Star-Configurations in ℙ² Having Generic Hilbert Function and the Weak Lefschetz Property

Abstract

We prove that a star-configuration 𝕏 in ℙ² is defined by general forms of degrees ≤2 if and only if 𝕏 has generic Hilbert function. We also show that if 𝕏 and 𝕐 are star-configurations in ℙ² defined by general forms of degrees ≤2 and σ(𝕏) ≠ σ(𝕐), then the ring R/(I _𝕏 + I _𝕐) has the Weak Lefschetz property. These two results generalize results of Ahn and Shin [3]. Furthermore, we find the Lefschetz element of the graded Artinian ring R/(I _𝕏 + I _ℤ) precisely when 𝕏 and ℤ are two star-configurations in ℙ² defined by general forms F ₁,…, F _s , and G ₁,…, G _s , L, respectively, with deg F _i  = deg G _i  = 2 for every i ≥ 1, and deg L = 1 with s ≥ 3.

Key Words:

• Hilbert functions
• Minimal free resolutions
• Lefschetz elements
• Star-Configurations
• Weak Lefschetz Property

2010 Mathematics Subject Classification:

• Primary 13P25
• Secondary 14Q99

ACKNOWLEDGMENTS

The author would like to express his sincere thanks to Professors Anthony V. Geramita, Jeaman Ahn, and Jung Pil Park with whom he has had interesting conversations about the problems discussed in this article. The author is also truly grateful to the reviewer, whose comments and suggestions enabled me to make improvements to the article. This research was supported by a grant from Sungshin Women's University in 2012 (2012-1-21-005).

Notes

Communicated by I. Swanson.