Differential theory of zero-dimensional schemes

• Published in: arXiv.org 2023-02
• Main Authors: Kreuzer, Martin, Linh, Tran N K, Long, Le N
• Format: Article
• Language: English
• Subjects: Modules; Polynomials; Regularity
• ISSN: 2331-8422

For a 0-dimensional scheme \(\mathbb{X}\) in \(\mathbb{P}^n\) over a perfect field \(K\), we first embed the homogeneous coordinate ring \(R\) into its truncated integral closure \(\widetilde{R}\). Then we use the corresponding map from the module of K\"ahler differentials \(\Omega^1_{R/K}\) to \(\Omega^1_{\widetilde{R}/K}\) to find a formula for the Hilbert polynomial \({\rm HP}(\Omega^1_{R/K})\) and a sharp bound for the regularity index \({\rm ri}(\Omega^1_{R/K})\). Additionally, we extend this to formulas for the Hilbert polynomials \({\rm HP}(\Omega^m_{R/K})\) and bounds for the regularity indices of the higher modules of K\"ahler differentials. Next we derive a new characterization of a weakly curvilinear scheme \(\mathbb{X}\) which can be checked without computing a primary decomposition of its homogeneous vanishing ideal. Moreover, we prove precise formulas for the Hilbert polynomial of \(\Omega^m_{R/K}\) of a fat point scheme \(\mathbb{X}\), extending and settling previous partial results and conjectures. Finally, we characterize uniformity conditions on \(\mathbb{X}\) using the Hilbert functions of the K\"ahler differential modules of \(\mathbb{X}\) and its subschemes.