Stable rationality of quadric and cubic surface bundle fourfolds

Abstract

We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in P 2 × P 3 is not stably rational. Via projections onto the two factors, X → P 2 is a cubic surface bundle and X → P 3 is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any n ≥ 4, new quadric surface bundle fourfolds Xn → P 2 with discriminant curve Dn ⊂ P 2 of degree 2n, such that Xn has nontrivial unramified Brauer group and admits a universally CH0-trivial resolution.

On the rationality of quadric surface bundles

“…where the map ι * : H 4 (P(E), C) → H 4 (X b , C) is induced by inclusion and is injective by the Lefschetz hyperplane theorem, provided that not all d j are simultaneously zero (1) . This construction is also applicable to the Hodge groups H p,q and gives a decomposition…”

“…for all p and hence ∇ gives rise to an O B -linear map 1) If d j = 0 for all j, Theorem 1.1 is trivial because a quadric surface bundle of type (0, 0, 0, 0) is the product of P 2 with a quadric surface in P 3 and hence rational.…”

“…Subsequently, other smooth families containing both rational and stably irrational fibres were identified, for example in [1,12,13,15,18,19]. Typically, it is easy to provide certain rational members in the studied families.…”

“…The fourfolds considered in [14] and [15] are (birational to) quadric surface bundles over P 2 of types (2, 2, 2, 2) and (0, 2, 2, 4), respectively. Here, a quadric surface bundle of type (d 0 , d 1 , d 2 , d 3 ) for integers d 0 , d 1 , d 2 , d 3 0 of the same parity is given by an equation of the form (1.1) 0 i,j 3 a ij y i y j = 0 where a ij = a ji is a homogeneous polynomial of degree 1 2 (d i + d j ) in the three coordinates of P 2 and y 0 , y 1 , y 2 , y 3 denote local trivializations of a split vector bundle E on P 2 of rank 4, see Section 3 for a more precise definition. The quadric surface bundle X ⊂ P(E) over P 2 defined by equation (1.1) is also called a standard quadric surface bundle.…”

Models of quadric surface bundles

“…Quadric surface bundles are objects of classical study [26]. A special role is played by quadric surface bundles over surfaces in diverse settings, e.g., [27], [7], [11], including questions about (stable) rationality [5], [16], [25].…”