Abstract
Let M d be the moduli space of stable sheaves on ℙ2 with Hilbert polynomial dm+1. In this paper, we determine the effective and the nef cone of the space M d by natural geometric divisors. Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem. We also present the stable base locus decomposition of the space M 6. As a byproduct, we obtain the Betti numbers of the moduli spaces, which confirm the prediction in physics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arcara D, Bertram A, Coskun I, et al. The minimal model program for the Hilbert scheme of points on ℙ2 and bridgeland stability. Adv Math, 2013, 235: 580–626
Bayer A, Macrì E. Projectivity and birational geometry of bridgeland moduli spaces. J Amer math Soc, 2014, 27: 707–752
Bertram A, Martinez C, Wang J. The birational geometry of moduli space of sheaves on the projective plane. Geom Dedicata, arXiv:1301.2011, 2013
Choi J, Chung K. Cohomology bounds for sheaves of dimension one. ArXiv:1302.3691, 2013
Choi J, Chung K. Moduli spaces of α-stable pairs and wall-crossing on ℙ2. ArXiv:1210.2499, 2012
Choi J, Katz S, Klemm A. The refined BPS index from stable pair invariants. Commun Math Phys, 2014, 328: 903–954
Choi J, Maican M. Torus action on the moduli spaces of plane sheaves of multiplicity four. J Geom Phys, 2014, 83: 18–35
Drézet J-M. Cohomologie des variétés de modules de hauteur nulle. Math Ann, 1988, 281: 43–85
Drézet J-M, Trautmann G. Moduli spaces of decomposable morphisms of sheaves and quotients by non-reductive groups. Ann Inst Fourier, 2003, 53: 107–192
Ellingsrud G, Strømme S A. On the Chow ring of a geometric quotient. Ann Math (2), 1989, 130: 159–187
Huang M-X, Kashani-Poor A-K, Klemm A. The Omega deformed B-model for rigid N=2 theories. Ann Henri Poincaré, 2013, 14: 425–497
Huybrechts D, Lehn M. The Geometry of Moduli Spaces of Sheaves. New York: Springer-Verlag, 1997
Iena O. Universal plane curve and moduli spaces of 1-dimensional coherent sheaves. Commun Algebra, arXiv:1103.1485, 2011
Katz S, Klemm A, Vafa C. M theory, topological strings and spinning black holes. Adv Theor Math Phys, 1999, 3: 1445–1537
Kouvidakis A. Divisors on symmetric products of curves. Trans Amer Math Soc, 1993, 337: 117–128
Le Potier J. Faisceaux semi-stables de dimension 1 sur le plan projectif. Rev Roumanine Math Appl, 1993, 38: 635–678
Maican M. A duality result for moduli spaces of semistable sheaves supported on projective curves. Rendiconti del Seminario Matematico della Università di Padova, 2010, 123: 55–68
Maican M. On the moduli spaces of semi-stable plane sheaves of dimension one and multiplicity five. Illinois J Math, 2011, 55: 1467–1532
Maican M. The classification of semi-stable plane sheaves supported on sextic curves. Kyoto J Math, 2013, 53: 739–786
Maican M. The homology groups of certain moduli spaces of plane sheaves. Int J Math, 2013, 24: Article ID 1350098, 42pp
Reineke M. The Harder-Narasimhan system in quantum groups and cohomology of quiver. Invent Math, 2003, 152: 349–368
Woolf M. Nef and effective cones on the moduli spaces of torsion sheaves on ℙ2. ArXiv:1305.1465, 2013
Yuan Y. Moduli spaces of semistable sheaves of dimension 1 on ℙ2. ArXiv:1206.4800, 2012
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Choi, J., Chung, K. The geometry of the moduli space of one-dimensional sheaves. Sci. China Math. 58, 487–500 (2015). https://doi.org/10.1007/s11425-014-4889-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-014-4889-9