On smooth surfaces in P4 containing a plane curve

TL;DR

This paper investigates smooth surfaces in projective 4-space containing a plane curve, establishing bounds on their degree and analyzing the linear systems associated with such surfaces, especially those lying on hypersurfaces with specific multiplicities.

Contribution

It provides new bounds on the degree of smooth surfaces in P^4 containing a plane curve and characterizes surfaces lying on hypersurfaces with high multiplicity along a plane.

Findings

Bounded degree of such surfaces in P^4

Explicit degree bound for quartic hypersurfaces

Characterization of surfaces containing a plane curve

Abstract

We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and prove some general result concerning the linear system $|H-P|$. We then look at regular surfaces lying on hypersurfaces of degree $s$ having a plane of multiplicity $(s-2)$. This implies that $S$ contains a plane curve. We prove that the degree of such surfaces is bounded and for $s=4$ we compute an actual bound.