Symbolic powers of generalized star configurations of hypersurfaces

TL;DR

This paper introduces sparse symmetric shifted monomial ideals, computes their Betti numbers, and proves that symbolic powers of generalized star configuration ideals are sequentially Cohen–Macaulay, addressing containment problems and bounds.

Contribution

It defines a new class of ideals, computes their algebraic invariants, and establishes Cohen–Macaulay properties for symbolic powers of star configurations.

Findings

Symbolic powers are sequentially Cohen–Macaulay under mild conditions

Betti numbers of the introduced ideals are explicitly computed

Demailly-like bounds are established for containment problems

Abstract

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.