Differential theory of zero-dimensional schemes

Abstract

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 $\tilde{R}$. Then we use the corresponding map from the module of Kähler differentials ${\Omega }_{R/K}^1$ to ${\Omega }_{\tilde{R}/K}^1$ to find a formula for the Hilbert polynomial $\mathrm{HP}({\Omega }_{R/K}^1)$ and a sharp bound for the regularity index $\mathrm{ri}({\Omega }_{R/K}^1)$. Additionally, we extend this to formulas for the Hilbert polynomials $\mathrm{HP}({\Omega }_{R/K}^m)$ and bounds for the regularity indices of the higher modules of Kähler 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 ${\Omega }_{R/K}^m$ 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 Kähler differential modules of $\mathbb{X}$ and its subschemes.

32 pages