Conic bundles and iterated root stacks

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

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Conic bundles and iterated root stacks

Cited by 12 publications

References 13 publications

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