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.