pn+m, where n = dimX, in which case we say X is × stably rational. Stable non-rationality of conic bundles in dimension 3 has been studied extensively in [1,2] and [8], giving a satisfactory answer. In higher dimensions almost nothing is known except for a few examples of stably non-rational conic bundles over P3 given in [1] and [9]. Throughout this article, by a conic bundle we mean a Mori fibre space of relative dimension 1 (see Definition 2.5 for details). The following is our main result. Theorem 1.1. Let n 3 and d be integers, and let be a direct sum of three invertible sheaves on Pn 1. Let X ≥ E − be a very general member of a complete linear system 2D + dF on PPn 1 ( ), where D is the tautological divisor j j − E and F is the pullback of the hyperplane on Pn 1. Suppose that the natural projection X Pn 1 is a conic bundle. − ! − (1) If X is singular, then X is rational. (2) If X is non-singular and K is not ample, then X is not stably rational. − X This result covers the following varieties as a special case. Corollary 1.2. Let X be a very general hypersurface of bi-degree (d;2) in Pn 1 P2. If d n 3, then X is not − × ≥ ≥ stably rational. This can be thought of as a higher dimensional generalisation of the main result of [2]. Corollary 1.3. Let X be a double cover of Pn 1 P1 branched along a very general divisor of bi-degree (2d;2). − × If 2d n 3, then X is not stably rational. ≥ ≥ By a result of Sarkisov [16], a conic bundle is birational to a standard conic bundle which is by definition a nonsingular conic bundle flat over a smooth base. The following criterion for rationality in terms of the discriminant was conjectured by Shokurov [17] (see also [10, Conjecture I]). Remarkabe progress toward this conjecture has been made in [10] and [13]. Conjecture 1.4. ([17, Conjecture 10.3]) Let X S be a 3-dimensional standard conic bundle and ∆ S the ! ⊂ discriminant divisor. If 2K + ∆ , , then X is not rational. j S j ; Although the statement becomes weaker than Theorem 1.1, we can restate our main result in terms of the discriminant: H. Ahmadinezhad and T. Okada, Stable rationality of conic bundles 3 Corollary 1.5. With notation and assumptions as in Theorem 1.1, assume in addition that X is nonsingular and let ∆ Pn 1 be the discriminant divisor of the conic bundle X Pn 1. ⊂ − ! − (1) If 3KPn 1 + ∆ , , then X is not stably rational. j − j ; n 1 (2) If n 7, π : X P − is standard and 2KPn 1 + ∆ , , then X is not stably rational. ≥ ! j − j ; This leads us to pose the following. Conjecture 1.6. Let π : X S be an n-dimensional standard conic bundle with n 3. If 2K + ∆ , , then ! ≥ S ; X is not rational. If in addition X is very general in its moduli, then X is not stably rational.j j The argument of stable non-rationality. It is known that a stably rational smooth projective variety is universally CH -trivial; see [5, Lemme 1.5] and [18, theorem 1.1] and references therein. Let be 0 X!B a flat family over a complex curve with smooth general fibre. Then, by the specialisation theorem of B Voisin [19, Theorem 2.1], the stable non-rationality of a very general fibre will follow if the special fibre X0 is not universally CH0-trivial and has at worst ordinary double point singularities. This was generalised by Colliot-Thélène and Pirutka [5, Théorème 1.14] to the case where 1. X admits a universally CH -trivial resolution ' : Y X such that Y is not universally CH -trivial, 0 0 ! 0 0 2. in mixed characteristic, that is, when = SpecA with A being a DVR of possibly mixed characteristic. B Thus it is enough to verify the existence of such a resolution ' : Y X over an algebraically closed field ! 0 of characteristic p > 0. In view of [18, Lemma 2.2], the core of the proof of universal CH0-nontriviality for Y in our case is done by showing that H0(Y;Ωi) , 0 for some i > 0, following Kollár [11] and Totaro [18]. This is done in Section3. Acknowledgements. We would like to thank Professor Jean-Louis Colliot-Thélène and Professor Vyach- eslav Shokurov for pointing out two oversights in an earlier version of this article. We would also like to thank the anonymous referee for helpful comments. The second author is partially supported by JSPS KAKENHI Grant Number 26800019. 2. Embedded conic bundles 2.A. Weighted projective space bundles In this subsection we work over a field k. Definition 2.1. A toric weighted projective space bundle over Pn is a projective simplicial toric variety with Cox ring Cox(P ) = k[u0;:::;un;x0;:::;xm]; which is Z2-graded as ! 1 1 λ λ ··· 0 ··· m 0 0 a a ··· 0 ··· m with the irrelevant ideal I = (u ;:::;u ) (x ;:::;x ), where λ ;:::;λ are integers and n, m, a ;:::;a are 0 n \ 0 m 0 m 0 m positive integers. In other words, P is the geometric quotient P = (An+m+2 V (I))=G2 ; n m where the action of G2 = G G on An+m+2 = SpecCox(P ) is given by the above matrix. m m × m 4 2. Embedded conic bundles Pn P The natural projection Π: P by the coordinates u0;:::;un realizes P as a (a0;:::;am)-bundle Pn ! P Pn over . In this paper, we simply call P the (a0;:::;am)-bundle over defined by 0 1 Bu0 un x0 xm C B ··· ··· C B 1 1 λ λ C: B ··· j 0 ··· mC @ 0 0 a a A ··· j 0 ··· m In the following, let P be as in Definition 2.1. Let p P be a point and q An+m+2 V (I) a preimage 2 2 n of p via the morphism An+m+2 V (I) P . We can write q = (α ;:::;α ;β ;:::;β ), where α ;β k. In n ! 0 n 0 m i j 2 this case we express p as (α : :α ;β : :β ). 0 ··· n 0 ··· m Remark 2.2. We will frequently use the following coordinate change. Consider a point p = (α : :α ;β : 0 ··· n 0 : β ) P and suppose for example that α , 0;β , 0 and a = 1 for some j. Then for l , j such that ··· m 2 0 j j λ =a λ , the replacement l l ≥ j λl al λj a λl al λj a x α − β l x β u − x l l 7! 0 j l − l 0 j induces an automorphism of P . By considering the above coordinate change, we can transform p (via an automorphism of P ) into a point for which the x -coordinate is zero for l with λ =a λ . l l l ≥ j We have the decomposition M Cox(P ) = Cox(P )(α,β); (α,β) Z2 2 where Cox(P ) consists of the homogeneous elements of bi-degree (α;β). An element f Cox(P ) (α,β) 2 (α,β) is called a (homogeneous) polynomial of bi-degree (α;β). The Weil divisor class group Cl(P ) is naturally isomorphic to Z2.
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