The K\"ahler Different of a Set of Points in $\mathbb{P}^m\times\mathbb{P}^n$

TL;DR

This paper investigates the Kähler different and Cayley-Bacharach property for ACM point sets in multiprojective spaces, establishing new characterizations and exploring their relationships beyond the classical case of  .

Contribution

It extends the understanding of the Kähler different and Cayley-Bacharach property to higher-dimensional multiprojective spaces, providing new characterizations for complete intersections.

Findings

Complete intersection characterization via Kähler different and Cayley-Bacharach property.

The Kähler different is non-zero at a specific degree for certain complete intersections.

Cayley-Bacharach property characterized through components under projections when -property holds.

Abstract

Given an ACM set $\mathbb{X}$ of points in a multiprojective space $\mathbb{P}^m\times\mathbb{P}^n$ over a field of characteristic zero, we are interested in studying the K\"ahler different and the Cayley-Bacharach property for $\mathbb{X}$. In $\mathbb{P}^1\times\mathbb{P}^1$, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the K\"ahler different. However, this result fails to hold in $\mathbb{P}^m\times\mathbb{P}^n$ for $n>1$ or $m>1$. In this paper we start an investigation of the K\"ahler different and its Hilbert function and then prove that $\mathbb{X}$ is a complete intersection of type $(d_1,...,d_m,d'_1,...,d'_n)$ if and only if it has the Cayley-Bachrach property and the K\"ahler different is non-zero at a certain degree. When $\mathbb{X}$ has the $(\star)$-property, we characterize the Cayley-Bacharach property of $\mathbb{X}$ in terms of its components under the canonical projections.