Differential theory of zero-dimensional schemes

Abstract

For a 0-dimensional scheme X in P n over a perfect field K, 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 differentialsto find a formula for the Hilbert polynomial HP(Ω 1 R/K ) and a sharp bound for the regularity index ri(Ω 1 R/K ). Additionally, we extend this to formulas for the Hilbert polynomials HP(Ω m R/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 X which can be checked without computing a primary decomposition of its vanishing ideal I X . Moreover, we prove precise formulas for the Hilbert polynomial of Ω m R/K of a fat point scheme X, extending and settling previous partial results and conjectures. Finally, we characterize uniformity conditions on X using the Hilbert functions of the Kähler differential modules of X and its subschemes.