Introduction

The classification of smooth surfaces of low degree d in P4 goes back to the Italian geometers at the turn of the century. They treated the cases d ≤ 6. More recently, Ionescu and Okonek have treated the cases d = 7 and d = 8 ([Io],[O1, O2]). Their work were complemented by a result of Alexander to give a classification ([A1]). In this paper we find the possible numerical invariants of surfaces of degree 9, and describe for each set of invariants the family of surfaces with the given invariants. Some of the results are mentioned in [Ra], which deals with the case d = 10.

We work over an algebraically closed field of characteristic 0.

The first result is the following

Theorem. Let S be a smooth nondegenerate surface of degree 9 in P4 with sectional genus π, Euler-Poincaré characteristic χ and canonical class K. Then S is a regular surface with K2 = 6χ – 5π + 23, where

π = 6 and χ = 1 and S is rational or S is the projection of an Enriques surface of degree 10 in P5 with center of projection on the surface, or

π = 7 and χ = 1 and S is a rational surface, or χ = 2 and S is a minimal properly elliptic surface, or

π = 8 and χ = 2 and S is a K3-surface with five (−1)-lines, or χ = 3 and S is a minimal surface of general type, or

π = 9 and χ = 4 and S is linked (3,4) to a cubic scroll (possibly singular/reducible), or

π = 10 and χ = 5 and S is a complete intersection (3,3), or

π = 12 and χ = 9 and S is linked (2,5) to a plane.