A natural Chow lemma for smooth threefolds and fourfolds
Abstract
Let X be a smooth proper integral variety of dimension at most four over a field of characteristic zero. We construct a finite sequence of blowups along smooth centers with projective endpoint, preserving the intersection of all maximal quasi-projective opens of X. The construction depends only on a fixed functorial principalization procedure and is natural under isomorphisms, including those over different ground fields. The proof has two parts. A colength argument on regular local surfaces gives, in every dimension, a finite preparation after which some maximal quasi-projective open has complement mapping into codimension at least three in X. In dimensions at most four these images are points or curves. Ample numerical classes on the normalized components of that complement give finite bounds for summing base ideals of linear systems. The resulting ideal descends to the original field; geometric integrality is not required.
Related
- Classification of Automorphism Groups of Smooth Cubic Threefolds and Fourfolds
- Ambient Chow Rings of Debarre--Voisin Fourfolds and the Beauville--Voisin Conjecture
- Remarks on Lattice Actions Corresponding to Automorphism Groups of Smooth Cubic Fourfolds
- Automorphism Groups of Smooth Cubic Fourfolds through Lattice Theory
- Non-symplectic Indices of Automorphism Groups of Smooth Cubic Fourfolds
BibTeX
@misc{bakhtary2026natural,
title = {{A natural Chow lemma for smooth threefolds and fourfolds}},
author = {Parsa Bakhtary},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05427},
eprint = {2610.05427},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05427},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A natural Chow lemma for smooth threefolds and fourfolds
AU - Parsa Bakhtary
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05427
UR - https://arxiv.org/abs/2610.05427
AB - Let X be a smooth proper integral variety of dimension at most four over a field of characteristic zero. We construct a finite sequence of blowups along smooth centers with projective endpoint, preserving the intersection of all maximal quasi-projective opens of X. The construction depends only on a fixed functorial principalization procedure and is natural under isomorphisms, including those over different ground fields. The proof has two parts. A colength argument on regular local surfaces gives, in every dimension, a finite preparation after which some maximal quasi-projective open has complement mapping into codimension at least three in X. In dimensions at most four these images are points or curves. Ample numerical classes on the normalized components of that complement give finite bounds for summing base ideals of linear systems. The resulting ideal descends to the original field; geometric integrality is not required.
ER -