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.
Categories: math.AC math.AG