Models of Brauer-Severi surface bundles

Abstract

This paper is motivated by the study of rationality properties of Mori fiber spaces. These are algebraic varieties, naturally occurring in the minimal model program, typically via contractions along extremal rays; they are fibrations with geometrically rational generic fiber. Widely studied are conic bundles π : X → S, when the generic fiber is a conic. According to the Sarkisov program [60], there exists a birational modification/ / S such that: (i) the general fiber ofπ is a smooth conic, (ii) the discriminant divisor ofπ is a simple normal crossing divisor, (iii) generally along the discriminant divisor, the fiber is a union of two lines, and (iv) over the singular locus of the discriminant divisor, the fiber is a double line in the plane.Rationality of conic bundles over surfaces has been classically studied by Artin and Mumford [11], who produced examples of nonrational unirational threefolds of this type, computing their Brauer groups and using its nontriviality as an obstruction to rationality. This was generalized to higher-dimensional quadric bundles in [20], bringing higher unramified cohomology into the subject and providing further examples of nonrational varieties based on these new obstructions.The specialization method, introduced by Voisin [64] and developed further by Colliot-Thélène-Pirutka [21], emerged as a powerful tool in the study of stable rationality. It allows to relate the failure of stable rationality of a very general member of a flat family to the existence of special fibers with nontrivial unramified cohomology and mild singularities; see also [13], [63]. In particular, the stable rationality problem for very general smooth rationally connected threefolds can be reduced to the case of conic bundles over rational surfaces [35], [42]. The case of very general families of conic bundles over rational surfaces is treated in [34].Date: August 20, 2017. 1 PreliminariesWe start with some algebraic results that will be used in our approach. Let k be a field. By a variety over k we mean a geometrically integral separated finite-type scheme over k. We will need, however, the generality of locally Noetherian schemes and Deligne-Mumford stacks. This additional generality allows us to construct models of Brauer-Severi bundles in which all of the fibers are smooth. This comes at the cost of imposing nontrivial stack structure on the base.2.1. Deligne-Mumford stacks. Deligne and Mumford [24] defined a class of stacks which includes all stack quotients of the form [X/G], where X is a variety and G a finite group, and which are now called Deligne-Mumford stacks. The stack [X/G] differs from the conventional quotient X/G (which exists as a variety, e.g., when X is quasi-projective) in that it keeps track of the stabilizers of the G-action. A further advantage is that

Conic bundles and iterated root stacks

“…An analogous result for Brauer-Severi surface bundles, i.e. fibrations whose generic fiber is a form of P 2 , has been proven by Kresch and Tschinkel in [12], also using root stack techniques.…”

Involution surface bundles over surfaces

“…For involution surfaces, as well as all of the singular degenerations in Definition 1, the dualizing sheaf has dual which is very ample with 9-dimensional space of global sections. As in [7,Rem. 4.2] and its consequences mentioned in Section 7 of op.…”

“…where the bottom morphism is proper birational and T → S is finite of degree 2. We conclude by applying Theorem 6 over the complement of the singular locus of the image in S of the degeneracy locus of the conic bundle filling in over the missing points as in the proof of Theorem 1.2 of [7].…”

“…In this paper, we continue our investigation of del Pezzo fibrations, initiated in [7]. Here we treat the case of fibrations in minimal del Pezzo surfaces of degree 8, also known as involution surfaces and described from the arithmetic viewpoint, e.g., in [1].…”

Stable rationality of Brauer–Severi surface bundles

“…The proof of Theorem 1 relies on the construction of good models of Brauer-Severi surface bundles in [12]. A new ingredient is a variant of the standard elementary transformation of vector bundles.…”