Symbolic powers of generalized star configurations of hypersurfaces

We introduce the class of sparse symmetric shifted monomial ideals. These ideals have linear quotients and their Betti numbers are computed. Using this, we prove that the symbolic powers of the generalized star configuration ideal are sequentially Cohen–Macaulay under some mild genericness assumption. With respect to these symbolic powers, we also consider the Harbourne–Huneke containment problem and establish the Demailly-like bound.

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Work Title Symbolic powers of generalized star configurations of hypersurfaces
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Open Access
Creators
  1. Kuei-Nuan Lin
  2. Yi-Huang Shen
License CC BY-NC-ND 4.0 (Attribution-NonCommercial-NoDerivatives)
Work Type Article
Publisher
  1. Elsevier BV
Publication Date March 2022
Publisher Identifier (DOI)
  1. 10.1016/j.jalgebra.2021.11.015
Source
  1. Journal of Algebra
Deposited January 13, 2022

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    Description
    • As a generalization of the ideals of star configurations of hypersurfaces, we<br> consider the a-fold product ideal I_a(f_1^{m_1} ···f_s^{m_s}) when f_1 ,...,f_s is a sequence of n-generic<br> forms and 1 ≤ a ≤ m_1 +· · ·+m_s. Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially linear. Then we study its symbolic powers while focusing on the uniform case with m_1 = ··· = m_s. For large a, we describe its resurgence and symbolic defect. And for general a, we also investigate the corresponding invariants for meeting-at-the-minimal-components version of symbolic powers.<br>
    • We introduce the class of sparse symmetric shifted monomial ideals. These ideals have linear quotients and their Betti numbers are computed. Using this, we prove that the symbolic powers of the generalized star configuration ideal are sequentially Cohen–Macaulay under some mild genericness assumption. With respect to these symbolic powers, we also consider the Harbourne–Huneke containment problem and establish the Demailly-like bound.
  • Updated