Abstract
Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface and moduli spaces of Higgs bundles as well. We estimate the dimension of the stack Mx (nC, χ) of pure sheaves supported at the non-reduced curve nC (n ≽ 2) with C an integral curve on X. We prove that the Hilbert-Chow morphism \({h_{L,\chi}}:{\cal M}_X^H\left({L,\chi} \right) \to \left| L \right|\) sending each semistable 1-dimensional sheaf to its support has all its fibers of the same dimension for X Fano or with the trivial canonical line bundle and |L| contains integral curves.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 21022107 and 11771229). I thank Lothar Göttsche for leading me to study 1-dimensional sheaves over surfaces when I was his PhD student. I also thank Junliang Shen for answering my questions on their paper [5]. I thank the referees for all the helpful comments and advices.
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Yuan, Y. Sheaves on non-reduced curves in a projective surface. Sci. China Math. 66, 237–250 (2023). https://doi.org/10.1007/s11425-021-1964-4
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DOI: https://doi.org/10.1007/s11425-021-1964-4