EXAMPLES OF SMOOTHNON - GENERAL TYPE SURFACES IN P 4 Sorin

Smooth projective varieties with small invariants have got renewed interest in recent years, primarily due to the ne study of the adjunction mapping by Reider, Sommese, Van de Ven and others. For the special case of smooth surfaces in P 4 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 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 classiication of smooth surfaces with 6, but his lists are incomplete since he misses the non-special rational surfaces of degree 9 and the minimal bielliptic surfaces of degree 10. Nowadays, through the eeort of several mathematicians (some references are given below), a complete classiication of smooth surfaces in P 4 has been worked out up to degree 10, and a partial one is available in degree 11. But, apart from the general framework of classiication problems concerning codi-mension two varieties, there is another strong motivation for the interest in these surfaces. Namely, in a recent paper Ellingsrud and Peskine 19] proved Hartshorne's conjecture that there are only nitely many families of special surfaces in P 4. More speciically, given an integer a < 6, they show that the degree of smooth surfaces with K 2 aa is bounded. In particular, there are only nitely many families of smooth surfaces in P 4 , not of general type. However, the question of an exact degree bound is still open. A recent work of Braun and Fllystad 10] improves the initial bound (10000) of Ellingsrud and Peskine to d 105, but it is believed that the degree of the smooth, non-general type surfaces in P 4 should be less than or equal to 15. A similar niteness result for 3-folds in P 5 was proved in 11], but the real degree bound is believed to be much higher in this case. Nevertheless, examples of smooth 3-folds in P 5 not of general type are known only up to degree 18 (see 18] for more details and a complete list of known examples). Another reason for the interest in studying surfaces in P 4 is the small number of known liaison classes of such surfaces. Each new specimen of liaison classes is of to whom …