# Differential theory of zero-dimensional schemes

## TL;DR

This paper develops a differential theory for zero-dimensional schemes in projective space, providing formulas for Hilbert polynomials and bounds for regularity indices of Kähler differentials, with applications to scheme characterization.

## Contribution

It introduces new formulas and bounds for Kähler differentials of zero-dimensional schemes, extending previous results and providing novel characterizations without primary decomposition.

## Key findings

- Formulas for Hilbert polynomials of Kähler differentials
- Bounds for regularity indices of differential modules
- Characterization of weakly curvilinear 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 $\widetilde{R}$. 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.