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\(K_{-1}\) and Higher \({\widetilde{K}}\)-Groups of Cones of Smooth Projective Plane Curves

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Abstract

This is a contribution toward calculating the K-theory of a certain class of affine algebraic surfaces which is made possible due to works by Cortiñas, Haesemeyer, Schlichting, Walker and Weibel. We discover that the negative K-theory and higher reduced \({\widetilde{K}}_n\)-group of the cone of a smooth projective plane curve over number field k, as vector spaces, are determined solely by the degree of the curve (and the ground field of course). In particular, the \({\widetilde{K}}_n\)-groups of the cone for \(n\ge 2\) alternate between zero and the Jacobian ring of the cone.

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Acknowledgements

Parts of the results in this paper already been stated and proved in the unpublished master dissertation of the author [16] during his master study at The University of Melbourne, which was jointly sponsored by Universiti Kebangsaan Malaysia and the Ministry of Higher Education of Malaysia via the scholarship program SLAB. The author personally would like to thank Christian Haesemeyer for his valuable guide and posing of the class of the rings for the author to compute during the author’s undertaking of master research project under his supervision. The author would also like to express his deepest gratitude to the anonymous referee whose series of comments and remarks on the original draft eventually led to the current version of the manuscript with greatly improved readability and content and to thank the referee for pointing out certain miscalculations by the author in some of the previous revisions.

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  1. Jabatan Sains Matematik, Fakulti Sains dan Teknologi, Universiti Kebangsaan Malaysia, 43600, UKM Bangi, Malaysia

    Taufik Yusof

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  1. Taufik Yusof
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Correspondence to Taufik Yusof.

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Communicated by Rosihan M. Ali.

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Yusof, T. \(K_{-1}\) and Higher \({\widetilde{K}}\)-Groups of Cones of Smooth Projective Plane Curves. Bull. Malays. Math. Sci. Soc. 45, 37–56 (2022). https://doi.org/10.1007/s40840-021-01175-y

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  • DOI: https://doi.org/10.1007/s40840-021-01175-y

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