Differential theory of zero-dimensional schemes
Martin Kreuzer, Tran N. K. Linh, Le Ngoc Long · Journal of Pure and Applied Algebra · 2024
To study a 0-dimensional scheme |X in |P^n over a perfect field K, we use the module of Kähler differentials (Omega)(^1)(_{R/K}) of its homogeneous coordinate ring R and its exterior powers, the higher modules of Kähler 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. Then we use the corresponding map from the module of Kähler 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 Kähler differentials. As a further application, we characterize uniformity conditions on |X using the Hilbert functions of the Kähler differential modules of |X and its subschemes.