2021/07/05 by Tran N. K. Linh, Le Ngoc Long, Le N. Long +11
Computer Science · Mathematics · #13C13 #13C40 #14M05 #14M10 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:13C13 #msc:13C40 #msc:14M05 #msc:14M10
paper · pdf · doi:10.48550/arxiv.2107.02231
18 pages, 1 figure
arxiv created 2021/07/05 · openalex publication_date 2021/07/05 · arxiv updated 2021/07/07 · openalex created_date 2021/07/19 · openalex updated_date 2026/07/28
Given an ACM set \mathbbX of points in a multiprojective space ℙm×ℙn over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for \mathbbX. In ℙ1×ℙ1, 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 ℙm×ℙn 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 \mathbbX is a complete intersection of type (d1,...,dm,d'1,...,d'n) if and only if it has the Cayley-Bachrach property and the Kähler different is non-zero at a certain degree. When \mathbbX has the (⋆)-property, we characterize the Cayley-Bacharach property of \mathbbX in terms of its components under the canonical projections.