Abstract
In this paper, we introduce the notion of a Klyachko diagram for a monomial ideal I in a certain multi-graded polynomial ring, namely the Cox ring R of a smooth complete toric variety, with irrelevant maximal ideal B. We present procedures to compute the Klyachko diagram of I from its monomial generators, and to retrieve the B −saturation Isat of I from its Klyachko diagram. We use this description to compute the first local cohomology module \({H}^{1}_{B}(I)\). As an application, we find a formula for the Hilbert function of Isat, and a characterization of monomial ideals with constant Hilbert polynomial, in terms of their Klyachko diagram.
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Acknowledgements
The first author was partially supported by PID2019-104844GB-I00. The second author is partially supported by MDM-2014-0445-18-2.
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Miró-Roig, R.M., Salat-Moltó, M. Klyachko Diagrams of Monomial Ideals. Algebr Represent Theor 26, 1497–1517 (2023). https://doi.org/10.1007/s10468-022-10146-1
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DOI: https://doi.org/10.1007/s10468-022-10146-1