An estimate of the regularity index of fat points in some cases

Abstract. We estimate the regularity index of a set of fat points \(Z=m_1P_1+\cdots +m_sP_s\) in three cases: all points \(P_1, \ldots, P_s\) are on two lines; \(Z\) consists at most five fat points; \(Z=m_1P_1+\cdots+P_{n+3}P_{n+3}\) is non-degenerate in \(\mathbb P^n\).