Abstract
We show the kernel sheaf associated to a sufficiently positive torsion-free sheaf of rank one is slope stable. Furthermore, we are able to give an explicit bound for “sufficiently positive.” This settles a conjecture of Ein–Lazarsfeld–Mustopa. The main technical lemma is a bound on the number of global sections of a globally generated, torsion-free sheaf in terms of its rank, degree, and invariants of the variety.
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Acknowledgements
The author is thankful to Rajesh Kulkarni and Yusuf Mustopa for many useful discussions. The author is also thankful to Federico Caucci, Peter Newstead, and Shitan Xu for comments on an earlier draft of this paper. Last, the author is thankful to the anonymous referee for feedback that has vastly improved this paper.
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The author was partially supported by the NSF grant DMS-2101761 during preparation of this article. The author is supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award Number DE-SC-SC0022134. This work is also partially supported by an OVPR Postdoctoral Award at Wayne State University.
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Rekuski, N. Stability of kernel sheaves associated to rank one torsion-free sheaves. Math. Z. 307, 2 (2024). https://doi.org/10.1007/s00209-024-03475-y
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DOI: https://doi.org/10.1007/s00209-024-03475-y