Abstract
This paper studies deformations and birational maps between singular moduli spaces of torsion free semistable sheaves with 2-divisible Mukai vectors on K3 surfaces. It is showed that when the greatest common divisor of the rank and the first Chern class is 2, two such moduli spaces of the same dimension can be connected by deformations and birational maps.
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References
Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. Math. (2) 106(1), 45–60 (1977)
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig (1997)
Huybrechts, D.: The Kähler cone of a compact hyperkähler manifold. Math. Ann. 326(3), 499–513 (2003)
Kaledin, D., Lehn, M., Sorger, C.: Singular symplectic moduli spaces. Invent. Math. 164(3), 591–614 (2006)
Maruyama, M.: Moduli of stable sheaves. I. J. Math. Kyoto Univ. 17(1), 91–126 (1977)
Maruyama, M.: Moduli of stable sheaves. II. J. Math. Kyoto Univ. 18(3), 557–614 (1978)
Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or \(K3\) surface. Invent. Math. 77(1), 101–116 (1984)
O’Grady, K.: Desingularized moduli spaces of sheaves on a \(K3\). J. Reine Angew. Math. 512, 49–117 (1999)
Perego, A.: A Note on Deformations of Moduli Spaces of Sheaves on K3 Surfaces. ArXiv preprint, February 2011. ArXiv:1102.1847 [math.AG]
Perego, A., Rapagnetta, A.: Deformation of the O’Grady Moduli Spaces. ArXiv, preprint, August 2010. ArXiv:1008.0190 [math.AG]
Yoshioka, K.: Irreducibility of Moduli Spaces of Vector Bundles on k3 Surfaces. Arxiv, preprint, July 1999. ArXiv:math/9907001v2
Yoshioka, K.: Some examples of Mukai’s reflections on \(K3\) surfaces. J. Reine Angew. Math. 515, 97–123 (1999)
Yoshioka, K.: A Note on Fourier–Mukai Transform. ArXiv preprint, December 2001. ArXiv:math/ 0112267
Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001)
Acknowledgments
I would like to express my deep gratitude to my Ph.D. advisor Jun Li, for all his support and encouragement. I would also like to thank Kōta Yoshioka for kindly answering my questions and pointing out references, and thank Jason Lo for helpful discussions. I also appreciate the help of Daniel Huybrechts and Manfred Lehn in the final stage of this work. Moreover, I would like to thank the referee for carefully reading the previous version of the manuscript and suggesting various improvements, which in particular leads to a few simplifications in Sect. 3.
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Zhang, Z. A note on singular moduli spaces of sheaves on K3 surfaces. Geom Dedicata 173, 347–363 (2014). https://doi.org/10.1007/s10711-013-9946-y
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DOI: https://doi.org/10.1007/s10711-013-9946-y