Advertisement

Geometriae Dedicata

, Volume 177, Issue 1, pp 385–400 | Cite as

Affine pavings for moduli spaces of pure sheaves on \(\mathbb {P}^2\) with degree \(\le 5\)

  • Yao Yuan
Original Paper
  • 86 Downloads

Abstract

Let \(M(d,r)\) be the moduli space of semistable sheaves of rank 0, Euler characteristic \(r\) and first Chern class \(dH~(d>0)\), with \(H\) the hyperplane class in \(\mathbb {P}^2\). In [14] we gave an explicit description of the class \([M(d,r)]\) of \(M(d,r)\) in the Grothendieck ring of varieties for \(d\le 5\) and \(g.c.d(d,r)=1\). In this paper we compute the fixed locus of \(M(d,r)\) under some \((\mathbb {C}^{*})^2\)-action and show that \(M(d,r)\) admits an affine paving for \(d\le 5\) and \(g.c.d(d,r)=1\). We also pose a conjecture that for any \(d\) and \(r\) coprime to \(d\), \(M(d,r)\) would admit an affine paving.

Keywords

Moduli spaces of 1-dimensional semistable sheaves on the projective plan Affine pavings Cellular decompositions 

Mathematics Subject Classification

14J 

Notes

Acknowledgments

I was partially supported by NSFC grant 11301292. When I wrote this paper, I was a post-doc at MSC in Tsinghua University in Beijing. Finally I thank Y. Hu for some helpful discussions.

References

  1. 1.
    Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. 98, 480–497 (1973)CrossRefGoogle Scholar
  2. 2.
    Choi, J., Maican, M.: Torus action on the moduli spaces of plane sheaves. arXiv:1304.4871
  3. 3.
    Drézet, J.-M., Maican, M.: On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics. Geom. Dedicata 152, 17–49 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Katz, S., Klemm, A., Vafa, C.: M-Theory, Topological Strings and Spinning Black Holes. Adv. Theor. Math. Phys. 3, 1445–1537 (1999)MathSciNetGoogle Scholar
  5. 5.
    Le Potier, J.: Faisceaux Semi-stables de dimension \(1\) sur le plan projectif. Rev. Roumaine Math. Pures Appl. 38(7–8), 635–678 (1993)MathSciNetGoogle Scholar
  6. 6.
    Maican, M.: On two notions of semistability. Pacific J. Math. 234, 69–135Google Scholar
  7. 7.
    Maican, M.: The homology groups of certain moduli spaces of plane sheaves. Int. J. Math. 24(12), 1350098 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Maican, M.: On the homology of the moduli space of plane sheaves with Hilbert polynomial 5m\(+\)3, arXiv:1305.5511
  9. 9.
    Pandharipande, R., Thomas, R.P.: Curve counting via stable pairs in the derived category. Invent. Math. 178, 407–447 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Toda, Y.: Curve counting theories via stable objects I. DT/PT correspondece. J. Am. Math. Soc. 23, 1119–1157 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Annalen 321, 817–884 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Yuan, Y.: Estimates of sections of determinant line bundles on Moduli spaces of pure sheaves on algebraic surfaces. Manuscripta Math. 137(1), 57–79 (2012)Google Scholar
  13. 13.
    Yuan, Y.: Determinant line bundles on Moduli spaces of pure sheaves on rational surfaces and strange duality. Asian J. Math. 16(3), 451–478 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yuan, Y.: Moduli spaces of semistable sheaves of dimension 1 on \(\mathbb{P}^2\). Pure Appl. Math. Q. arXiv: 1206.4800

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.MSCTsinghua UniversityBeijingChina

Personalised recommendations