Curves in projective spaces and their index of regularity

Abstract

For all integers $n \ge 3$ we show the existence of many triples $(d,g,\rho)$ such that there is a smooth non-degenerate curve $C \subset \mathbf{P}^n$ with degree $d$, genus $g$ and index of regularity $\rho$. The curve $C$ lies in a smooth $K3$ surface $S \subset \mathbf{P}^n$.