An improved bound for the degree of smooth surfaces in P4 not of general type.

TL;DR

This paper refines the upper bound on the degree of smooth surfaces in projective 4-space that are not of general type, establishing a maximum degree of 80 with potential reduction to 76.

Contribution

It improves previous bounds by leveraging constructions and regularity of curves, providing a tighter maximum degree for such surfaces in P4.

Findings

Maximum degree of 80 for non-general type surfaces in P4.

Potential reduction of the bound to 76.

Surfaces of degree at most 70 are also characterized.

Abstract

This is an addendum to the paper of Braun and Fl{\o}ystad ([BF]) on the bound for the degree of a smooth surface in $\pfour$ not of general type. Using their construction and the regularity of curves in $\pthree$, one may lower the bound a little more. We will prove the following: Theorem. Let $S$ be a smooth surface of degree $d$ in $\pfour$ not of general type. Then either $d \leq 70$ or $S$ lies on a hypersurface in $\pfour$ of degree 5 and $d \leq 80$. Thus $d \leq 80$. We will also indicate how to lower the bound to 76.