Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 3 min ago
PreprintCC BY 4.0arXiv

Canonical Projectivization of Algebraic Varieties in Characteristic Zero

  • Parsa Bakhtary
Open at sourcePDFDOICite

Abstract

We construct a canonical projectivization of every smooth proper algebraic variety X in characteristic zero. Once a principalization procedure is fixed, the construction gives a finite sequence of blowups in smooth centers with smooth projective endpoint, natural under semilinear isomorphisms and preserving the quasi-projective core, the intersection of all maximal quasi-projective open subsets. At each stage we select finitely many ideals near the generic points of the image in X of the boundary of a maximal quasi-projective open. Principalizing their product strictly lowers the dimension of that image. The selection is defined intrinsically, without a polarization or a chosen projective modification; a Hilbert scheme argument proves its finiteness. Naturality embeds the automorphism group in that of a smooth projective variety of the same dimension. We then extend the construction to smooth proper integral algebraic spaces, obtaining canonical projectivizations of all compact connected Moishezon manifolds.

Related

  1. Polarized varieties without arithmetically Cohen-Macaulay bundles and section rings without graded maximal Cohen-Macaulay modules in characteristic zero
    Cristian Anghel
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. Sharp Bounds and Canonical Blow-ups for Toric Projectivization
    Parsa Bakhtary
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Characteristic-free characterizations of projective space for smooth toric varieties
    Osamu Fujino, Hiroshi Sato
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Grothendieck ring of varieties and at-the-characteristic numerical invariants
    Toni Annala, Matthew Satriano, Evan Sundbo
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Flag Varieties of General Quiver Representations: Characteristic Classes and Semi-Invariants
    Jiarui Fei
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{bakhtary2026canonical,
  title = {{Canonical Projectivization of Algebraic Varieties in Characteristic Zero}},
  author = {Parsa Bakhtary},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04314},
  eprint = {2611.04314},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04314},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Canonical Projectivization of Algebraic Varieties in Characteristic Zero
AU  - Parsa Bakhtary
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04314
UR  - https://arxiv.org/abs/2611.04314
AB  - We construct a canonical projectivization of every smooth proper algebraic variety X in characteristic zero. Once a principalization procedure is fixed, the construction gives a finite sequence of blowups in smooth centers with smooth projective endpoint, natural under semilinear isomorphisms and preserving the quasi-projective core, the intersection of all maximal quasi-projective open subsets. At each stage we select finitely many ideals near the generic points of the image in X of the boundary of a maximal quasi-projective open. Principalizing their product strictly lowers the dimension of that image. The selection is defined intrinsically, without a polarization or a chosen projective modification; a Hilbert scheme argument proves its finiteness. Naturality embeds the automorphism group in that of a smooth projective variety of the same dimension. We then extend the construction to smooth proper integral algebraic spaces, obtaining canonical projectivizations of all compact connected Moishezon manifolds.
ER  -