Graded betti numbers and shifts of the symbolic power of a general star configuration in ℙⁿ

Abstract

In [37] the author finds a graded minimal free resolution of the 2-nd order symbolic power of a general star configuration in ${\mathrm{\mathbb{P}}}^n$ of any codimension r, which generalizes the result of Galetto et al. in [18, Theorem 5.3], and of Geramita et al. in [20, Theorem 3.2]. In [33] we find a graded minimal free resolution of the m-th order symbolic power of a general star configuration in ${\mathrm{\mathbb{P}}}^n$ of codimension r for m = 3, 4 with $m\le r,$ and also find them for codimension 2 with any $m\ge 2.$ In this paper, we show how to find a closed formula of graded Betti numbers and shifts of the symbolic power of a general star configuration in ${\mathrm{\mathbb{P}}}^n$ of any codimension r based on representation theory and matroid configurations for $5\le m\le r,$ which generalizes the result of Biermann et al. in [3, Corollary 5.7] and [31, Proposition 7.9]. We also show an example of our methods to find a closed formula of their graded Betti numbers and shifts for any codimension r with $m>r\ge 2.$

Keywords and phrases:

• symbolic powers
• regular powers
• star configurations
• a graded minimal free resolution
• Betti numbers
• shifts

2010 Mathematics Subject Classification:

• 13A15
• 14M05

Acknowledgement

The authors are grateful to the reviewer for their meticulous comments and suggestions on this paper, which are helpful to substantially improve the original paper.

Additional information

Funding

This research was supported by a grant from Sungshin Women’s University.