Abstract
Given a fat point scheme \(\mathbb {W}=m_{1}P_{1}+\cdots +m_{s}P_{s}\) in the projective n-space \(\mathbb {P}^{n}\) over a field K of characteristic zero, the modules of Kähler differential k-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of \(\mathbb {W}\) when \(k\in \{1,\dots , n+1\}\). In this paper, we determine the value of its Hilbert polynomial explicitly for the case k = n + 1, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme \(\mathbb {Y} = (m_{1}-1)P_{1} + {\cdots } + (m_{s}-1)P_{s}\). For n = 2, this allows us to determine the Hilbert polynomials of the modules of Kähler differential k-forms for k = 1,2,3, and to produce a sharp bound for the regularity index for k = 2.
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The second author thanks the University of Passau for its hospitality and support during part of the preparation of this paper. The authors thank the referee for his/her careful reading of the paper.
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The authors were supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.07.
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Kreuzer, M., Linh, T.N.K. & Long, L.N. Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes. Acta Math Vietnam 46, 441–455 (2021). https://doi.org/10.1007/s40306-021-00432-3
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DOI: https://doi.org/10.1007/s40306-021-00432-3