M ar 1 99 7 EXAMPLES OF SMOOTH NON - GENERAL TYPE SURFACES IN P 4

Smooth projective varieties with small invariants have got renewed interest in recent years, primarily due to the fine study of the adjunction mapping by Reider, Sommese, Van de Ven and others. For the special case of smooth surfaces in P the method goes back to the Italian geometers, who at the turn of the century used it for the study of the surfaces of degree less than 7, or sectional genus π ≤ 3. Later on, for larger values of the invariants, there are contributions by Commesatti and especially Roth. For example, in [38], Roth tried to establish a classification of smooth surfaces with π ≤ 6, but his lists are incomplete since he misses the nonspecial rational surfaces of degree 9 and the minimal bielliptic surfaces of degree 10. Nowadays, through the effort of several mathematicians (some references are given below), a complete classification of smooth surfaces in P has been worked out up to degree 10, and a partial one is available in degree 11.