Resurgences for ideals of special point configurations in PNPN coming from hyperplane arrangements

• We study two asymptotic invariants for ideals of point sets in projective space.
• The square of the ideals studied do not contain their symbolic cubes.
• We give the first examples where the two invariants are not equal.

Symbolic powers of ideals have attracted interest in commutative algebra and algebraic geometry for many years, with a notable recent focus on containment relations between symbolic powers and ordinary powers; see for example [3], [7], [13], [16], [18], [19] and [20] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [3] and [15].There have been exciting new developments in this area recently. It had been expected for several years that I(Nr−N+1)⊆IrI(Nr−N+1)⊆Ir should hold for the ideal I   of any finite set of points in PNPN for all r>0r>0, but in the last year various counterexamples have now been constructed (see [11], [17] and [8]), all involving point sets coming from hyperplane arrangements. In the present work, we compute their resurgences and obtain in particular the first examples where the resurgence and the asymptotic resurgence are not equal.