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$\mathbb{P}^{n}$, 
$n\geqslant 8$
We prove the existence of a smooth and non-degenerate curve 
$X\subset \mathbb{P}^{n}$, 
$n\geqslant 8$, with 
$\deg (X)=d$, 
$p_{a}(X)=g$, 
$h^{1}(N_{X}(-1))=0$, and general moduli for all 
$(d,g,n)$ such that 
$d\geqslant (n-3)\lceil g/2\rceil +n+3$. It was proved by C. Walter that, for 
$n\geqslant 4$, the inequality 
$2d\geqslant (n-3)g+4$ is a necessary condition for the existence of a curve with 
$h^{1}(N_{X}(-1))=0$.
