The initial degree of symbolic powers of Fermat-like ideals of planes and lines arrangements

Demailly's Conjecture and the containment problem

“…Chang and Jow [CJ20] verified the conjecture for s = k N general points, where k ∈ N. The authors along with Grifo and Hà [BGHN22a] proved that for any fixed m ∈ N, any set of at least (2m + 2) N general points verifies Demailly's conjecture. It was also shown to be true for special configurations of points including the star configurations of points [MSS18,BGHN22a], (a more elementary proof for) the Fermat configurations and other configurations of points in P 2 [Ngu23b], and even beyond the points setting including determinantal ideals [BGHN22a], star configurations of hypersurfaces [BGHN22a] (also known as the uniform a−fold product ideal, see, Lin and Shen [LS22]), and special configurations of lines [Ngu23a].…”

“…Besides classical applications in complex analysis, see for instance [Wal77,Mor80], Demailly's bound was shown to be beneficial for the study of the ideal containment problem [TX20], as well as computing the values α(I (m) X ) for special configurations [Ngu23a,Ngu23b]. Demailly's bound also gives rise to upper bounds for the regularity values of a quotient by ideals generated by powers of linear forms [DNS23], which is useful to study dimensions of certain spline spaces [DV21].…”