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

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 Kähler 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 $(d_1,...,d_m,d_1^{\prime },...,d_n^{\prime })$ if and only if it has the Cayley-Bachrach 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.

18 pages, 1 figure

References in corpus (1)

• Multigraded regularity of complete intersections