Differential theory of zero-dimensional schemes

To study a 0-dimensional scheme X in P n over a perfect field K, we use the module of Kahler differentials Omega(1)(R/K) of its homogeneous coordinate ring R and its exterior powers, the higher modules of Kahler differentials Omega(m)(R/K). One of our main results is a characterization of weakly curvilinear schemes X by the Hilbert polynomials of the modules Omega(m)(R/K) which allows us to check this property algorithmically without computing the primary decomposition of the vanishing ideal of X. Further main achievements are precise formulas for the Hilbert functions and Hilbert polynomials of the modules Omega(m)(R/K) for a fat point scheme X which extend and settle previous partial results and conjectures. Underlying these results is a novel method: we first embed the homogeneous coordinate ring R into its truncated integral closure (R) over tilde . Then we use the corresponding map from the module of Kahler differentials Omega(1)(R/K) to Omega(1)(R/K) to find a formula for the Hilbert polynomial HP(Omega(1)(R/K))and a sharp bound for the regularity index ri(Omega(1)(R/K)). Next we extend this to formulas for the Hilbert polynomials HP(Omega(m)(R/K)) and bounds for the regularity indices of the higher modules of Kahler differentials. As a further application, we characterize uniformity conditions on X using the Hilbert functions of the Kahler differential modules of X and its subschemes. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org /licenses/by-nc-nd/4.0/).