Skip to content
DraconiticResearch
Search Sources API
197,785 studies31 sourcesUpdated 21 min ago
PreprintarXiv licencearXiv

Moduli of Conics on General Plucker Linear Sections of Grassmannians

  • Zheyuan Fu
Open at sourcePDFDOICite

Abstract

Let G=Gr(k,V) be a complex Grassmannian in its Plücker embedding, and let YE be a general codimension-r linear section. We study the open Hilbert scheme R2(YE) of smooth conics using the kernel and span of a conic. These define a flag variety B=Fl(k−2,k+2;V) and a relative Grassmannian of three-planes in a rank-six bundle. For 0≤ r≤3, we prove that R2(YE) is nonempty, smooth, irreducible and rational of dimension e=2dim V+k(dim V−k)−3−3r, with birational model B×Gr(3,6−r). For r≥4, a general-position argument gives a componentwise birational correspondence with a rank-three degeneracy locus and two smooth projective incidence resolutions. Every component meets the locus of smooth minimal-envelope conics; no additional density assumption is needed for general E. If 0≤ e≤ r, the degeneracy locus itself is smooth. In dimension zero we obtain a top Chern class formula, recovering the known counts $1225$ and $1176$ by exact localization.

Related

  1. On Calabi-Yau varieties that are linear sections of Grassmannians of lines
    Michele Bolognesi, Ciro Ciliberto, Alessandro Verra
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. On Fano varieties of index one that are linear sections of Grassmannians of lines
    Michele Bolognesi, Ciro Ciliberto, Alessandro Verra
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Quantum error correcting codes from quiver moduli: Grassmannians, cyclic monodromy, and argyle collision divisors
    Steven Rayan, Evan Sundbo
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  4. Ramification of the moduli map for hyperplane sections of K3s and hypersurfaces
    Dario Faro, Frank Gounelas
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. Integrals of general geometric random variables on the moduli space of hyperbolic surfaces
    Victor Le Guilloux
    HALPreprintCC BY 4.0Full textGeometry & topologyHALarXiv

Cite

BibTeXRIS
BibTeX
@misc{fu2026moduli,
  title = {{Moduli of Conics on General Plucker Linear Sections of Grassmannians}},
  author = {Zheyuan Fu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04727},
  eprint = {2609.04727},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04727},
}
RIS
TY  - UNPB
TI  - Moduli of Conics on General Plucker Linear Sections of Grassmannians
AU  - Zheyuan Fu
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.04727
UR  - https://arxiv.org/abs/2609.04727
AB  - Let G=Gr(k,V) be a complex Grassmannian in its Plücker embedding, and let Y_E be a general codimension-r linear section. We study the open Hilbert scheme R_2(Y_E) of smooth conics using the kernel and span of a conic. These define a flag variety B=Fl(k−2,k+2;V) and a relative Grassmannian of three-planes in a rank-six bundle. For 0≤ r≤3, we prove that R_2(Y_E) is nonempty, smooth, irreducible and rational of dimension e=2dim V+k(dim V−k)−3−3r, with birational model B×Gr(3,6−r). For r≥4, a general-position argument gives a componentwise birational correspondence with a rank-three degeneracy locus and two smooth projective incidence resolutions. Every component meets the locus of smooth minimal-envelope conics; no additional density assumption is needed for general E. If 0≤ e≤ r, the degeneracy locus itself is smooth. In dimension zero we obtain a top Chern class formula, recovering the known counts $1225$ and $1176$ by exact localization.
ER  -