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 $h^1(\mathcal{I}_X(t))=0$.

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