2023/02/23 by Martin Kreuzer, Kreuzer, Martin, Tran N. K. Linh +3 · 1 citation
Mathematics · #13N05 (Primary) 13D40 #14N05 (Secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2302.11903
openalex publication_date 2023/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a 0-dimensional scheme \mathbbX in ℙn over a perfect field K, we first embed the homogeneous coordinate ring R into its truncated integral closure \widetildeR. Then we use the corresponding map from the module of Kähler differentials Ω1R/K to Ω1_\widetildeR/K to find a formula for the Hilbert polynomial \rm HP(Ω1R/K) and a sharp bound for the regularity index \rm ri(Ω1R/K). Additionally, we extend this to formulas for the Hilbert polynomials \rm HP(ΩmR/K) and bounds for the regularity indices of the higher modules of Kähler differentials. Next we derive a new characterization of a weakly curvilinear scheme \mathbbX which can be checked without computing a primary decomposition of its homogeneous vanishing ideal. Moreover, we prove precise formulas for the Hilbert polynomial of ΩmR/K of a fat point scheme \mathbbX, extending and settling previous partial results and conjectures. Finally, we characterize uniformity conditions on \mathbbX using the Hilbert functions of the Kähler differential modules of \mathbbX and its subschemes.