On the Hilbert function of lines union one non-reduced point (Q2807092)

If \(P\) is a point in \(\mathbb{P}^n\) whose ideal is \(I_P\subset k[x_0,\dots,x_n]\), with \(k\) algebraically closed of characteristic zero, the scheme supported on \(P\) defined by \((I_P)^{m}\) is called an \(m\)-multiple point supported on \(P\).NEWLINENEWLINEIn this paper, the authors completely solve the study of determining the Hilbert function of schemes \(X\subset \mathbb{P}^n\) which are the generic union of \(s\) lines and one \(m\)-multiple point for any \(s\) and \(m\) when \(n\geq 4\) (Theorem 3.2).NEWLINENEWLINESection 4 is devoted to the case \(n=3\) (see Theorem 4.2).