Skip to content
DraconiticResearch
Search Sources API
193,349 studies31 sourcesUpdated 56 min ago
PreprintCC BY 4.0arXiv

Geometry of some determinantal arrangements

  • Marek Janasz
Open at sourcePDFDOICite

Abstract

Let Dm,n⊂Cm× n be the union of the zero loci of the maximal minors of a generic m× n matrix, and let Um,n be its complement. We show that Um,n≅GLm(C)×Gr(m,n)∘, where Gr(m,n)∘ is the open stratum of the Grassmannian on which all Plücker coordinates are nonzero. For m∈{2,n−2} this gives closed formulas for the topology of Um,n and recovers results of Yim. Our main result, which is computer-assisted, is that D3,5 is \emph{not} a free divisor. This answers a question of Yim. We determine Der(−log D3,5) explicitly. It has projective dimension one, and it fails to be free exactly on the matrices of rank at most two. There the failure comes from a non-free transversal arrangement: the ten planes spanned by pairs of five general vectors in C3. Computations in further cases support the conjecture that Dm,n is free only for m≤ 2 or n=m+1.

Related

  1. Topological inequalities for conic-line arrangements with some prescribed singularities
    Marco Golla, Piotr Pokora
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXivHAL
  2. Determinantal sampling on general spaces: fast approximation via diffusion geometry
    Hoang-Son Tran, Pranav Gupta, Subhroshekhar Ghosh
    arXivPreprintarXiv licenceFull textProbabilityStatisticsarXiv
  3. Some Results on Lame-Heun Spectral Geometry
    Ubong Sam Idiong
    arXivPreprintCC0Full textAnalysis & PDEsarXiv
  4. Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem
    Xiyu Hu
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  5. The λ-chain condition for unit interval determinantal fiber rings
    Takayuki Hibi, Sora Miyashita
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{janasz2026geometry,
  title = {{Geometry of some determinantal arrangements}},
  author = {Marek Janasz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13291},
  eprint = {2610.13291},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13291},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Geometry of some determinantal arrangements
AU  - Marek Janasz
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13291
UR  - https://arxiv.org/abs/2610.13291
AB  - Let D_m,n⊂C^m× n be the union of the zero loci of the maximal minors of a generic m× n matrix, and let U_m,n be its complement. We show that U_m,n≅GL_m(C)×Gr(m,n)^∘, where Gr(m,n)^∘ is the open stratum of the Grassmannian on which all Plücker coordinates are nonzero. For m∈{2,n−2} this gives closed formulas for the topology of U_m,n and recovers results of Yim. Our main result, which is computer-assisted, is that D_3,5 is \emph{not} a free divisor. This answers a question of Yim. We determine Der(−log D_3,5) explicitly. It has projective dimension one, and it fails to be free exactly on the matrices of rank at most two. There the failure comes from a non-free transversal arrangement: the ten planes spanned by pairs of five general vectors in C^3. Computations in further cases support the conjecture that D_m,n is free only for m≤ 2 or n=m+1.
ER  -