Abstract. Inspired by results of Guardo, Van Tuyl and the second author for lines in P 3 , we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in P n for n ≥ 2r + 1. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P 2 is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.

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Abstract. 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 [BH1, Cu, ELS, HaHu, HoHu, Hu1, Hu2] to cite just a few. Several invariants have been introduced and studied in the latter context, including the resurgence and asymptotic resurgence [BH1, GHvT].There have been exciting new developments in this area recently. It had been expected for several years that I Nr−N+1 ⊆ I r should hold for the ideal I of any finite set of points in P N for all r > 0, but in the last year various counterexamples have now been constructed (see [DST, HS, C. et al]), 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.