On the Hilbert Function of General Unions of Curves in Projective Spaces

Let \(X=X_1\cup \cdots \cup X_s\subset \mathbb {P}^n\), \(n\ge 4\), be a general union of smooth non-special curves with \(X_i\) of degree \(d_i\) and genus \(g_i\) and \(d_i\ge \max \{2g_i-1,g_i+n\}\) if \(g_i>0\). We prove that X has maximal rank, i.e., for any \(t\in \mathbb {N}\) either \(h^0(\mathcal {I}_X(t))=0\) or \(h^1(\mathcal {I}_X(t))=0\) if it is so in a few explicit cases in \(\mathbb {P}^4\). We also prove an unconditional weaker result, maximal rank up to a positive integer \(\delta _n\).