Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online March 16, 2022

Copyright © The Korean Mathematical Society.

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

Nguyen T. Hoa, Tran N.K. Linh, Le N. Long, Phan T.T. Nhan, and Nguyen T.P. Nhi

University of Education - Hue university, University of Passau

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 Kähler 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 Kaehler 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 Kähler 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-Bacharach property and
the Kähler 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.

Keywords: ACM set of points, complete intersection, Cayley-Bacharach property, Kähler different

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

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