Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 18 min ago
PreprintarXiv licencearXiv

Categorical genus for log del Pezzo surfaces with cyclic quotient singularities

  • Alex Junior Gomez Saltachin
Open at sourcePDFDOICite

Abstract

For the canonical stack š’³ of a log del Pezzo surface with cyclic quotient singularities, we compute its categorical genus as gcat(š’³)= 1+1/2āˆ‘j wj(ā„“jāˆ’1), where wj=gcd(nj,qj+1) and ā„“j=nj/wj are the local widths and Gorenstein indices of the singularities 1/nj(1,qj). Serre duality and inertial Riemann--Roch separate the smooth contribution from the local canonical characters, and identify categorical genus with 1+dim Hš’³age<1. Passing to a sufficiently general ā„š-Gorenstein deformation defines the residual invariant gcatqG(X); its difference from gcat(š’³) is exactly the weighted T-content contribution. For any surface admitting a toric ā„š-Gorenstein degeneration, this identifies the residual categorical invariant with Tveiten's mutable genus, the genus of a general maximally mutable Laurent-polynomial fiber for the chosen degeneration. The locally ā„š-Gorenstein rigid comparison is the special case in which the drop vanishes. For toric surfaces, the categorical genus of the given canonical stack counts all interior lattice points and equals the genus of a general Laurent-polynomial fiber. Explicit Oneto--Petracci mirrors give direct checks.

Related

  1. Resolutions of linear p-cyclic quotient singularities
    Linghu Fan, Hongmin Li
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. The Kemper--Malle conjecture and isolated linear quotient singularities
    Yang Chen
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. K-regularity for finite quotient singularities
    Matthew Satriano, Wanchun Shen
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Matrix factorization of quotient singularities of type A and D
    Ayse Sharland, Meral Tosun
    arXivPreprintCC BY-SA 4.0Full textAlgebra & number theoryarXiv
  5. Categorical reconstruction of del Pezzo surfaces: Hochschild--Serre algebras and spinor modifications
    Xun Lin, Marco Rampazzo, Shizhuo Zhang
    arXivPreprintCC BY-NC-ND 4.0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{saltachin2026categorical,
  title = {{Categorical genus for log del Pezzo surfaces with cyclic quotient singularities}},
  author = {Alex Junior Gomez Saltachin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13102},
  eprint = {2609.13102},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13102},
}
RIS
TY  - UNPB
TI  - Categorical genus for log del Pezzo surfaces with cyclic quotient singularities
AU  - Alex Junior Gomez Saltachin
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.13102
UR  - https://arxiv.org/abs/2609.13102
AB  - For the canonical stack š’³ of a log del Pezzo surface with cyclic quotient singularities, we compute its categorical genus as g_cat(š’³)= 1+1/2āˆ‘_j w_j(ā„“_jāˆ’1), where w_j=gcd(n_j,q_j+1) and ā„“_j=n_j/w_j are the local widths and Gorenstein indices of the singularities 1/n_j(1,q_j). Serre duality and inertial Riemann--Roch separate the smooth contribution from the local canonical characters, and identify categorical genus with 1+dim H_š’³^age<1. Passing to a sufficiently general ā„š-Gorenstein deformation defines the residual invariant g_cat^qG(X); its difference from g_cat(š’³) is exactly the weighted T-content contribution. For any surface admitting a toric ā„š-Gorenstein degeneration, this identifies the residual categorical invariant with Tveiten's mutable genus, the genus of a general maximally mutable Laurent-polynomial fiber for the chosen degeneration. The locally ā„š-Gorenstein rigid comparison is the special case in which the drop vanishes. For toric surfaces, the categorical genus of the given canonical stack counts all interior lattice points and equals the genus of a general Laurent-polynomial fiber. Explicit Oneto--Petracci mirrors give direct checks.
ER  -