Curves in projective spaces and their index of regularity (Q2504131)

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