An Artinian Quotient of the Coordinate Ring of a Complete Intersection in ℙⁿ Having the Weak Lefschetz Property

Abstract

For a complete intersection set of points in a projective space ℙ ⁿ over an algebraically closed field, we prove that the h-vector of the graded Artinian ring R/(I _𝕏 + I _𝕐) is an SI-sequence provided 𝕏 is also a complete intersection subset of ℤ and 𝕐 = ℤ \ 𝕏. Furthermore, we also show that if {𝕏, 𝕐} is a regular division of a general basic configuration ℤ in ℙ ⁿ , then R/(I _𝕏 + I _𝕐) has the Weak Lefschetz property. These two results give positive answers to the interesting question of [39].

Key Words:

• AI-sequence
• Generic initial ideal
• Gorenstein sequence
• Hilbert function
• h-vector
• Unimodal
• Weak Lefschetz property

2010 Mathematics Subject Classification:

• Primary: 13A02
• Secondary: 16W50

ACKNOWLEDGMENT

The authors are truly thankful to the reviewer, whose comments and suggestions enabled us to make substantial improvements in the paper.

Notes

Communicated by I. Swanson.