The K\"ahler different of a 0-dimensional scheme

TL;DR

This paper investigates the Kähler different of 0-dimensional schemes in projective space, characterizing properties like generic position and Cayley-Bacharach, and extending classical theorems to this context.

Contribution

It introduces the use of the Kähler different to analyze 0-dimensional schemes, providing new characterizations and generalizations of classical results.

Findings

Characterization of generic position using the Kähler different

Generalized Apéry-Gorenstein-Samuel theorem for arithmetically Gorenstein schemes

Characterization of 0-dimensional complete intersections via the Kähler different

Abstract

Given a 0-dimensional scheme $\mathbb{X}$ in the projective $n$-space $\mathbb{P}^n_K$ over a field $K$, we are interested in studying the K\"ahler different of $\mathbb{X}$ and its applications. Using the K\"ahler different, we characterize the generic position and Cayley-Bacharach properties of $\mathbb{X}$ in several certain cases. When $\mathbb{X}$ is in generic position, we prove a generalized version of the Ap\'{e}ry-Gorenstein-Samuel theorem about arithmetically Gorenstein schemes. We also characterize 0-dimensional complete intersections in terms of the K\"ahler different and the Cayley-Bacharach property.