Torsion Cycles in Projective Threefolds:From Serre's Construction to Artin-Mumford

Source abstract

The torsion subgroup of third integral cohomology is a birational invariant of smooth projective complex threefolds; its nonvanishing obstructs stable rationality. We recall a proof using Hironaka's elimination of indeterminacies and the integral blowup formula. The Godeaux--Serre construction and Poincaré duality imply that every finite abelian group is the third integral cohomology group of a smooth projective complex threefold. This classical realization also has a description through families of concurrent plane lines. Their incidence threefolds have smooth normalizations, obtained by an explicit blowup and described as projective-line bundles over smooth surfaces. The Lefschetz theorem determines the fundamental group of these surfaces, and the projective bundle formula computes the third cohomology. The resulting threefolds are birational to products of surfaces with the projective line, and their torsion three-cycles are represented by sphere bundles over loops. Examples and the comparison with the unirational Artin--Mumford threefold explain the role of the base surface and the distinction between smooth blowups and resolution of a singular space.