Genus bound of curves on surfaces of almost minimal degree

Let $C\subset P^r$, $r\ge 3$, be a nondegenerate projective integral curve of degree d and arithmetic genus g. Castelnuovo theory says that

• (i)
  if $g>{\pi }_1(d,r)$ then $C$ is contained in a surface of minimal degree, and
• (ii)
  if $g>{\pi }_2(d,r)$ then $C$ is contained in a surface of degree ≤r.

In this paper, we prove that if $g>{\pi }_0(d,r+1)+1$ then $C$ is contained in a surface of minimal degree or a del Pezzo surface. To this aim, we show that ${\pi }_0(d,r+1)+1$ is the upper bound of g when $C$ lies on a surface of degree r which is not a del Pezzo surface. We also provide a specific construction of curves with genus equal to the upper bound ${\pi }_0(d,r+1)+1$.