Differential theory of zero-dimensional schemes

Abstract

To study a 0-dimensional scheme $X$ in $P^n$ over a perfect field K, we use the module of Kähler differentials ${\Omega }_{R/K}^1$ of its homogeneous coordinate ring R and its exterior powers, the higher modules of Kähler differentials ${\Omega }_{R/K}^m$. One of our main results is a characterization of weakly curvilinear schemes $X$ by the Hilbert polynomials of the modules ${\Omega }_{R/K}^m$ 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 ${\Omega }_{R/K}^m$ 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 $\tilde{R}$. 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 $\mathrm{HP}\left({\Omega }_{R/K}^1\right)$ and a sharp bound for the regularity index $\mathrm{ri}\left({\Omega }_{R/K}^1\right)$. Next we extend this to formulas for the Hilbert polynomials $\mathrm{HP}\left({\Omega }_{R/K}^m\right)$ 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.