Bulletin of the
Korean Mathematical Society BKMS

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 $(d_1,\ldots,d_m,d'_1,\ldots,d'_n)$ if and only if it has the Cayley-Bacharach property and the K\"ahler different is non-zero at a certain degree. We characterize the Cayley-Bacharach property of $\mathbb{X}$ under certain assumptions.

Keywords: ACM set of points, complete intersection, Cayley-Bacharach property, K\"ahler different

MSC numbers: Primary 13C40, 14M05; Secondary 13C13, 14M10

Supported by: The first author and the last two authors were supported by University of Education - Hue University under grant number T.20-TN.SV-01. The second author was partially supported by the Program for Research Activities of Senior Researchers of VAST under the grant number NVCC01.11/21-21. The second and third authors were partially supported by Hue University, Grant No. NCM.DHH.2020.15 and DHH2021-03-159.