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Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2

  • Yao Yuan ORCID logo EMAIL logo
Published/Copyright: November 30, 2023
Forum Mathematicum
From the journal Forum Mathematicum Volume 36 Issue 1

Abstract

Let M ⁢ ( d , χ ) , with ( d , χ ) = 1 , be the moduli space of semistable sheaves on ℙ 2 supported on curves of degree d and with Euler characteristic χ. The cohomology ring H * ⁢ ( M ⁢ ( d , χ ) , ℤ ) of M ⁢ ( d , χ ) is isomorphic to its Chow ring A * ⁢ ( M ⁢ ( d , χ ) ) by Markman’s result. Pi and Shen have described a minimal generating set of A * ⁢ ( M ⁢ ( d , χ ) ) consisting of 3 ⁢ d - 7 generators, which they also showed to have no relation in A ≤ d - 2 ⁢ ( M ⁢ ( d , χ ) ) . We compute the two Betti numbers b 2 ⁢ ( d - 1 ) and b 2 ⁢ d of M ⁢ ( d , χ ) , and as a corollary we show that the generators given by Pi and Shen have no relations in A ≤ d - 1 ⁢ ( M ⁢ ( d , χ ) ) , but do have three linearly independent relations in A d ⁢ ( M ⁢ ( d , χ ) ) .

Keywords: motivic measures; Betti numbers; generators of the Chow rings
MSC 2020: 14D22; 14J26

Communicated by Jan Bruinier

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 21022107

Funding statement: The author is supported by NSFC 21022107.

Acknowledgements

I would like to thank Weite Pi and Junliang Shen for their paper [18] which motivated me for this work. I thank the referees for their attention reading the paper.

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Received: 2023-03-31
Revised: 2023-08-14
Published Online: 2023-11-30
Published in Print: 2024-01-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On Mukhin’s necessary and sufficient condition for the validity of the local limit theorem
  3. Non-commutative Khinchine-type inequality for dependent random variables and overview of its applications in data science
  4. Connections and genuinely ramified maps of curves
  5. Caustics of pseudo-spherical surfaces in the Euclidean 3-space
  6. On liftings of modular forms and Weil representations
  7. Approximation via statistical measurable convergence with respect to power series for double sequences
  8. Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems
  9. Gem-induced trisections of compact PL 4-manifolds
  10. Categories of modules, comodules and contramodules over representations
  11. On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions
  12. On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms
  13. Congruence subgroups and crystallographic quotients of small Coxeter groups
  14. Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2
  15. Hardy and BMO spaces on Weyl chambers
  16. De Branges–Rovnyak spaces and local Dirichlet spaces of higher order
https://doi.org/10.1515/forum-2023-0111
Keywords for this article
motivic measures; Betti numbers; generators of the Chow rings

Articles in the same Issue

  1. Frontmatter
  2. On Mukhin’s necessary and sufficient condition for the validity of the local limit theorem
  3. Non-commutative Khinchine-type inequality for dependent random variables and overview of its applications in data science
  4. Connections and genuinely ramified maps of curves
  5. Caustics of pseudo-spherical surfaces in the Euclidean 3-space
  6. On liftings of modular forms and Weil representations
  7. Approximation via statistical measurable convergence with respect to power series for double sequences
  8. Global existence, scattering, rigidity and inverse scattering for some quasilinear hyperbolic systems
  9. Gem-induced trisections of compact PL 4-manifolds
  10. Categories of modules, comodules and contramodules over representations
  11. On the Iwasawa invariants of BDP Selmer groups and BDP p-adic L-functions
  12. On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms
  13. Congruence subgroups and crystallographic quotients of small Coxeter groups
  14. Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ2
  15. Hardy and BMO spaces on Weyl chambers
  16. De Branges–Rovnyak spaces and local Dirichlet spaces of higher order
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