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

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