The smooth surfaces of degree 9 in P4

Abstract

Introduction The classification of smooth surfaces of low degree d in P 4 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 P 4 with sectional genus π, Euler-Poincare characteristic χ and canonical class K . Then S is a regular surface with K 2 = 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 P 5 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 K 3-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.