The regularity index of up to \(2n-1\) equimultiple fat points of \(\mathbb{P}^n\)
For any \(P\in \mathbb {P}^n\) and any integer \(m>0\) let \(mP\) be the closed subscheme of \(\mathbb {P}^n\) with \((\mathcal {I}_P)^m\) as its ideal sheaf. A fat scheme \(X\subset \mathbb {P}^n\) is a finite disjoint union of schemes \(m_iP_i\). Trung and, independently, Fatabbi and Lorenzini, conjectured a sharp upper bound for the regularity index of a fat scheme (it is called Segre's bound). In this paper the authors give an affermative answer to the equimultiple case (all \(m_i\) are the same) for a low number of points. Later, Segre's conjecture was solved by \textit{U. Nagel} and \textit{B. Trok} [``Segre's regularity bound for fat points schemes'', Preprint, \url{arXiv:1611.06279}].
scientific article; zbMATH DE number 6676884