Projective Curves With Nice Normal Bundles and Containing a Prescribed Subset of a Hyperplane

Abstract

Fix a hyperplane $H\subset \mathbb{P}^n$, $n>3$, and a finite set $S\subset H$. We give conditions on the integers $d$, $g$ and $\sharp (S)$ such that there exists a smooth and connected curve $X\subset \mathbb{P}^n$ with $\deg (X) =d$, $p_a(X)=g$ and $S\subset X\cap H$. When $d=\sharp (S)$ we may take $g$ up to order $2d/n$, $d\gg 0$, when $S$ is in linear general position. We also prove the existence of $X$ with $h^1(N_X(-1))=0$ if $n\ge 8$, $g$ is odd and $2d\ge (n-3)g+n+11$.