Upper bounds for the regularity index of fat points (Q1899091)

Let \(P_1, \ldots, P_s\) be \(s\) points in \(\mathbb{P}^n\). Given positive integers \(m_1 \geq m_2 \geq \cdots \geq m_s\), we can consider the cycle \(Z = \sum m_i P_i\). That is, take the subscheme \(Z\) of \(\mathbb{P}^n\) defined by the ideal sheaf \(I = \prod {\mathcal I}^{m_j}_j\) where \({\mathcal I}_j\) is the sheaf of ideals defining \(P_j\). The smallest integer \(r(Z)\) such that \(H^1 (\mathbb{P}^n, I(r(Z))) = 0\) is called the regularity index of \(Z\). It is of considerable interest to find good upper bounds for \(r(Z)\) in terms of the \(m_i\)'s and a lot of work has been done in this direction. The study of these cycles is classical, but it has bloomed in the recent times. If the points were arbitrary, then the following bound is known: \(r(Z) \leq (\sum m_i)-1\) and equality if and only if the points lie on a line. The interested reader can find the references in the paper. If one assumes that the points are in general position, then the bound can be improved. Let \(\delta (Z) = [{1 \over n} (\sum^n_{i = 1} m_i + n - 2)]\), where as usual, \([x]\) stands for the integral part of \(x\). Then \(r(Z) \leq \max \{m_1 + m_2 - 1, \delta (Z)\}\). This, the authors call the Segre bound, since Segre proved it for \(n = 2\). This bound is attained if the points lie on a rational normal curve. In this paper, the authors prove that under some mild hypotheses, the converse also holds. i.e. if this bound is attained, then the points lie on a rational normal curve. They also give a better bound if the points are assumed to be in uniform position.