Narrow Results

The purpose of this paper is to study the containment problem for the corresponding ideal $I_Z$ of a special zero dimensional subscheme $Z$ in ${\mathbb{P}}^2$, what we call a fat nearly-complete intersection. We show that for every positive integer $m$, the equality $I_Z^{(m)}=I_Z^m$ holds.

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg.220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of $(N-2)$-planes $Z$ in ${\mathbb{P}}^N$ with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals $I(Z)$ of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of $I(Z)$, as well as the Hilbert function of $Z$.