Abstract
In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus g is greater than \(2g+2k+p\) for nonnegative integers k and p, then the k-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen–Macaulay, and satisfies the property \(N_{k+2, p}\). In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo–Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.
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Acknowledgements
The authors would like to thank Robert Lazarsfeld for helpful suggestions and useful comments. The authors also wish to express their gratitude to Adam Ginensky for bringing the problems considered in this paper to our attention and to Jürgen Rathmann for his work in the paper [16]. The authors are very grateful to the referee for careful reading of the paper and valuable suggestions to help improve the exposition of the paper.
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L. Ein was partially supported by NSF Grant DMS-1801870. J. Park was partially supported by NRF-2016R1C1B2011446 and the Sogang University Research Grant of 201910002.01.
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Ein, L., Niu, W. & Park, J. Singularities and syzygies of secant varieties of nonsingular projective curves. Invent. math. 222, 615–665 (2020). https://doi.org/10.1007/s00222-020-00976-5
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DOI: https://doi.org/10.1007/s00222-020-00976-5