Torsion Cycles in Projective Threefolds:From Serre's Construction to Artin-Mumford
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.
Related
- Reflection groups on smooth projective threefolds
- Examples of descendent generating series for Pandharipande--Thomas stable pairs on smooth projective Fano threefolds via one-dimensional wall-crossing
- Weighted projective spaces admitting ℚ-Gorenstein smoothings to Fano threefolds with Picard number one
- Power-of-two cycles in projective cubic graphs of large edge-width
- Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles
BibTeX
@misc{verjovsky2026torsion,
title = {{Torsion Cycles in Projective Threefolds:From Serre's Construction to Artin-Mumford}},
author = {Alberto Verjovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25417},
eprint = {2610.25417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25417},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Torsion Cycles in Projective Threefolds:From Serre's Construction to Artin-Mumford
AU - Alberto Verjovsky
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25417
UR - https://arxiv.org/abs/2610.25417
AB - 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.
ER -